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Full text of "Mathematics Course 2"

f^ HOLT McDOUGAL 



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Indiana 



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Course 2 



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in every book issued. 

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New. Good, Fair; Poor; Bad. 



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any time you need it. 



yo.hrw.com 



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Premier Online Edition 

• Complete Student Edition 

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> Course 1 : 287 videos 

> Course 2: 317 videos 

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• Interactive practice with feedback 

Extra Practice 

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in 2011 witii funding from 
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http://www.archive.org/details/mathematicscoursOObenn 



t. 



INDIANA 




^ HOLT McDOUGAL 



Course 2 



Jennie M. Bennett 
Edward B. Burger 
David J. Chard 
Earlene J. Hall 
Paul A. Kennedy 
Freddie L Renfro 
Tom W. Roby 
Janet K. Scheer 
Bert K. Waits 



^K HOLT McDOUGAL 

V\- \ a division of Houghton Mifflin Harcourt 




Cover Photo: Colorful soap bubbles in wand 
HMH/Sam Dudgeon ,,r . 



Copyright © 2011 Holt McDougal, a division of Houghton Mifflin Harcourt Publishing 
Company. All rights reserved. 

Warning: No part of this publication may be reproduced or transmitted in any form or 
by any means, electronic or mechanical, including photocopy, and recording, or by any 
information storage or retrieval system without the prior written permission of Holt 
McDougal unless such copying is expressly permitted by federal copyright law. 

Requests for permission to make copies of any part of the work should be mailed to the 
following address: Permissions Department, Holt McDougal. 10801 N. MoPac Expressway, 
Building 3, Austin, Texas 78759. 

Microsoft and Excel are registered trademarks of Microsoft Corporation in the United 
States and/or other countries. 

HOLT MCDOUGAL is a trademark of Houghton Mifflin Harcourt Publishing Company, 
registered in the United States of America and/or other jurisdictions. 

Printed in the United States of America 



If you have received these materials as examination copies free of charge. Holt 
McDougal retains title to the materials and they may not be resold. Resale of 
examination copies is strictly prohibited. 



Possession of this publication in print format does not entitle users to convert 
this publication, or any portion of it. into electronic format. 



ISBN-13 978-0-554-03328-0 
ISBN-10 554-03328-3 



123456 0690 13 12 11 10 09 



Authors 






Jennie M. Bennett, Ed.D., is a 

recently retired mathematics teacher 
at Hartman Middle School m Houston, 
Texas. She is past president of the 
Benjamin Banneker Association, the 
former First Vice-President of NCSM, 
and a former board member of NCTM. 



Edward B. Burger, Ph.D., is 

Professor of Mathematics and Chair at 
Williams College and is the author of 
numerous articles, books, and videos. 
He has won many prestigious writing 
and teaching awards offered by the 
Mathematical Association of America. 
In 2006, Dr. Burger was named 
Reader's Digest's "Best Math Teacher" 
in its "100 Best of America" issue. He 
has made numerous television and 
radio appearances and has given 
countless mathematical presentations 
around the world. 

David J. Chard, Ph.D., is the 

Leon Simmons Dean of the School of 
Education and Human Development 
at Southern Methodist University. He 
IS a Past President of the Division for 
Research at the Council for Exceptional 
Children, a member of the International 
Academy for Research on Learning 
Disabilities, and has been the Principal 
Investigator on numerous research 
projects for the U.S. Department 
of Education. He is the author of 
several research articles and books on 
instructional strategies for students 
struggling in school. 

Earlene J. Hall, Ed.D., is the 

Middle School Mathematics Supervisor 
for the Detroit Public Schools district. 
She teaches graduate courses in 
Mathematics Leadership at University 
of Michigan Dearborn. Dr. Hall has 
traveled extensively throughout Africa 
and China and has made numerous 
presentations including topics such 
as Developing Standards Based 
Professional Development and Culture 
Centered Education. She is a member 
of the NCTM 2009 Yearbook Panel. 







■y 



-A 



iU 



Paul A. Kennedy, Ph.D., 

is a professor in the Department 
of Mathematics at Colorado State 
University. Dr. Kennedy is a leader 
in mathematics education. His 
research focuses on developing 
algebraic thinking by using multiple 
representations and technology He is 
the author of numerous publications. 

Freddie L. Renfro, MA, 

has 35 years of expenence in Texas 
education as a classroom teacher and 
director/coordinator of Mathematics 
PreK-12 for school districts in the 
Houston area. She has served as a 
reviewer and TXTEAM trainer for Texas 
Math Institutes and has presented at 
numerous math workshops. 

Tom W. Roby, Ph.D., is Associate 
Professor of Mathematics and Director 
of the Quantitative Learning Center 
at the University of Connecticut. He 
founded and co-directed the Bay 
Area-based ACCLAIM professional 
development program. He also 
chaired the advisory board of the 
California Mathematics Project and 
reviewed content for the California 
Standards Tests. 

Janet K. Scheer, Ph.D., 

Executive Director of Create A 
Vision"*', IS a motivational speaker 
and provides customized K-12 math 
staff development. She has taught 
and supervised internationally and 
nationally at all grade levels. 



Bert K. Waits, Ph.D., s a 

Professor Emeritus of Mathematics 
at The Ohio State University and 
cofounder of T^ (Teachers Teaching with 
Technology), a national professional 
development program. Dr. Waits is 
also a former board member of NCTM 
and an author of the original NCTM 
Standards. 



Indiana Teacher Reviewers 



David Cotner 

Highland Middle School 
Highland, IN 

Alice Craig 

Carniel Middle School 
Carmel, IN 



Jason Hunt 

Selma Middle School 
Selma, IN 



Samantha McGlennen 

Summit Middle School 
Fort Wayne, IN 



Field Test Participants 



Wendy Black 

Southmont Ir. High 
Crawfordsville, IN 

Barbara Broeckelman 

Oakley Middle School 
Oakley KS 

Cindy Busli 

Riverside Middle School 
Greer, SC 

Cadian Coiiman 

Cutler Ridge Middle School 
Miami, FL 

Dora Corcini 

Eisenhower Middle School 
Oregon, OH 

Deborali Drinkwalter 

Sedgefiekl Middle School 
Goose Creek, SC 

Susan Gomez 

Glades Middle School 
Miami, FL 

LaChandra Hogan 

Apollo Middle School 
HolK-wood, FL 

ly Inlow 

Oaklev Middle School 
Oakley KS 



Leighton Jenlcins 

Glades Middle School 
Miami, FL 

Heather King 

Clever Middle School 
Clever, MO 

Dianne Marrett 

Pines Middle School 
Pembroke Pines, FL 

Angela J. McNeal 

Audubon Middle School 
Los Angeles, CA 

Wendy Misner 

Lakeland Middle School 
LaGrange. IN 

Vanessa Nance 

Pines Middle School 
Pembroke Pines, FL 

Teresa Patterson 

Damonte Ranch High School 
Reno. NV 

Traci Peters 

Carlo Middle School 
Mount Pleasant, SC 

Ashley Piatt 

East Forsyth Middle School 
Kernersville, NC 



leannine Quigley 

Wilbur Wright Middle School 
Da\1on, OH 

Shioban Smith-Haye 

Apollo Middle School 
Holl>'wood, FL 

Jill Snipes 

Bunn Middle School 
Bunn, NC 

Cathy Spencer 

Oakridge lunior High 
Oakridge, OR 

Connie Vaught 

K.D.Waldo School 
Aurora, IL 

Shelley Weeks 

Lewis Middle School 
Valparaiso, FL 

lennie Woo 

Gaithersburg Middle School 
Gaithersburg, MD 

Reggie Wright 

West Hopkins School 
Nebo, KT 



Program Reviewers 



ludy Broughton 

Math Teacher 

Poplar Street Sixth Grade Center 

North Little Rock, Arlcansas 

Troy Deckebach 

Matli Teacher 
Tred\'ffrin-Easttomi NHddle 

School 
Berwyn, PA 

Maridith Gebhart 

Math 'ieacher 

Ramay Junior High School 

Fayette\ille. AR 

Ruth Harbin-IVIiles 

District Math Coordinator — 

Retired 
Instructional Resource Center 
Olathe, KS 

Kim Hayden 

Math Teacher 

Clermont County School District 

Milford, OH 

Rhoni Herell 

Math Teacher 

Enid Waller Junior High 

Enid. Oklahoma 

Becky Lowe 

Math readier 
Bartlesville Mid-High 
Bartles\ille, Oklahoma 

George Maguschak 

Math Teacher/ Building 

Chairperson 
Wilkes- Barre Area 
Wilkes- Barre, PA 



Samantha McGlennen 

Math Teacher/Department 

Coordinator 
Summit Middle School 
Fort Wayne, Indiana 

Diane Mclntire 

Math Teacher 
Garfield School 
Kearny, NJ 

Kenneth Mclntire 

Math Jeacher 
Lincoln School 
Kearny NJ 

Tim IVlessal 

Math Teacher/Math Department 

Chair 
Woodside Middle School 
Fort Wayne, Indiana 

Vicki Ferryman Petty 

Math Ieacher 
Central Middle School 
Murfreesboro, TN 

Laronda Raines- Langham 

Math Teacher 

North Jefferson Middle School 

Kimberly, Alabama 

Rene Rush 

Math Teacher 

Colonial Heights Middle School 

Colonial Heights, VA 

Jennifer Sawyer 

Math Teacher 
Shawboro, NC 



Shelly Schram 

Math Teacher 

East Grand Rapids Middle School 

East Grand Rapids, Michigan 

Richard Seavey 

Math Teacher— Retired 
Metcalf lunior High 
Eagan, MN 

Gail M. Sigmund 

Math Teacher — Retired 
Charles A. Mooney Preparatory 

School 
Cleveland, OH 

Jeffrey Slagel 

Math Department t^hair 
South Eastern Middle School 
Fawn Grove, PA 

Paul Turney 

Math Teacher 
Ladue School District 
St. Louis, MO 

Dave Warren 

Math Teacher 
Meridian Middle School 
Meridian. Idaho 

Marilyn Wheeler 

Math Teacher 
Cityside Middle School 
Zeeland, Michigan 



Indiana 

The Hoosier State 



State Capital, 
Indiana 




correlated to 

Indiana's 

Academic Standards for 
Mathematics Grade 7 

CONTENTS 

Indiana Academic Standards for Mathematics, 

Grade 7 IN3 

Process Standards IN6 

Indiana Countdown to ISTEP+ IN 12 



■■55!55!SS5S!S?S>SSSBB^!!5S^S!S55SSS55SS!mS5!5!!!^^ 




IN2 



IndianaPs^ 
Academic Standards for 
Mathematics, Grade 7 



Indiana's 

Academic Standards for 

Mathematics, Grade 7 



STANDARD 1: 



NUMBER SENSE AND COMPUTATION 

7.1.1 Read, write, compare and solve 
problems using whole numbers in 
scientific notation. 

7.1.2 Recognize and compute whole 
number powers of whole numbers. 

7.1.3 Find the prime factorization of whole 
numbers and write the results using 
exponents. 

7.1.4 Recognize or use prime and 
composite numbers to solve 
problems. 

7.1.5 Recognize and use the inverse 
relationship between squaring and 
finding the square root of a perfect 
square integer. 

7.1.6 Identify, write, rename, compare 
- and order rational and common 

irrational numbers and plot them on 
a number line. 

7.1.7 Solve problems that involve 
multiplication and division with 
integers, fractions, decimals and 
combinations of the four operations. 

7.1.8 Solve problems involving percents. 
Find the whole given a part and the 
percentage. Find percentage increase 
or decrease. 




Indiana 

The Hoosier state 








) 




7.1.9 Solve problems involving ratios and 

proportions. Express one quantity as a 
ifraction of another, given their ratio, 
and vice versa. Find how many times 
one quantity is as large as another, 
given their ratio, and vice versa. 
Express one quantity as a fraction of 
another given the two quantities. 
Find the whole, or one part, when a 
whole is divided into parts in a given 
ration. Solve problems involving two 
pairs of equivalent ratios. 



STANDARD 2: 



ALGEBRA AND FUNCTIONS 

7.2.1 Use variables and appropriate 
operations to write an expression, 
equation or inequality that represents 
a verbal description. 

7.2.2 Write and solve two-step linear 
equations and inequalities in one 
variable. 

7.2.3 Evaluate numerical expressions 
and simplify algebraic expressions 
involving rational and irrational 
numbers. 

7.2.4 Solve an equation or formula with 
two variables for a particular variable. 

7.2.5 Find the slope of a line from its graph 
and relate the slope of a line to 
similar triangles. 

7.2.6 Draw the graph of a line given its 
slope and one point on the line or 
two points on the line. 

7.2.7 Identify situations that involve 
proportional relationships, draw 
graphs representing these situations, 
and recognize that these situations 
are described by a linear function in 
the form y = mx where the unit rate 
m is the slope of the line. 



^IfnTTiRTTif 



STANDARD 3: 



STANDARD 4: 



GEOMETRY AND MEASUREMENT 

7.3.1 Identify and use basic properties 
of angles formed by transversals 
intersecting pairs of parallel lines. 

7.3.2 Identify, describe, and use 
transformations (translations, 
rotations, reflections and 
simple compositions of these 
transformations) to solve 
problems. 

7.3.3 Draw two-dimensional patterns 
(nets) for three-dimensional 
objects, such as right prisms, 
pyramids, cylinders and cones. 

7.3.4 Recognize, describe, or extend 
geometric patterns using tables, 
graphs, words, or symbols. 

7.3.5 Identify, describe, and construct 
similarity relationships and solve 
problems involving similarity 
(including similar triangles) 
and scale drawings by using 
proportional reasoning. 

7.3.6 Solve simple problems involving 
distance, speed and time. 
Understand concepts of speed and 
average speed. Understand the 
relationship between distance, 
time and speed. Find speed, 
distance or time given the other 
two quantities. Write speed in 
different units (km/h, m/s, cm/s, 
mi/hr, ft/sec). Solve simple problems 
involving speed and average 
speed. 



DATA ANALYSIS AND PROBABILITY 

7.4.1 Create, analyze and interpret 
data sets in multiple ways using 
bar graphs, frequency tables, line 
plots, histograms and circle graphs. 
Justify the choice of data display. 

7.4.2 Make predictions from statistical 
data and use proportions to make 
estimates about a population 
based on a sample. 

7.4.3 Describe how additional data, 
particularly outliers, added to a 
data set may affect the mean, 
median and mode. 

7.4.4 Analyze data displays, including 
ways that they can be misleading. 
Analyze ways in which the wording 
of questions can influence survey 
results. 

7.4.5 Understand that when all 
outcomes of an experiment are 
equally likely, the theoretical 
probability of an event is the 
fraction of outcomes in which 
the event occurs. Use theoretical 
probability and proportions to 
make approximate predictions. 



INS 



Indiana 

The Hoosier State 



PROCESS STANDARDS 

(* denotes NCTM process standards) 



Problem Solving^ 



• Build new mathematical knowledge through problem solving. 



• Solve problems that arise in mathematics and in other contexts. 



• Apply and adapt a variety of appropriate strategies to solve problems. 



Monitor and reflect on the process of mathematical problem solving. 



Reasoning and Proof* 



Recognize reasoning and proof as fundamental aspects of mathematics. 



• Make and investigate mathematical conjectures. 



• Develop and evaluate mathematical arguments and proofs. 



• Select and use various types of reasoning and methods of proof. 



Communication" 



• Organize and consolidate their mathematical thinking through communication. 



• Communicate their mathematical thinking coherently and clearly to peers, teachers, 
and others. 



• Analyze and evaluate the mathematical thinking and strategies of others. 



Use the language of mathematics to express mathematical ideas precisely. 



Connections* 



• Recognize and use connections among mathematical ideas. 



• Understand how mathematical ideas interconnect and build on one another to 
produce a coherent whole. 



• Recognize and apply mathematics in contexts outside of mathematics. 



Representation" 



• Create and use representations to organize, record, and communicate 
mathematical ideas. 



• Select, apply, and translate among mathematical representations to solve problems. 



• Use representations to model and interpret physical, social, and mathematical 
phenomena. 



IN6 



rtiK^FnTTiFrmf 






Estimation and Mental Computation 



• Know and apply appropriate methods for estimating tine results of computations. 

• Use estimation to decide whether answers are reasonable. 

• Decide when estimation is an appropriate strategy for solving a problem. 

• Determine appropriate accuracy and precision of measurement in problem 
situations. 

• Use properties of numbers and operations to perform mental computation. 

• Recognize when the numbers involved in a computation allow for a mental 
computation strategy. 

Technology 

• Technology should be used as a tool in mathematics education to support and 
extend the mathematics curriculum. 

• Technology can contribute to concept development, simulation, representation, 
communication, and problem solving. 

• The challenge is to ensure that technology supports-but is not a substitute for 
the development of skills with basic operations, quantitative reasoning, and 
problem-solving skills. 



1N7 




The National Council of Teachers of Mathematics recommends 
the following curriculum focal points and related connections as 
the most important math topics to be learned in Grade 7. You can 
learn more about the focal points by visiting ^^ttp://www.nctm.org. 

Grade 7 Curriculum Focal Points 

7.FP.1: Grade 7 Focal Point 1 

Number and Operations and Algebra and Geometry: Developing an 
understanding of and applying proportionality, including similarity. 

Students extend their work with ratios to develop an understanding of 
proportionality that they apply to solve single and multistep problems in 
numerous contexts. They use ratio and proportionality to solve a wide variety 
of percent problems, including problems involving discounts, interest, 
taxes, tips, and percent increase or decrease. They also solve problems 
about similar objects (including figures) by using scale factors that relate 
corresponding lengths of the objects or by using the fact that relationships 
of lengths within an object are preserved in similar objects. Students graph 
proportional relationships and identify the unit rate as the slope of the 
related line. They distinguish proportional relationships (4 = k, or y — kx) 
from other relationships, including inverse proportionality {xy = k, or y = |). 

CHAPTERS 4, 5, AND 6 

* 7.FP.2: Grade 7 Focal Point 2 

Measurement and Geometry and Algebra: Developing an understanding 
of and using formulas to determine surface areas and volumes of three- 
dimensional shapes. 

By decomposing two- and three-dimensional shapes into smaller, 
component shapes, students find smface areas and develop and justify 
formulas for the surface areas and volumes of prisms and cylinders. As 
students decompose prisms and cylinders by slicing them, they develop and 
understand formulas for their volumes (Volume = Area of base x Height). 
They apply these formulas in problem solving to determine volumes 
of prisms and cylinders. Students see that the formula for the area of a 
circle is plausible by decomposing a circle into a number of wedges and 
rearranging them into a shape that approximates a parallelogram. They 
select appropriate two- and three-dimensional shapes to model real-world 
situations and solve a variety of problems (including multistep problems) 
involving surface areas, areas and circumferences of circles, and volumes of 
prisms and cylinders, chapters 9 and 10 

■ 7.FP.3: Grade 7 Focal Point 3 

Number and Operations and Algebra: Developing an understanding of 
operations on all rational numbers and solving linear equations. 

Students extend understandings of addition, subtraction, multiplication, and 
division, together with their properties, to all rational numbers, including 
negative integers. By applying properties of arithmetic and considering 



INS Focal Points 



negative numbers in everyday contexts (e.g.. situations of owing money 
or measuring elevations above and below sea level), students explain why 
the rules for adding, subtracting, multiplying, and dividing with negative 
numbers make sense. They use the arithmetic of rational numbers as 
they formulate and solve linear equations in one variable and use these 
equations to solve problems. Students make strategic choices of procedures 
to solve linear equations in one variable and implement them efficiently, 
understanding that when they use the properties of equality to express an 
equation in a new way, solutions that they obtain for the new equation also 
solve the original equation, chapters 1, 2, 3, and 12 

Connections to the Focal Points 

7.FPC.4: Grade 7 Focal Point Connection 4 

Measurement and Geometry: Students connect their work on 
proportionality w ith their work on area and volume by investigating similar 
objects. They understaird that if a scale factor describes how corresponding 
lengths in two similar objects are related, then the square of the scale 
factor describes how corresponding areas are related, and the cube of the 
scale factor describes how corresponding volumes are related. Students 
apply their work on proportionality to measurement in different contexts, 
including converting among different units of measurement to solve 
problems involving rates such as motion at a constant speed. They also 
apply proportionality when they work with the circumference, radius, and 
diameter of a circle; when they find the area of a sector of a circle; and when 
they make scale drawings, chapters 4, 9, and 10 

7.FPC.5: Grade 7 Focal Point Connection 5 

Number and Operations: In grade 4, students used equivalent fractions 
to determine the decimal representations of fractions that they could 
represent with terminating decimals. Students now use division to express 
any fraction as a decimal, including fractions that they must represent 
with infinite decimals. They find this method useful when working with 
proportions, especially those involving percents. Students connect their 
work with dividing fractions to solving equations of the form ax = b, where 
a and b are fractions. Students continue to develop their understanding of 
multiplication and division and the structure of numbers by determining if a 
counting number greater than 1 is a prime, and if it is not, by factoring it into 
a product of primes, chapters 2, 3, and 6 

7.FPC.6: Grade 7 Focal Point Connection 6 

Data Analysis: Students use proportions to make estimates relating to a 
population on the basis of a sample. They apply percentages to make and 
interpret histograms and circle graphs, chapter 7 

7.FPC.7: Grade 7 Focal Point Connection 7 

Probability: Students understand that when all outcomes of an experiment 
are equally likely, the theoretical probability of an event is the fraction of 
outcomes in which the event occurs. Students use theoretical probability and 
proportions to make approximate predictions, chapter 1 1 



Focal Points IN9 



Countdown to ISTEP+ 



Holt Mathematics provides many opportunities for you to 
prepare for standardized tests, such as the Indiana Statewide 
Testing for Educational Progress-Plus Test. 



Test Prep Exercises 

Use the Test Prep Exercises for daily 
practice of standardized test questions 
in various formats. 



' Multiple Choice — choose your answer. 

I Gridded Response — write your answer 
j in a grid and fill in the corresponding 
bubbles. 

Short Response — write open-ended 
responses that are scored with a 
2-point rubric. 

Extended Response — wTite open- 
ended responses that are scored with a 
4-point rubric. 








J S"''" ^.00 ^"^, ^, 




«»■«■ 





















<P 9 






Test Tackier 

Use the Test Tackier to 
become familiar with 
and practice test-taking 
strategies. 



The first page of this 
feature explains and 
{ shows an example of 
a test-taking strategy. 



*Bg'^" 






TO of (he c^.!,^ 



^■•- 602,9 [„; 



The second page 
guides you through 
applications of the 
test- taking strategy. 




IN10 Countdown to ISTEP+ 



Countdown to ISTEP+ 



Standardized 
Test Prep 

Use the Standardized Test Prep to 
apply test-taking strategies. 



L" 



The Hot Tip provides test- 
taking tips to help you 
succeed on your tests. 



These pages include practice with 
multiple choice, gridded response 
short response, and extended 
response test items. 





Countdown to ISTEP+ 



Use the Comitdown to ISTEP+ to practice 
for the hidiana Statewide Testing for 
Educational Progress- Plus Test every day. 



There are 24 pages of practice for 
the 1STEP+. Each page is designed to 
be used in a week so that all practice 
will be completed before the ISTEP+ 


IS given. 





Each week's page has five 
practice test items, one for 
each day of the week. 



Test-Taking Tips 



^^OM/v^ 



<^0,T>0^ 



Get plenty of sleep the night before the 
test. A rested mind thinks more clearly 
and you won't feel like falling asleep 
while taking the test. 

Draw a figure when one is not provided 
with the problem. If a figure is given, 
write any details from the problem on 
the figure. 

Read each problem carefully. As you 
finish each problem, read it again to 
make sure your answer is reasonable. 



■1^: Review the formula sheet that will be 

^""^ supplied with the test. Make sure you 

know when to use each formula. 

\f/^ First answer problems that you know 
how to solve. If you do not know how to 
solve a problem, skip it and come back 
to it when you have finished the others. 

1|^ Use other test-taking strategies that can 
be found throughout this book, such 
as working backward and eliminating 
answer choices. 



Countdown to ISTEP+ IN11 




..J.,-; 



Countdown to ISTEP+ 




DAY 1 



Constructed Response 

What is the value of the expression 

3(15 - 6) + (18 - 12)2? 5how your worl< or 

provide an explanation for your answer. 



DAY 2 


1 


f 




Willy is 25 inches tall. His brother Carlos is 
2 i- times as tall. Which is the best estimate 
of Carlos's height? 

A. 23 inches C. 25 inches 

B. 50 inches D. 56 inches 



DAY 3 



Gridded Response 

Derek made this sketch for a bulletin board in his room. 



2.8 m 



If Derek is using 8.4 square meters of wood to build the board, what is the best 
estimate of the board's length in meters? 



DAY 4 


■ 










f 


Estimate the vo 


lume of the square prism. 








centimeters 


/ 




y 










B. 400 cubic 










centimeters 










C. 500 cubic 19.7 cm 










centimeters 










D. 500 cubic 










centimeters 


J* 




A 


.2 cm 






S.2 cm 





DAY 5 



Extended Response 

Kyle received $100 for his birthday. He used 
it to buy 3 CDs for $7 each and 2 DVDs for 
$18 each. He paid $3.42 in tax. 

What was Kyle's total cost? Explain your 
answer. 

How much change would Kyle receive? 
Explain your answer. 

Kyle wants to buy new shirts that cost 
$15.99 each with the remaining money. 
How many shirts could he buy? Explain 
your answer. 



IN12 Countdown to ISTEP+ 




Countdbwri to ISTEH 



DAY 1 




Rose's Bakery uses these apples to make 
one small apple tart. 




Which expression represents the number of 
apples used in 4 small tarts? 



A. 4-2-1 



4-2J 



C. 4 + 2^ 



D. 



DAY 2 



Extended Response 

Gil wants to fill his fish tank with water 



l4f^ 



2 ft 



lift 



Estimate the volume of water he needs. 
Explain your answer. 

Find the exact volume of the fish tank. 
How much does your exact answer differ 
from your estimate? Explain your answer. 



DAY 3 



Tim's pet frog will grow 2.25 times in size in one month. If the frog is 
4.7 centimeters long right now, which is the best estimate of its length 
after one month? 



A. 


5 centimeters 


C. 


10 centimeters 


B. 


8 centimeters 


D. 


15 centimeters 



DAY 4 



Constructed Response 

What is the value of the expression 

4(8 - 3)^ - 10 • (25 - 5)? Show your work 

or provide an explanation for your answer. 



DAY 5 



Gridded Response 

The window box measures 4| inches x 
4g inches « 10^ inches. Which is the best 
estimate for the cubic inches of soil that 
will fill the window box? 



Countdown to ISTEP+ INI 3 



Countdown to ISTEP+ 




DAY 1 



Constructed Response 

Write an expression that shows the fraction 
of flowerpots that have polka dots in both 
groups? 



DAY 2 



Jake estimates that the answer to 25 • 10.6 
is between 250 and 275. Which of the 
following shows that Jake's estimate 
is reasonable? 

A. 250 4- 10 = 25; 275 ^ 10 = 27.5 

B. 250 + 275 = 525 

C. 25 • 10 = 250; 25-11 =275 

D. 250 ^ 11 = 23 



DAY 3 



June surveyed her class and found that 45% of her classmates have visited the Grand Canyon. 
With 20 students in her class, June calculated that 9 students have visited the Grand Canyon. 
Which of the following shows that June's answer is reasonable? 

A. 0.45 • 100 = 45 

B. 4.5 • 20 = 9 

C. 9 -20 -4.5 = 81 

D. 0.45 • 20 = 9 



DAY 4 



Extended Response 

Use the following expression to answer the 
following questions. 

(16 - 8) • 3 + (10 = 100) 

Evaluate the expression. Explain your 
answer. 

Describe how you used the order of 
operations to evaluate the expression. 



DAY 5 


\ 






' 


Gridded Respc 


>nse 


Estimate the volume in cubic centimeters 


of the figure below. 






^ 






9.75 cm 




^,'' 


^ A Qi; /-m 


10.25 cm 



IN14 Countdown to ISTEP+ 




DAY 1 



Which value does NOT make the following 
statement true? 



0.028 < 



< 0.064 



A. 


0.027 


B. 


0.029 


C. 


0.043 


D. 


0.062 



DAY 2 



Constructed Response 

Six friends equally share the cost of a 
breakfast. The breakfast costs $42.30. Write 
an expression that shows each person's 
share? 



DAY 3 



Ann buys 3 black candles, 2 white candles, and 4 striped candles. She gives the 
cashier a $50 bill and estimates that she will get about $8 in change. Which of the 
following shows that Ann's estimate is reasonable? 




A. 15 + 11 + 16 = 42 

B. 26 - 18 = 8 



C. 50 - 8 = 42 

D. 12 + 10 + 12 = 34 



DAY 4 



Gridded Response 

Miguel recorded the distances he ran each 
month. What is the total number of miles 
he ran? 



Month 



Miles 



May 



June 



July 



22.5 



20.8 



25.2 



DAY 5 



Extended Response 

Martin is filling a trough with water. 



21ft 



gfft 



3|ft 



What is the volume of the trough? Explain 
your answer. 

If the pail he is using can hold 9 cubic feet 
of water, how many times will he need 
to empty his pail into the trough in order 
to fill the trough completely? Explain 
your answer. 



Countdown to ISTEP+ INI 5 




DAY 1 



Rosie visited her grandmotlner by train. 
The train traveled 588 miles in 5| hours, so 
Rosie estimates that the train traveled 100 
miles per hour. Which equation shows that 
her estimate is reasonable? 

A. 500 • 100 = 6 

B. 6 • 600 = 100 

C. 6 - 600 = 100 

D. 600 - 6 = 100 



DAY 2 


1 


f 


Constructed Response 

Jackie used the Associative Property to find 
that 6 • 14.3 • 0.5 = 85.8 • 0.5 = 42.9. Write 
an expression that would also work. 



DAY 3 



You are multiplying this recipe for pesto so that you use 2j cups of basil leaves. 
Which expression shows the amount of olive oil you need? 



Pesto 



1 cup basil leaves 

1/4 cup parmesan cheese 

1/2 cup olive oil 

5 tbsp pine nuts 

Blend ingredients until they form 
a smooth paste. 



A. 
B. 



|(1 . 2l) 

l-2l 
4 2 



c. 



'2 2 
"2 ■ 2 



DAY 4 



Extended Response 

A canal boat went through a series of 
locks with the following rises and drops. A 
positive number shows a rise. A negative 
number shows a drop. At which lock 
was there the greatest rise? Explain your 
answer. 



Lock 


1 


2 ; 3 


4 


Rise or 
Fall (ft) 


-17 


11 -8 ' 6 



After traveling through all four locks, 
what is the net change of the water level? 
Explain your answer. 



DAY 5 



Gridded Response 

Ryan is making 7^ cups of rice to serve 
at dinner with his friends. If he wants to 
give I cup of rice to each guest, how many 
people will the rice serve? 



IN16 Countdown to I5TEP+ 




Countdown to ISTEP+ 




WEEK 



DAY 1 



Jeff runs 8.077 miles in an hour. Tina runs 
8.102 miles in an hour. Jade runs 8.05 miles 
in an hour. Andy runs 8.032 miles in an 
hour. If they all started a race at the same 
time, who will finish first? 



A. 
B. 



Tina 
Andy 



C. 
D. 



Jade 
Jeff 



DAY 2 



+ 3 - 1^ 



Constructed Response 

Kevin simplified the problem ^ , ^ .^ 
on the chalkboard. Where was his first 
mistake? Show your work or provide an 
explanation for your answer. 

Step 1: 
Step 2: 
Step 3: 



+ 3 



^+3(^ 
4 ^ l4 

3+12 



3 
2 

m 



step 4: 15 



Step 5: ^, which is 1^ 



DAY 3 



Jon has 4 shelves with 52 CDs on each shelf. He multiplies 50 by 4 and 2 by 4 to find 
that he has 208 CDs in all. Which property justifies Jon's solution? 

A. Associative C. Distributive 

B. Commutative D. Identity 



DAY 4 



Extended Response 

Nate is buying a shirt that is on sale, but 
part of the tag is ripped off. 




What is the amount of the discount 
written as a percent? Explain your answer. 

If the price of Nate's shirt is $26.75, what 
would be the sale price of this shirt? 
Explain your answer. 



DAY 5 



Gridded Response 

Tom is creating a model of a building. 
What is the height in feet of the real 
building? 



6ft 



3ft 



xft 



3ft 



63 ft 



63 ft 



Countdown to ISTEP+ IN17 



DAY 1 



Gridded Response 

Which of the following is the least 
number? 

0.305 0.02 0.10 0.081 



Extended Response 

If it takes 5 buses to carry 225 passengers, 
how many passengers will 3 buses carry? 
Explain your answer. 

How many buses will be needed to 
transport 687 students on a school trip? 
Explain your answer. 



DAY 3 



Sandra read a survey that found that 82.5% of people polled believed that 
volunteering one's time was the best way to serve one's community. What is this 
percent written as a fraction? 



A. 
B. 



82 



10 
100 



c. 

D. 



33 
40 



DAY 4 



Constructed Response 

Peter and a friend share a pizza. Peter eats 
2 slices and his friend eats 3 slices. What 
fraction represents the amount of pizza 
both boys ate? Show your work or provide 
an explanation for your answer. 




DAY 5 



What is the best estimate of the volume of 
this figure? 



1.9 cm 



12.2 cm 

A. 24 cubic centimeters 

B. 36 cubic centimeters 

C. 72 cubic centimeters 

D. 80 cubic centimeters 



3.3 cm 



INI 8 Countdown to ISTEP+ 




Countdown to IStEP+ 



DAY 1 



Extended Response 

The table shows the number of students in 
four different classes at Park Street Middle 
School who take the bus to school. 



Class 


A B CD 


Students Who 
Take Bus 


15 20 1 12 12 
20 25 ! 18 24 



Which class has the greatest fraction of 
students who take the bus to school? 
Explain your answer. 

Order the classes according to the fraction 
of students who take the bus from the 
greatest to the least. Explain your answer. 



DAY 2 



Tim and Sue are setting up a tent at a 
campground. Tim estimates that the tent 
will cover an area of 190 square feet, while 
Sue estimates the area will be 220 square 
feet. Whose estimate is better and why? 




19.75 ft 



10.65 ft 

A. Tim's; 19 • 10 = 190 

B. Sue's; 20 • 1 1 =220 

C. Tim's; 10 - 19 = 190 

D. Sue's; 2(11 + 20) = 220 



DAY 3 



Constructed Response 

Mrs. Robbins is knitting a scarf for her niece. She knitted l| feet yesterday and l| 
feet today. How many feet did Mrs. Robbins knit in both days? Show your work or 
provide an explanation for your answer. 



DAY 4 



Gridded Response 

What decimal completes this equivalency? 
I = 75% = ? 



DAY 5 



Mr. Reyes wants to fence in the area 
behind his house. How many meters of 
fencing does he need to buy? 
30.75 m 



20.5 m 




A. 51.25 meters 

B. 71.75 meters 

C. 102.5 meters 

D. 630.38 meters 



Countdown to ISTEP+ IN19 



DAY 1 



Gridded Response 

What is the value of this expression? 
3 + 4- (2^ + 21 H- 3) 



DAY 2 


1 


' 


Diane is buying 


4 DVDs for $15.40 each. 


She calculates that she will spend $61.60. 


Which of the following justifies Diane's 


solution? 


A. 4(15 + 0.40) = 60 + 1.60 = 61.60 


B. 61.60 ~ 0.4 = 15.4 


C. 4-15 + 2- 0,40 = 62 - 0.40 = 61 .60 


D. 4(15.40+15.40+15.40+15.40) = 


61.60 



DAY 3 



Constructed Response 

Marc needs | pound of blueberries to make a batch of muffins and another ^ pound 
to make blueberry pancakes. How many pounds of blueberries does Marc need? 
Show your work or provide an explanation for your answer. 



DAY 4 



Extended Response 

In the morning, Steve drives to his job at 
the bookstore. After work, he drives to the 
college where he takes classes. Then he 
drives back home. 
College 



2.6 km 



Home 



6.3 km 




4.7 km 



Bookstore 



What is the total distance Steve travels 
each day? Explain your answer. 

If Steve works 20 days this month, how 
many kilometers will he have traveled by 
the end of month? Explain your answer. 



DAY 5 



Brian is building a small reflecting pool. 
Which is the best estimate of the amount 
of water the pool will hold? 



4" 



7l« 




A. 
B. 
C. 
D. 



84 cubic feet 
1 12 cubic feet 
140 cubic feet 
160 cubic feet 



IN20 Countdown to ISTEP+ 



CouhtdGliiKi^o ISTEP+ 




DAY 1 


I 






f 

described 


by the ordered 


Which point 


is 


pair (-4, 2)? 












, 


yy 




1 




A - 




C 











_ — 






• 


A 

• 




2 




X 


-4 


' ' o 

• 
D 




2 4 




"" 


• - 
B 

















' -4 

1 


1 










A. A 




c. 


C 


B. B 




D. 


D 



DAY 2 



Extended Response 

Use the table to answer the following 
questions. 



Input X 


5 10 15 20 


Output y 


25 50 75 100 



Based on the pattern in the table, if the 
input value was 50, what would be the 
output value? Explain your answer. 

Write a sentence using x and y to describe 
the pattern in the table. 



DAY 3 



Which of the following describes the relationship between the numbers in 
this sequence? 

2, 8, 32, 128, ... 

A. A number is four more than the number preceding it. 

B. A number is four times greater than the preceding number. 

C. A number is one-fourth the preceding number. 

D. A number is the square of the preceding number. 



DAY 4 



Constructed Response 

Mrs. Reese is taking a trip to visit her sister. 
If she drives 162 miles in 3 hours, what 
is her average rate of speed? Show your 
work or provide an explanation for your 
answer. 



DAY 5 



Gridded Response 

Sandy and his father built a tree house for 
Sandy's sister. How tall in feet is the tree? 



y 18.2 ft 




- 15.6 ft 



Countdown to ISTEP+ IN21 



Countdown to.lSTEP.+ . 



...^-..~.—.. .^^^-^aa 




DAY 1 



Constructed Response 

Write a description of the relationship 
between the numbers in this sequence? 

145, 115, 85, 55, ... 



DAY 2 



Using the following pattern, which figure 
comes next? 




A. 




B. 



D. 




DAY 3 



Which point is described by the ordered pair (-3, 2)? 

A. fi CD 

B. C D. £ 



*y 



4- 



-< — I — I — (- 



-4 -2 O 

D 
• • -2 



-4 



DAY 4 



Gridded Response 

Dante recorded the following information 
about a seedling's growth for science class. 
How many inches did the seedling grow in 
three weeks? 



Week 


1 2 ' 3 


Inches 
Grown 


7 5 7 

8 1 6 24 



DAY 5 



Extended Response 

Olivia read 125 pages of her medical 
textbook in 4 hours. 

What is Olivia's average rate of reading 
in pages per hour? Explain your answer. 

If Olivia has a 335 page medical textbook 
to read, approximately how long will it 
take her to complete the reading? Explain 
your answer. 



y 



IN22 Countdown to ISTEP+ 



Countdow 



"•"•---'^•'■r'-i'if 




DAY 1 



Gridded Response 

What number best completes the pattern? 
2, 5, 11, , 47, 95 



DAY 2 


1 


f 


Wh 


ch of the following describes the 


relationship between the numbers in 


this 


sequence? 




243, 81, 27, 9, ... 


A. 


A number is three more than the 




preceding number. 


B. 


A number is three less than the 




preceding number. 


C. 


A number is one-third of the 




preceding number. 


D. 


A number is three times more than 




the preceding number. 



DAY 3 



Which two of the figures below are similar? 
Figure A Figure B 

10ft 



5ft 



4ft 



4ft 



3ft 




Figure D 
4ft 



2ft 



A. Figures A and D 

B. Figures A and B 



C. Figures B and D 

D. Figures B and C 



DAY 4 



Constructed Response 

A discount store is selling a case of 24 
bottles of water for $12.99. What is the 
unit price of a bottle of water to the 
nearest cent? Show your work or provide 
an explanation for your answer. 



DAY 5 


1 




W 
ponse 




Extended Res 




The shadow of 


a 4-foot-tall mailbox 


is 


2 feet long. Th 


e shadow of a tree is 


16 feet 


long. 






Write a 


A> 




proportion 


'idpfe* 




you could 
use to 






determine 


-^:^'--' 




the height 


"• >'i^K^ — 


4ftT 


of the tree. 
Explain your 
answer. 


1 


16ft 


2ft 


Find the heigh! 


of the tree. 




Explain your answer. 





Countdown to ISTEP+ IN23 





DAY 1 



Constructed Response 

A model car and a real car have the given 
dimensions. What is the length of the real 
car if the scale factor is 1:30? Show your 
work or provide an explanation for your 
answer. 




4 ft 



DAY 2 



Which pair of triangles are similar? 




DAY 3 



Which point is described by the ordered pair (-1, -1)? 

A. e c. f 

B. D D. F 




DAY 4 



Extended Response 

Randy wants to buy an MP3 player for 
$98.99, and it is on sale for 37% off. 

How much money will Randy pay for the 
MP3 player before tax? Explain your answer. 

What would be Randy's total cost including 
6% sales tax? Explain your answer. 



DAY 5 



Gridded Response 

Julie goes mountain biking every Saturday. 
Last week, she rode 36 kilometers in 3 hours. 
What was her average rate of speed in 
kilometers per hour? 



IN24 Countdown to ISTEP+ 



Countdbiinf ri^f b ISTEP+ 



DAY 1 



Which point is described by the ordered 
pair (2, -2)? 




-< — I — ♦ — I — h 



-4 . O 

C 

• - 

D 



H i 1 1 *■ 



2 4 



A. e 

B. D 



C. E 

D. F 



DAY 2 



Extended Response 

Use the sequence to answer the following 
questions. 

1 1 1 J_ 
2' 4' 8' 16' ■•• 

Describe the relationship between the 
numbers in this sequence. Explain your 
answer. 

Name the next three terms in the 
sequence. Explain your answer. 



DAY 3 



Two similar figures 

A. have the same size. 

B. have the same shape. 

C. have the same size and shape. 

D. are congruent. 



DAY 4 



Gridded Response 

April is standing next to a tree. The length 
of April's shadow is 4 feet, and the length 
of the tree's shadow is 32 feet. If April is 
5 feet tall, how tall in feet is the tree? 




DAY 5 



Constructed Response 

Susan buys leather purses from the 
manufacturer for $11 .90 each and sells 
them to the public at 425% the price 
she paid. About how much do Susan's 
customers pay for a purse? 



Countdown to ISTEP+ IIM25 




mB 



^^w^y-i'-iiy-ry 



Countdo 




DAY 1 



Gridded Response 

What is the median of this data set? 




Louis received the following scores on his 
English quizzes this semester: 95, 95, 80, 
70, 60. Which description of this data set 
would make Louis' results look best? 

A. the mean of his scores 

B. the median of his scores 

C. the mode of his scores 

D. the range of his scores 



DAY 3 



Constructed Response 

Nora wants to display data about the amount of time it took each runner to 
complete a race. What type of graph should she use? 



DAY 4 



The price of a meal came to $1 1.82 without 
tax or tip. Which is the best estimate of the 
cost of the meal if the tip is 1 5% and the 
tax is 8%? (Figure the tax and the tip on 
the base price of the meal.) 

A. $10 C. $15 

B. $12 D. $20 



DAY 5 



Extended Response 

Use the data to answer the following 
questions. 

X 
XX X 

X X X X 

-H — \ — \ — \-* 



12 3 4 



What is the mode of this set of data? 
Explain your answer. 

What is the mean of this set of data? 
Explain your answer. 



IN26 Countdown to ISTEP+ 



Countdbvi/ri to ISTEP+ 




DAY 1 



You are conducting a survey to see if tine 
amount of hours of sleep that people need 
each night is related to their age. What 
type of diagram would you use to display 
some of the data you found? 

A. line plot 

B. circle graph 

C. stem-and-leaf plot 

D. scatter plot 



DAY 2 



What kind of data is most likely 
represented by this plot? 



Stems 



Leaves 



2 2 4457 

135578889 

02 



A. cost of a movie ticket at local theaters 

B. average height (in.) of students in 
a class 

C. average daily temperatures at the 
beach 

D. ages of students in a class 



_J 



DAY 3 



Constructed Response 

What is the measure 33 for this set of data? 
33, 33, 56, 33 



DAY 4 



Extended Response 

Use the packages to answer the following 
questions. 





/ 


/ 










5.75 

kg 


X 


/^~ 


-7 




/ 


y 




32.5 
kg 






12.1 
kg 


X 









0.5 

kg \y 



&1.2S kg 



What is the mean weight of these 
packages? Explain your answer. 

If shipping costs $0.08 per kilogram, how 
much would it cost to ship the largest 
package? Explain your answer. 



DAY 5 



Gridded Response 

You buy a book for $24.75 and pay 6.25% 
sales tax. What is the total cost of the book? 



Countdown to ISTEP+ IN27 




htciowri to iSTEP+ 



DAY 1 



Which of the following is the greatest 
number for this data set? 

32, 35, 19, 26, 40, 32, 18, 32, 16, M 

A. median B. mean 

C. mode D. range 




DAY 2 



Constructed Response 

Naomi surveyed a group of people about 
their favorite movie genre: comedy, drama, 
action, musical, or science-fiction. What 
type of graph or plot would be the best 
way for Naomi to display her results? 



DAY 3 



Which two angles are complementary? 
A. A 



C. A 





145° 



C 





D. A 




95° 




DAY 4 


■ 




f 


Extended Res 


ponse 


Jason recorded the number of cardinals 


he saw each month. What is the mean 


number of cardinals Jason saw? Round to 


the nearest whole number. Explain your 


answer. 


Stems 


Leaves 





6 689 


1 


2458889 


2 


1 


What is the range of the number of 


cardinals Jason saw? Explain your answer. 



DAY 5 



Gridded Response 

The median of 4 numbers is 48. If three of 
the numbers are 42, 45, and 52, what is the 
other number? 



■K^S^^^^^!^55^^^^^ 



IN28 Countdown to ISTEP+ 



Countdown to ISTEI*^ 




DAY 1 



Constructed Response 

What type of triangle is formed when you 
connect the three points? Show your work 
or provide an explanation for your answer. 

♦ y 




DAY 2 


1 


f 


Alex kept track of the number of 


telemarketing calls he received each 


month for 6 months. 


14, 10, 17, 12, 11, 15 


Which of the following would not change 


if Alex decided to add the data value 1 1 


for a seventh month? 


A. median C. mode 


B. range D. mean 



DAY 3 



Which two angles are supplementary? 
A. A D 






D. A 




120° 




DAY 4 



Gridded Response 

The line plot shows the daily low 
temperatures during one week. What is the 
mean low temperature in degrees 
fahrenheit for the entire week? 

X 

X 

X X X X X 

-H — \ — \ — \ — \ — h* 



57 58 59^ 50' 61 62° 



DAY 5 



Extended Response 

A car is traveling at a speed of 48 miles 
per hour 

If the car continues at this rate, how far 
can the car travel in l| hours? Explain your 
answer. 

If the car traveled 312 miles in a day, how 
many hours did it take the car to travel this 
far? Explain your answer. 



Countdown to I5TEP+ IN29 



Countdown to lStEP+ 




DAY 1 



Henry is designing the lobby of an office 
building. He wants a tile pattern that will 
tessellate. Which tile can he use? 





DAY 2 



Constructed Response 

Ellis listed the following shapes as 
parallelograms: square, rectangle, 
trapezoid, and rhombus. He made one 
mistake. Which shape is not a 
parallelogram? Show your work or provide 
an explanation for your answer. 



DAY 3 



Extended Response 

Use the graph to answer the following 
questions about figure ABCD. 

What is the area of figure ABCD7 Explain 
your answer. 

If figure ABCD is reflected across the y-axis, 
what will the new coordinates of D be? 
Explain your answer. 



ky 




-+- 



H 1 h- 



-4 -2 O 

— l^:^ -2 



X 



DAY 4 








P 


g angle measures 


Which of the 


fc 


)llowir 


is complementary to the 


measure of 


angle ABC? 


A 

1 








/74° 
B 




~C 


A. 5° 




C. 


36° 


B. 16° 




D. 


106° 



DAY 5 



Gridded Response 

What is the mean of this set of data? 
90, 108, 67, 84, 90, 82, 73, 90 



# 



IN30 Countdown to ISTEP+ 



Countdown to ISTEP+ 



WEEK 20 



DAY 1 



If figure FGHJ is reflected across the x-axis, 
what will the new coordinates of J be? 



A. 
B. 




DAY 2 



Constructed Response 

Each of the four triangles has the same 
area. If one bag of stones will cover an area 
of 25 square feet, how many bags will it 
take to cover the large triangle? Show your 
work or provide an explanation for your 
answer. 

50 ft 




100 ft 



DAY 3 



Extended Response 

Suppose that m^ABC = 65°. 

If ZABC and /IDEF are supplementary, what is the measure of ^DER 
Explain your answer 

If ilABC and zDff are complementary, what is the measure of ^DEFl 
Explain your answer. 



DAY 4 



Which of the following figures does not 
belong in the group if the triangles are 
classified by angles? 

A. C. 




DAY 5 



Gridded Response 

Four shovels of sand are mixed with 
5 shovels of gravel to make cement. 
About how many shovels of gravel are 
needed for 45 shovels of sand? 



Countdown to ISTEP+ IN31 



J 



Couiltdown ta i5tEP+ 



DAY 1 



Constructed Response 

If figure ABCDE is reflected across the x-axis, 
what will the new coordinates of E be? 
Show your worl< or provide an explanation 
for your answer. 

*y 




DAY 2 



Which of the following figures is a 
parallelogram? 

A. / 7 C. 




DAY 3 



Which of the following best describes the angles below? 




B C E 

A. They are congruent. 

B. They are supplementary. 



C. They are complementary. 

D. Not here. 



DAY 4 



Gridded Response 

What is the price of the most expensive TV? 




DAY 5 



Extended Response 

Carrie is designing a mosaic wall for her 
school's library. The wall measures 4 meters 
by 8 meters. The tiles she is using are 10 
centimeters by 10 centimeters. 

How many tiles will Carrie need to cover 
the wall? Explain your answer. 

If the tiles come 600 to a package, how 
many packages will Carrie need to cover 
the wall? Explain your answer. 



IN32 Countdown to ISTEP+ 






L- 



Countdbwit to 1STEP+ 




DAY 1 



Which point is described by (-4, 3)? 




DAY 2 


1 






f 


Gridded Response 


William received the following blueprint 


for a building. What is the area in square 


feet of this building? 




20 ft 






/ 


10 ft 


/ 


5 ft 




35 ft' 




5ft 


1 


10 ft 


35 ft 





DAY 3 



Which of the following is an isosceles triangle that is not equilateral? 
A. , , B. , , C. „ D. 






DAY 4 



Extended Response 

Kenny is building a compost bin. 




What is the volume of Kenny's compost 
bin? Explain your answer. 

If there are 7.5 gallons per cubic foot, 
could Kenny's compost bin hold the 
contents of five 39-gallon lawn and leaf 
bags? Explain your answer. 



DAY 5 



Constructed Response 

What is the value of this expression? Show 
your work or provide an explanation for 
your answer. 

(12 - 3)^ + 50 - 2.5 - 10 



Countdown to ISTEP+ IIM33 








HAY 1 



Extended Response 

Use the table to answer the following 
questions. 



X 


1 


2 


3 ! 4 


y 


1 


4 7 10 



What is the rule for the pattern in the 
table? Explain your answer. 

If you continued the table to x = 12, what 
would be the value of y? Explain your 
answer. 



DAY 2 



Which pair contains similar figures? 





C. 



D. 



DAY 3 



Which of the following describes this figure? 

A. triangular prism 

B. triangular pyramid 

C. rectangular pyramid 

D. cone 




DAY 4 



Constructed Response 

Danny needs to add the following lengths 
together so that he can buy enough wood 
for a project. What decimal number should 
Danny use to replace 12| m? Show your 
work or provide an explanation for your 
answer. 

2.5 m, 6.75 m, 10.425 m, 12|m 



DAY 5 



Gridded Response 

What angle is supplementary to the 
measure of angle ABC7 



92° 






IN34 Countdown to ISTEP+ 



Countdown to ISTEP+ 




DAY 1 



Isaac had to draw four different pyramids 
for math class. He drew the figures below. 
Which figure is not a pyramid? 

A. A\ c. 






DAY 2 


1 


f 


What object is 


represented by this net? 


A. cone C. cylinder 


B. sphere D. prism 



DAY 3 



Constructed Response 

Record the scores you've received on science quizzes this semester. If you want 
to see the shape of the data set, which of the following is the best way to display 
the data? 



DAY 4 


■ 




f 1 


Gridded Resp< 

Mrs. Minato's rr 
yesterday. Any 
76 will have to 
many students 
take the make- 

Sterr 


}n 

la 

StL 

ta 
n 
up 

s 


se 

th class took a test 
jdent who scored below 
<e a make-up test. How 
the class will not have to 
test? 

Leaves 


9 
8 

7 
6 


2 44 6 
003479 
2 2 5 6 
38 



DAY 5 



Extended Response 

Tamara uses 0.8 pound of mango to make 
a mango-banana fruit shake. 

How many shakes can Tamara make with 
3.6 pounds of mango? Explain your answer 

Tamara invites 6 friends over for mango- 
banana fruit shakes. How many pounds of 
mango will she need to make fruit shakes 
for herself and her friends? Explain your 
answer. 



Countdown to ISTEP+ IN35 



I 



Math Testing and 
Critical Thinking Skills 



What Are 
Critical 
Thinking Skills? 



Critical thinking skills are not a new phenomenon on 
the education scene. In 1956, Benjamin Bloom published 
a book that listed critical thinking skills in the form of a 
taxonomy as shown in the illustration below. 

Bloom's Taxonomy of Educational Objectives 



Evaluation 



Synthesis 



Analysis 



Application 
Comprehension 
Knowledge 



Knowledge is the simplest level of education objectives 
and is not considered a higher-order thinking skill. It 
requires the learner to remember information without 
having to fully understand it. Tasks that students perform 
to demonstrate knowledge are recalling, identifying, 
recognizing, citing, labeling, listing, reciting, and stating. 

EXAMPLES 

1 . \i''hat IS the formitld for the area of a trapezoid? 

2. What quadrant is the point (2, -6) located m? 

3. What IS the reciprocal of y? 



IN36 



• Comprehension is not considered a higher-order 
thinking skill either. Learners demonstrate compre- 
hension when they paraphrase, describe, summarize, 
illustrate, restate, or translate. Information isn't use- 
ful unless it's understood. Students can show they've 
understood by restating the information in their own 
words or by giving an example of the concept. 

EXAMPLES 

1 . Explain the difference between the points (4, 5) 
and (5, 4). 

2. Interpret the information in the graph below. 

3. Give an example of an irrational number. 

Many teachers tend to focus the most on knowledge and 
comprehension — and the tasks performed at these levels 
are important because they provide a solid foundation for 
the more complex tasks at the higher levels of Bloom's 
pyramid. 

However, offering students the opportunity to perform 
at still higher cognitive levels provides them with more 
meaningful contexts in which to use the information 
and skills they have acquired, thus allowing them to 
more easily retain what they have learned. 

When teachers incorporate application, analysis, 
synthesis, and evaluation as objectives, they allow 
students to utilize higher-order thinking skills. 

• Application involves solving, transforming, determining, 
demonstrating, and preparing. Information becomes 
useful when students apply it to new situations — 
predicting outcomes, estimating answers — this is 
application. 

EXAMPLES 

1. Organize the forms of pollution frorn most damaging 
to least damaging. 

2. Using the scale of 1 inch equals 200 miles, determine 
the point-to-point distance between Boston and 
Atlanta. 

3. Put the information below into a bar graph. 



IN37 



• Analysis includes classifying, comparing, making asso- 
ciations, verifying, seeing cause-and-effect relationships, 
and determining sequences, patterns, and consequences. 
You can think of analysis as taking something apart in 
order to better understand it. Students must be able to 
thmk in categories in order to analyze. 

EXAMPLES 

1 . What math skills do you use when reading a circle 
graphs 

2. Use the function table to write a rule for y in terms 
ofx. 

3. How can you use the LCM of 3 and 5 to find the 
sum of y and t ? 

• Synthesis requires generalizing, predicting, imagining, 
creating, making inferences, hypothcsizmg, making deci- 
sions, and drawing conclusions. Students create some- 
thing which is new to them when thev use synthesis. 
It's important to remember, though, that students can't 
create until thev have the skills and information thev have 
received in the comprehension through analvsis levels. 

EXAMPLES 

1. Make a scale drawing of your classroom. 

2. Write a word problem that can be represented by the 
equation 3 + y = 5. 

3. Poll your classmates about their favorite breakfast 
food and display your results in an appropriate graph. 

• Evaluation involves assessing, persuading, determining 
value, ludging, validating, and solving problems. Evaluation 
is based on all the other levels. When students evaluate, 
they make judgments, but not judgments based on per- 
sonal taste. These judgments must be based on criteria. It 

is important for students to evaluate because they learn to 
consider different points of view and to know how to 
validate their |udgments. 

EXAMPLES 

1 . Which of the following describes the correct way to 
round "3-^r' 

2. Based on the ratios of protein to serving size and 
fat to serving size, which muffin do you think is 
healthier? Explain. 

3. Do you think the statistics given in the article are 
accurate? Why or why not? 



IN38 



Why is it 
Important 
for Students 
to Work with 
Higher-Order 
Thinl<ing Skills? 



For one thing, if students can determine the levels of 
questions that will appear on their tests, they will be able 
to study using appropriate strategies. Bloom's leveling 
of questions provides a useful structure in which to 
categorize test questions, since tests will characteristically 
ask questions within particular levels. 

Also, thinking is a skill that can be taught. When vou 
have students practice answering questions at all the 
levels of Bloom's taxonomy, you are helping to scaffold 
then' learning. Information just becomes trivia unless that 
information is understood well enough to build more 
complicated concepts or generalizations. When students 
can comprehend — not just recall — the information, it 
becomes useful for future problem solving or creative 
thought. Think of information as a building material — 
like a board. It could be used to build something, but 
it is just useless litter unless you understand how to 
make use of it. 

Below are some question stems you — or your students- 
could use to create questions for each of the levels of 
higher-order thinking: 

Application 

1. Make a diagram to show . 

2. Use (a formula, manipulatives, mental math, a 
problem solving strategy, etc.) to find . 



3. (Find, determine, calculate, compute, etc.) . 

4. Explain how the (prmciple, theorem, concept) is 
evident in . 

5. In what way is a ? 

Analysis 

1. Which (strategies, operations, etc.) would you use 
to solve this problem? 

2. Find a pattern in . 

3. What other (properties, rules, definitions) are similar? 
Explain. 

4. Compare and contrast . 

5. How does the value of affect the value 

of ? 



IN39 



Synthesis 

1 . Write a problem that can be solved by . 

2. Use information in (your science book, a newspaper 
article, etc.) to write a problem. 

3. Create a new way to classify . 

4. Design your own to show . 

5. Create a new way to . 



Evaluation 

1. Is (an answer, an estmiate, etc.) reasonable? Explain. 

2. Do you have enough information to solve this 
problem? 

3. Which best represents ? 

4. Which solution method (is most efficient, is most 
accurate, gives the most information, etc.)? 

5. What is the nnportant mformation in this problem? 



IN40 



Multiple-Choice Questions 



The most common type of test question is multiple choice. 
To answer questions on a multiple-choice test, you will 
most likely fill in an answer sheet. It is very important to 
fill in your answer sheet correctly. When shading in circles, 
make your marks heavy and dark. Fill in the circles com- 
pletely, but do not shade outside the circles. Do not make 
any stray marks on your answer sheet. 

Questions on a multiple-choice test may require an under- 
standing of number and operations, algebra, geometiy, mea- 
surement, and data analysis and probability. Drawings, grids, 
or charts mav be included for certain tvpes of questions. 

Read each question carefullv and work the problem. You 
may be allowed to use blank space in the test booklet to 
write your calculations. Choose your answer from among 
the answer choices given, and fill in the corresponding 
circle on your answer sheet. 

If your answer is not one of the choices, read the question 
again. Be sure that vou understand the problem. Check 
your work for possible errors. 

Sample Question 

Try the following practice question to prepare for taking 
a multiple-choice test. Choose the best answer from the 
choices given. 

In a group of 30 students, 17 are middle school students, 
and the others are high school students. If one person is 
selected at random from this group, what is the probability 
that the person selected will be a high school student? 



I 



B. 



C. 



D. 



10 



"• 30 "-10 ^-10 

Think About the Solution 

There are 30 people in the group. If T? are middle school 
students, how many are high school students? (3) If one 
person is selected, there is a probability of 3 out of 30 that 
the person will be a high school student. This can be writ- 
ten as a ratio (3:30), a fraction (-^j, a decimal (0.1), or a 
percent (10%). None of these solutions is listed, so look 
for an equivalent solution. The fraction -r^ can be simpli- 
fied to tq . Since jq is given as one of your answer 
choices, B is the correct response. 



IN41 



Gridded-Response Questions 



Some questions require you to place your answer in 
a special grid. This type of question is called "gridded 
response" and may be identified by a special logo on 
your test. Answers to these questions may be whole 
numbers, fractions, or decimals. 

Work the problem and find an answer. Then write your 
answer in the grid provided. There is often more than one 
correct way to write your answer in the response grid. 

When filling in your grid, make your marks heavy and 
dark. Fill in the circles completely, but do not shade 
outside the circles. Do not make any stray marks on 
or outside of your grid. 

If your answer does not fit in the grid, you may need to 
write vc^ur answer in another form. If your answer still 
does not fit, read the question again. Be sure that you 
understand the problem. Check your work for possible 
errors. 

Sample Question 

A bowl of fiTut contains 3 oranges, 4 apples, and 3 bananas. 
If Amv chooses 1 piece of fioiit at random, what is the 
probability that she will choose an apple? 



Sample Correct Answers 

\V)7ft" your —*■ 
answer in the 
answer boxes 
at the top of 
the grid. 

Fill in the —>■ 
corresponding 
circle under 
each box. 





4 


J 


1 







f 


® 


(f> 




(•') 


• 


f.-> 


,.-, 


(•) 


•i) 
® 

® 

5 
6 
7 


p 

12, 
13) 
« 

d 

'2 


S' 

® 
® 
® 
n 

6 




9 

(21 
® 

5 
6 


® 

1 

® 

(31 
6 
7 

a 
W 



2 


/ 


5 






•« 










5 




m 


(/) 


(fj 








<7> 


(/"I 


1/ 




(^) 


!•) 


(•) 


(•) 


(•) 


» 


(•1 


(•) 


« 


(•' 


(0( 

® 

@ 

5 
6 


P 

J) 

4 

5 

e 

19) 


'01 

® 

ID 

@ 

4 

® 

6 

i 


(3) 

® 

4 
5 

i 


® 
® 

® 

iS 

® 


® 

(21 

4 

® 
® 
® 


® 
® 

(2 
5 

® 
® 
® 


® 

(21 

4 

® 
ffl 


® 
® 

21 

'1 

(8 

(9) 


® 

121 

® 

® 

(T! 

® 
® 



Write a 
decimal point 
or fraction 
bar in the 
answer boxes 
if It IS part of 
your answer 
Shade the 
decimal point 
or fraction 
bar circle 
below this 
answer box. 

Notice in the sample answers above that you may write 

your answers as either fractions or decimals. However, 

you mav not write mixed numbers, such as 137, in a 

^ - - I ■ 

response grid. If vou tried to fill in 13^, it wc)uld be read 

as -J- and would be counted as wrong. If your answer is a 

mixed number, write it as an improper fraction (^j or as 

a decimal (13.25) before filling in your grid. 



IN42 



Test-Taking Tip 

When filling m a response grid, 

DO: 

• Write your answer in the answer boxes at the top of 
the grid. 

• Write the decimal point or fraction bar in an answer 
box if it is part of the answer. 

• Fill in the corresponding circle under each of the boxes 
in which you wrote your answer. 

• Completely fill in each circle with a solid black mark. 
DO NOT: 

• Place spaces between the digits or symbols in the 
answer boxes. 

• Fill in more than one cnxle below each answer box 
in which you have written a number or svmbol. 

• Fill in a cnxle below an empty answer box. 

• Enter symbols besides the fraction bar or decimal 
point (such as $ or %) in the answer boxes. 

• Enter commas in numbers that have four or more 
digits. 

• Enter a mixed number into a grid. Write your answer 
as an improper fraction or a decimal. 

• Make stray marks on or outside of your grid. 

It is not important where you place your answer in the 
answer boxes, as long as your entire answer fits. You may 
left align, right align, or center your answer: 



2/5 2/5 2/5 



Also, you may include a leading zero for decimals less 
than one, but it is not necessary. 












4 












4 



IN43 




Algebraic Reasoning 



Are You Ready? 3 



■* OnlineResoiircesgo.hrw.com, 



7.1.2 
7.1.1 



1-1 
1-2 
1-3 




7.1.7 1-4 



Patterns and Relationships 

Numbers and Patterns 

Exponents 

Scientific Notation 

Scientific Notation with a Calculator 

Order of Operations 

Explore Order of Operations ^^. . . 

1-5 Properties of Numbers 

Ready to Go On? Quiz 

Focus on Problem Solving: Solve 



. 6 
10 
14 
18 
19 
23 
24 
28 
29 



7.2.3 
7.2.1 
7.2.3 
7.2.1 



1-6 
1-7 
1-8 
1-9 




Algebraic Thinking 

Variables and Algebraic Expressions 30 

Translating Words into Math 34 

Simplifying Algebraic Expressions 38 

Equations and Their Solutions 42 

Model Solving Equations 46 

7.2.1 1-10 Solving Equations by Adding or Subtracting 48 

7.2.1 1-11 Solving Equations by Multiplying or Dividing 52 

Ready to Go On? Quiz 56 

Real-World Connection: Illinois 57 

Study Guide; Preview 4 

Reading and Writing Math 5 

Game Time: Jumping Beans 58 

It's in the Bag! Step-by-Step Algebra 59 

Study Guide: Review 60 

Chapter Test 63 



Tools for Success 



and WrvtUta 
Math -^ 



Reading Math 5,10,42 

Writing Math 9,13,15,17,22,27, 
33,37,41,45,51,55 

Vocabulary 6, 10, 14, 19, 24, 30, 38, 

42, 48, 52 




Know-It Notebook Chapter 1 

Homework Help Online 8, 12, 16, 

21,26,32,36,40,44,50,54 

Student Help 7, 14, 20, 24, 25, 38, 39 







«% 



Test Prep 



Countdown to Testing Weeks 1, 2, 3 

Test Prep and Spiral Review 9, 

13,17,22,27,33,37,41,45,51,55 

Test Tackier 64 
ISTEP+ Test Prep 66 



iJWWt„,j.^.^,MU»W4:<t-VU. 'I V4W i lM,*W^ 



IN44 



--■- •*-, 







Integers and 
Rational lUumbers 



Are You Ready? 69 

Integers 

7.1.6 2-1 Integers 72 

EXT Negative Exponents 76 

fj^^ Model Integer Addition 78 

2-2 Adding Integers 80 

^03 Model Integer Subtraction 84 

2-3 Subtracting Integers 86 

^^^ Model Integer Multiplication and Division 90 

7.1.7 2-4 Multiplying and Dividing Integers 92 

^^3 Model Integer Equations 96 

7.2.1 2-5 Solving Equations Containing Integers 98 

Ready to Go On? Quiz 102 

Focus on Problem Solving: Make a Plan 103 

Factors and Multiples 

7.1.4 2-6 Prime Factorization 104 

7.1.3 2-7 Greatest Common Factor 108 

2-8 Least Common Multiple 112 

Ready to Go On? Quiz 116 

Focus on Problem Solving: Look Back 117 

Rational Numbers 

2-9 Equivalent Fractions and Mixed Numbers 118 

2-10 Equivalent Fractions and Decimals 122 

7.1.6 2-11 Comparing and Ordering Rational Numbers 126 

Ready to Go On? Quiz 130 

Real-World Connection: Arizona 131 

Study Guide: Preview 70 

Reading and Writing IVIath 71 

Game Time: Magic Squares 132 

It's in the Bag! Flipping Over Integers and Rational Numbers . . 133 

Study Guide: Review 134 

Chapter Test 137 



Tools for Success 




«^Reading 

- and WriHita 
Math -^ 

Reading Math 73,123 

Writing Math 71, 75, 77, 83, 95, 101, 
104,107, 111,115,121,125, 129 

Vocabulary 72, 104,108,112,118, 
122, 126 



Know-It Notebook Chapter 2 

Homework Help Online 74, 82, 8£ 

94, 106, 110,114, 120,124,128 

Student Help 72, 73, 76, 81, 86, 92, 

100, 104, 119,122,126, 127 







OiOl 



7.FP.3: Number and Operations 
and Algebra 

7.FPC.5: Number and 
Operations 

See pp. C2-C3 for text 



<% 



Test Prep 



Countdown to Testing Weeks 3, 
4,5,6 

Test Prep and Spiral Review 75, 

83,89,95, 101, 107,111, 115,121, 
125,129 

ISTEP+Test Prep 138 



IN45 




►^ 



7.FP.3: Number and Operations 
and Algebra 

7.FPC.5: Number and Operations 

See pp C2-C3 for text 




Applying Rational 
Rlumbers 

Are You Ready? i4i 

Decimal Operations and Applications 

Estimating with Decimals 144 

Adding and Subtracting Decimals 148 

Model Decimal Multiplication 152 

Multiplying Decimals 154 

Model Decimal Division 158 

Dividing Decimals 160 

Solving Equations Containing Decimals 164 

Ready to Go On? Quiz 168 

Focus on Problem Solving: Look Back 169 

Fraction Operations and Applications 

Estimating with Fractions 170 

Model Fraction Addition and Subtraction 174 

Adding and Subtracting Fractions 176 

Adding and Subtracting Mixed Numbers 180 

Model Fraction Multiplication and Division 184 

Multiplying Fractions and Mixed Numbers 186 

Dividing Fractions and Mixed Numbers 190 

Solving Equations Containing Fractions 194 

Ready to Go On? Quiz 198 

Real-World Connection: Virginia 199 

Study Guide: Preview 142 

Reading arid Writing Math 143 

Game Time: Number Patterns 200 

It's in the Bag! Operation Slide Through 201 

Study Guide: Review 202 

Chapter Test 205 






.v^Reading 

and WrbtUta 
Math -^ 

Writing Math 147, 151, 157, 163, 
157, 173, 179,183,189, 192, 197 

Vocabulary 144, 190 




Know-It Notebook Chapter 3 

Study Strategy 143 

Homework Help Online 146, 
156, 152, 156, 172, 178, 182,188, 
192, 196 

Student Help 144,145, 149,155, 
150, 151, 164, 166, 167,171,175, 
180, 181, 186, 194, 195,205 



50, 







Test Prep 



Countdown to Testing Weeks 6, 
7,8 

Test Prep and Spiral Review 147, 
151,157,163,167,173,179,183, 
189, 193,197 

Test Tackier 206 

ISTEP+ Test Prep 208 




Proportional 
Relationships 

n Are You Ready? 211 

Ratios, Rates, and Proportions 

7.1.9 4-1 Ratios 214 

7.1.9 4-2 Rates 218 

4-3 Identifying and Writing Proportions 222 

4-4 Solving Proportions 226 

Ready to Go On? Quiz 230 

Focus on Problem Solving: Make a Plan 231 

Measurements 

4-5 Customary Measurements 232 

4-6 Metric Measurements 236 

4-7 Dimensional Analysis 240 

Ready to Go On? Quiz 244 

Focus on Problem Solving: Solve 245 

Proportions in Geometry 

IVg^J Make Similar Figures 246 

7.3.5 4-8 Similar Figures and Proportions 248 

7.3.5 4-9 Using Similar Figures 252 

7.3.5 4-10 Scale Drawings and Scale Models 256 

Make Scale Drawings 260 

Ready to Go On? Quiz 262 

Real-World Connection: Minnesota 263 

Study Guide: Preview 212 

Reading and Writing Math 213 

Game Time: Water Works 264 

It's in the Bag! Paper Plate Proportions 265 

Study Guide: Review 266 

Chapter Test 269 



CHAPTER 



.^Reading 



»"«< WrbtUia 



Math 



Reading Math 222, 237, 248 

Writing Math 213,221,225,229, 
235,239,243,248,251,255,258 

Vocabulary 214, 218, 222, 226, 240, 
248, 252, 256 




Know-It Notebook Chapter 4 

Homework Help Online 216, 220, 
224, 228, 234, 238, 242, 250, 254, 258 

Student Help 215,232,236,240, 
249, 256 




Graphs and 
Functions 



Are You Ready? 273 

Tables and Graphs 

The Coordinate Plane 276 

Interpreting Graphs 280 

Functions, Tables, and Graphs 284 

Sequences 288 

Ready to Go On? Quiz 292 

Focus on Problem Solving: Understand the Problem 293 

Linear Functions 

Explore Linear Functions 294 

Graphing Linear Functions 296 

Nonlinear Functions 300 

Slope and Rates of Change 302 

Generate Formulas to Convert Units 307 

Slope-Intercept Form 308 

Direct Variation 313 

Inverse Variation 318 

Ready to Go On? Quiz 320 

Real-World Connection: Alabama 321 

Study Guide: Preview 274 

Reading and Writing Math 275 

Game Time: Clothes Encounters 322 

It's in the Bag! Graphs and Functions Fold-A-Books 323 

Study Guide: Review 324 

Chapter Test 327 



Tools for Success 



Writing Math 275, 279, 283, 287, 
299,306,312,317 

Vocabulary 276, 284, 288, 296, 300, 
302,308,313,318 




Know-It Notebook Chapter 5 

Homework Help Online 278, 282, 
286,290,298,304,310,316 

Student Help 285, 297, 300, 303, 
309,310,314,315 



0^ Test Prep _^ 



Countdown to Testing Weeks 11, 
12 

Test Prep and Spiral Review 279, 
283,287,291,299,306,312,317 

Test Tackier 328 

ISTEP+ Test Prep 330 






Percents 



7,1.9 

7.1.9 
7.1.9 
7.1.9 
7.1.9 



7.1.8 
7.1.9 



Are You Ready? 333 

Proportions and Percents 

6-1 Percents 336 

^3 Model Percents 339 

6-2 Fractions, Decimals, and Percents 340 

6-3 Estimating with Percents 344 

6-4 Percent of a Number 348 

6-5 Solving Percent Problems 352 

Ready to Go On? Quiz 356 

Focus on Problem Solving: Make a Plan 357 

Applying Percents 

6-6 Percent of Change 358 

6-7 Simple Interest 362 

Ready to Go On? Quiz 366 

Real-World Connection: Nebraska 367 

Study Guide: Preview 334 

Reading and Writing Math 335 

Game Time: Lighten Up 368 

It's in the Bag! Percent Strips 369 

Study Guide: Review 370 

Chapter Test 373 








oiol 



Tools for Success 



^^ Reading 
Math -^ 

Reading Math 335, 359 

Writing Math 338, 343, 347, 351, 
355,365 

Vocabulary 336, 358, 362 




Know-It Notebook Chapter 6 

Study Strategy 335 

Homework Help Online 337, 342, 
346, 350, 354, 360, 364 

Student Help 340, 344, 348, 353, 358 



7.FP.1 : Number and Operations | 
and Algebra and Geometry 

7.FPC.5: Number and 
Operations 

See pp C2-C3 for text 






B 

7.4.4 
7.4.3 
7.4.4 
7.4.4 
7.4.4 



7-1 
7-2 
7-3 
7-4 
7-5 



7.4.4 7-6 





IW 


7.4.1 


7-7 




11^) 


7.4.4 


7-8 


7.4.4 


7-9 




@ 


7.4.4 


7-10 



Collecting, Displaying, 
and Analyzing Data 

Are You Ready? 377 

Organizing and Displaying Data 

Frequency Tables, Stem-and-Leaf Plots, and Line Plots 380 

Mean, Median, Mode, and Range 385 

Bar Graphs and Histograms 390 

Reading and Interpreting Circle Graphs 394 

Box-and-Whisker Plots 398 

Explore Box-and-Whisker Plots %, 402 

Ready to Go On? Quiz 404 

Focus on Problem Solving: Solve 405 

Representing and Analyzing Data 

Line Graphs 406 

Use Venn Diagrams to Display Collected Data 410 

Choosing an Appropriate Display 412 

Use Technology to Display Data ^^^ 416 

Populations and Samples 418 

Scatter Plots 422 

Samples and Lines of Best Fit ^ 426 

Misleading Graphs 428 

Ready to Go On? Quiz 432 

Real-World Connection: Utah 433 

Study Guide: Preview 378 

Reading and Writing IVIath 379 

Game Time: Code Breaker 434 

It's in the Bag! Graph Match 435 

Study Guide: Review 436 

Chapter Test 439 



^^ Reading 

Reading IVIath 379 

Writing IVIath 384, 389, 393, 397, 
401,409,415,421,431 

Vocabulary 380, 385, 390, 394, 398, 
406,418,422 



Tools for Success 




Know-It Notebook Chapter 7 

Homework Help Online 382, 388, 
392, 395, 400, 408, 414, 420, 424, 430 

Student Help 380, 381, 385, 387, 
398,406,418,419 







Test Prep 



Countdown to Testing Weeks 

15,16,17 
Test Prep and Spiral Review 

389,393,397,401,409,415,421 
425, 431 

Test Tackier 440 

ISTEP+ Test Prep 442 



14, 
384, 




IN50 



B 



8-1 



8-2 



7.3.1 8-3 



8-4 



Geometric Figures 



Are You Ready? 445 

Lines and Angles 

Building Blocks of Geometry 448 

Explore Complementary and Supplementary Angles 452 

Classifying Angles 454 

Explore Parallel Lines and Transversals 458 

Line and Angle Relationships 460 

Construct Bisectors and Congruent Angles 464 

Ready to Go On? Quiz 466 

Focus on Problem Solving: Understand the Problem 467 

Circles and Polygons 

Properties of Circles 468 

Construct Circle Graphs 472 

Classifying Polygons 474 

Classifying Triangles 478 

Classifying Quadrilaterals 482 

Angles in Polygons 486 

Ready to Go On? Quiz 490 

Focus on Problem Solving: Understand the Problem 491 

Transformations 

Congruent Figures 492 

Translations, Reflections, and Rotations 496 

Explore Transformations ^^ 501 

Dilations 502 

Symmetry 504 

Create Tessellations 508 

Ready to Go On? Quiz 510 

Real-World Connection: Maine 511 

Study Guide: Preview 446 

Reading and Writing Math 447 

Game Time: Networks 512 

It's in the Bag! Brochure Book of Geometric Figures 513 

Study Guide: Review 514 

Chapter Test 517 



8-5 
8-6 
8-7 
8-8 



.3.4 


8-9 


.3.2 


8-10 




lUJ 

EXT 


.3.4 


8-11 



^v Reading 

, and WrbtiAta 
Math -^ 



Reading Math 449, 455, 460, 461, 
468, 474, 497 

Writing Math 447,451,454,457, 
463, 471, 481, 485, 489, 495, 500, 507 

Vocabulary 448, 454, 460, 468, 474, 
478, 482, 485, 492, 496, 502, 504 



Tools for Success 



StudV 



Know-It Notebook Chapter 8 

Homework Help Online 450,456, 
462, 470, 475, 480, 484, 488, 494, 
498, 506 

Student Help 448, 455, 475, 493, 
498, 502 




0. Test Prep 

Countdown to Testing Weeks 17, 
18, 19 

Test Prep and Spiral Review 451, 
457,463,471,477,481,485,489, 
495, 500, 507 

ISTEP+Test Prep 51 



lil^p"— 



CHAPTER 







B 



£1 



Online Resources go.hrw.com 





7.1.5 



o«al 



^ 



7.FP.2: Measurement and 
Geometry and Algebra 

7.FPC.4: Measurement and 
Geometry 

i See pp C2-C3 for te>.i 



Measurement: Two- 
Dimensional Figures 

Are You Ready? 521 

Perimeter, Circumference, and Area 

9-1 Accuracy and Precision 524 

^^p Explore Perimeter and Circumference 528 

9-2 Perimeter and Circumference 530 

^^p Explore Area of Polygons 534 

9-3 Area of Parallelograms 536 

9-4 Area of Triangles and Trapezoids 540 

^^3 Compare Perimeter and Area of Similar Figures 544 

9-5 Area of Circles 546 

9-6 Area of Irregular Figures 550 

Ready to Go On? Quiz 554 

Focus on Problem Solving: Understarnd the Problem 555 

Using Squares and Square Roots 

QI3 Explore Square Roots and Perfect Squares 556 

9-7 Squares and Square Roots 558 

EXT Identifying and Graphing Irrational Numbers 562 

Explore the Pythagorean Theorem 564 

The Pythagorean Theorem 566 

Ready to Go On? Quiz 570 

Real-World Connection: Indiana 571 

Study Guide: Preview 522 

Reading and Writing Math 523 

Game Time: Shape Up 572 

It's in the Bag! Bag o' Measurement 573 

Study Guide: Review 574 

Chapter Test 577 



9-8 




Tools for Success 



.N^Reading 

-=^ and i^irUrl^ 

Math -^ 



Reading Math 523, 540, 541, 558, 
561 

Writing Math 527, 533, 537, 539, 
543, 549, 553, 559, 569 

Vocabulary 524, 530, 536, 558, 552, 
566 




Know-It Notebook Chapter 9 

Homework Help Online 526, 532, 
536, 538, 542, 548, 552, 560, 568 

Student Help 524, 531, 546, 547, 
551,562 



B 



Test Pr«p^^ 



Countdown to Testing Weeks 1 9, 
20,21 

Test Prep and Spiral Review 527, 
533,537,539,543,549,561,559 

Test Tackier 578 

ISTEP+ Test Prep 580 



Measurement: Three- 
Dimensional Figures 

Are You Ready? 583 

Volume 

^^3 Sketch Three-Dimensional Figures from Different Views .... 586 

10-1 Introduction to Three-Dimensional Figures 588 

EXT Cross Sections 592 

^^3 Explore the Volume of Prisms and Cylinders 594 

10-2 Volume of Prisms and Cylinders 596 

10-3 Volume of Pyramids and Cones 600 

Ready to Go On? Quiz 604 

Focus on Problem Solving: Solve 605 

Surface Area 

^^) Use Nets to Build Prisms and Cylinders 606 

10-4 Surface Area of Prisms and Cylinders 607 

^^J Explore the Surface Area of Pyramids and Cones 612 

10-5 Surface Area of Pyramids and Cones 614 

^^p Explore the Surface Areas of Similar Prisms 618 

10-6 Changing Dimensions 620 

Explore Changes in Dimensions ^g^ 625 

Ready to Go On? Quiz 626 

Real-World Connection: Kentucky 627 

Study Guide: Preview 584 

Reading and Writing Math 585 

Game Time: Blooming Minds 628 

It's in the Bag! CD 3-D 629 

Study Guide: Review 630 

Chapter Test 633 




Tools for Success 



^v^ Reading 

3"" WrbtiAta 
Math -^ 



Reading Math 596 

Writing Math 599,503,611,617 

Vocabulary 588, 592, 596, 607, 614 




_^,Jii'i 



Know-It Notebook Cliapter 10 

Study Strategy 585 

Homework Help Online 590, 598, 
602,610,616,623 

Student Help 588,601,620,521 



7.FP.2: Measurement and 

Geometry and Algebra 

7.FPC.4: Measurement and 
Geometry 

See pp- C2-C3 for text. 




Countdown to Testing Weeks 22, 23 



Test Prep and Spiral Review 591, 
599,503,511,517,524 

ISTEP-i- Test Prep 534 




7.FPC.7: Probability 

See pp C2-C3 for text 

m 




Probability 



Are You Ready? 637 

Introduction to Probability 

Probability 640 

Experimental Probability 644 

Find Sample Spaces 648 

Theoretical Probability 652 

Simulations 656 

Making Predictions 658 

Experimental and Theoretical Probability 662 

Ready to Go On? Quiz 664 

Focus on Problem Solving: Understand the Problem 665 

Applications of Probability 

Probability of Independent and Dependent Events 666 

Combinations 670 

Permutations 674 

Ready to Go On? Quiz 678 

Real-World Connection: Delaware 679 

Study Guide: Preview '. 638 

Reading and Writing Math 639 

Game Time: Buffon's Needle 680 

It's in the Bag! The Business of Probability 681 

Study Guide: Review 682 

Chapter Test 685 



^^^ Reading 

and WrbtiKa 
Math -^ 



Reading Math 639, 645, 652, 666 

Writing Math 641, 643, 644, 651, 
655,661,669,677 

Vocabulary 640, 644, 648, 652, 658, 
666, 670, 674 



Tools for Success 



Know-it Notebook Chapter 11 

Homework Help Online 642, 646, 
650, 654, 660, 668, 672, 676 

Student Help 653, 675 







Test Prep 



Countdown to Testing Week 24 

Test Prep and Spiral Review 643, 
647,651,655,661,669,673,677 

Test Tackier 686 

ISTEP+ Test Prep 688 



_>^ 




Multi-step Equations 
and Inequalities 

U Are You Ready? 69i 

Multi-Step Equations 

Model Two-Step Equations 694 

7.2.2 12-1 Solving Two-Step Equations 696 

7.2.1 12-2 Solving Multi-Step Equations 700 

7.2.1 12-3 Solving Equations with Variables on Both Sides 704 

Ready to Go On? Quiz 708 

Focus on Problem Solving: Solve 709 

Inequalities 

12-4 Inequalities 710 

7.2.1 12-5 Solving Inequalities by Adding or Subtracting 714 

7.2.1 12-6 Solving Inequalities by Multiplying or Dividing 718 

7.2.2 12-7 Solving Multi-Step Inequalities 722 

EXT Solving for a Variable 726 

Ready to Go On? Quiz 728 

Real-World Connection: New Hampshire 729 

Study Guide: Preview 692 

Reading and Writing Math 693 

Game Time: Flapjacks 730 

It's in the Bag: Wired for Multi-Step Equations 731 

Study Guide: Review 732 

Chapter Test 735 



CHAPTER 



Tools for Success 



Reading 

and Wri4d4Ui 
Math -' 

Reading Math 711 

Writing Math 703,707,711,712, 
717,721,725 

Vocabulary 710 




Know-It Notebook Chapter 12 

Study Strategy 693 

Homework Help Online 698, 702, 
706,712,716,720,724 

Student Help 696,700,714,715 



Focus on Problem Soliring 



The Problem Solving Process 

In order to be a good problem solver, you first need a good problem- 
solving process. A process or strategy will help you to understand the 
problem, to work through a solution, and to check that your answer 
makes sense. The process used in this book is detailed below. 



UNDERSTAND the Problem 



I What are you asked to find? 
I What information is given? 

I What information do you need? 

I Is all the information given? 



Restate the problem in your own words. 

Identify the important facts in the 

problem. 

Determine which facts are needed to 

solve the problem. 

Determine whether all the facts are 

given. 



Have you ever solved a similar 

problem? 

What strategy or strategies 

can you use? 



Think about other problems like this 
that you successfully solved. 
Determine a strategy that you can 
use and how vou will use it. 



SOLVE 

■ Follow your plan. 

LOOK BACK 

■ Have you answered the question? 

■ Is your answer reasonable? 

■ Is there another strategy you 
could use? 

■ Did you learn anything while 
solving this problem that could 
help you solve similar problems 
in the future? 



Show the steps in your solution. Write 
your answer as a complete sentence. 



Be sure that you answered the question 

that is being asked. 

Your answer should make sense 

in the context of the problem. 

Solving the problem using another 

strategy is a good way to check 

your work. 

Try to remember the problems you have 

solved and the strategies you used to 

solve them. 



■^:^^?yy?v!^^^yy^^vyg:$y:f ^^^w^^N r ; ?yy^^ 



IN 56 Focus on Problem Solving 




Using the Problem Solving Process 

During summer vacation, Ricardo wll go to space 
camp and then to visit his relatives. He will be gone 
for 5 weeks and 4 days and will spend 1 1 more 
days with his relatives than at space camp. 
How long will Ricardo stay at each place? 




^ 



UNDERSTAND the Problem 

List the important information. 

* Ricardo will be gone for 5 weeks and 4 days. 

• He will spend 1 1 more days with his relatives than at space camp. 
The answer will be how long Ricardo stays at each place. 



You can draw a diagram to show how long he will stay at each place. 
Use boxes for the length of each stay. The length of each box will 
represent the length of each stay. 

SOLVE 

Think: There are 7 days in a week, so 5 weeks and 4 days is a total of 
39 days. Your diagram might look like this: 

Relatives 



' days 



11 days 



Space camp 



? days 



= 39 days 



39 — 1 1 = 28 Subtract 1 1 days from the total number of days. 
28 -^ 2 = 14 Divide this number by 2 for the 2 places he visits. 



Relatives 



Space camp 



14 days 



11 days 



= 25 days 



14 days 



= 14 days 



So Ricardo will stay with his relatives for 25 days and at space camp 
for 14 days. 

LOOK BACK 

Twenty- five days is 1 1 days longer than 14 days. The total length of the 
two stays is 25 + 14 = 39 days, or 5 weeks and 4 days. This solution fits 
the information given in the problem. 



Focus on Problem Solving IN 57 



Using Your Book for Success 

This book has many features designed to help you learn and study 
math. Becoming familiar with these features will prepare you for 
greater success on your exams. 



Learn 



Preview new vocabulary 
terms listed at the 
beginning of every lesson. 



Look for the 
Student Help 

for hints and 
reminders. 



Practice 



Look back at 
examples from the 
lesson to solve the 
Guided Practice 
and Independent 
Practice exercises. 



Review 






'""■'■'•■'—' 


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j.^^.«|.^l 


'■ 1 i.nr*oi 


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t-™-'--'- 


n^p 


at ^ cm « Un. lOu.T.X t. 


rzr^™ 




ami 




'I'svw,— ■mITITI 'Iu^i '-^["^i 



jji.i.ii.j.M.iji.iijjin»» 



Study the examples to 
learn new math ideas 
and skills. The examples 
include step-by-step 
solutions. 



Use the internet for 
Homework Help 
Online. 



Review the 
vocabulary from 
the entire chapter. 




Review important 
examples and 
test yourself with 
practice problems 
from every lesson in 
the chapter. 



IN 58 Using Your Book for Success 



Scavenger Hlrnt 



% 



\ 



I. 

2. 

3. 

4. 

5. 

6. 
7. 



Holt McDoiigal MatJiematics is your resource to 

help you succeed. Use this scavenger hunt to discover some of 

the many tools Holt provides to help you be an independent learner. 

On a separate sheet of paper, fill in the blanks to answer each question 
below. In each answer, one letter will be in a yellow box. Wlien you 
have answered every question, use the letters to fill in the blank at the 
bottom of the page. 

What is the first key vocabulary term in the Study Guide: Preview for chapter 8? 



-m 



Wliat is the last key vocabulary term in the Study Guide: Review for cliapter 7? 



What game is featured in chapter 2 Game Time? 



What keword should you enter for Learn It Online on page 368? 



What project is outlined in chapter 7 It's in the Bag? 



What structure is spotlighted on page 444? 



#^ 






i 



What building is featured in chapter 1 Real World Connections? 



8. 



The chapter 5 Test Tackier gives strategies for what kind of standardized 
test item? 



m 



Why did the chicken add its opposite to itself? To get to the 
other side of the... 



% 



Scavenger Hunt IIM59 




1A 


Patterns and 
Relationships 




1-1 


Numbers and Patterns 




1-2 


Exponents 


7.1.2 


1-3 


Scientific Notation 


7.1.1 


LAB 


Scientific Notation with 
a Calculator 




1-4 


Order of Operations 


7.1.7 


LAB 


Explore Order of 
Operations 




1-5 


Properties of Numbers 




IB 


Algebraic Thinking 




1-6 


Variables and Algebraic 
Expressions 


7.2.3 


1-7 


Translating Words 
into Math 


7.2.1 


1-8 


Simplifying Algebraic 
Expressions 


7.2.3 


1-9 


Equations and Their 
Solutions 


7.2.1 


LAB 


Model Solving Equations 




1-10 


Solving Equations by 
Adding or Subtracting 


7.2.1 


1-11 


Solving Equations by 
Multiplying or Dividing 


7.2.1 



■f^fg^^s^-^rrj^f-v^^ 



£?. 



Chapter 1 




Why Learn This? 



Yellowstone National Park was created by 
Congress in 1872. An algebraic expression 
can model the current age of the park. 



Learn It Online 

Chapter Project Online go.hrw.com, 



■'•i 




'^m;% W'l'^*'^- ^-^^^ 



t-jf^!i 







Are You Ready? 



(L 



y Learn It Online 

Resources Online go.hrw.com 

l yyf^ijyfMSIO AYR1 ^G^ 



0^ Vocabulary 



Choose the best term from the Hst to complete each sentence. 

1 . The operation that gives the quotient of two numbers 
is ? . 

2. The ? of the digit 3 in 4,903,672 is thousands. 

3. The operation that gives the product of tvvo numbers 
is ? . 

? is 5. 



4. In the equation 15^3 = 5, the . 



division 
multiplication 
place value 
product 
quotient 



Complete these exercises to review skills you will need for this chapter. 

0^ Find Place Value 

Give the place value of the digit 4 in each number. 
5. 4,092 6. 608,241 7. 7,040,000 8. 4,556,890,100 

9. 3,408,289 10. 34,506,123 11. 500,986,402 12. 3,540,277,009 

Use Repeated Multiplication 

Find each product. 

13. 2-2-2 14.9-9-9-9 15.14-14-14 16.10-10-10-10 

17. 3-3-5-5 18.2-2-5-7 19. 3-3- 11 -11 20.5-10-10-10 



Division Facts 






Find each quotient. 






21. 49^7 22. 54^9 


23. 96^ 12 


24. 88 4- 8 


25. 42 H- 6 26. 65 ^ 5 


27. 39 ^ 3 


28. 121 ^ 11 



whole Number Operations 



Add, subtract, multiply, or divide. 








29. 425 30. 619 


31. 


62 


32. 373 


+12 + 254 




- 47 


+ 86 


33. 62 34. 122 


35. 


7)623 


36. 24)149 



X 42 



X 15 




Algebraic Reasoning 



CHAPTER 




Study Guide^rW^^n 



Where You've Been 

Previously, you 

• used order of operations 
to simplify whole number 
expressions without exponents. 

• used multiplication and 
division to solve problems 
involving whole numbers. 

• wrote large numbers in 
standard form. 



In This Chapter 



You will study 

• simplifying numerical 
expressions involving order of 
operations and exponents. 

• using concrete models to solve 
equations. 

• writing numbers in scientific 
notation. 



Where You're Going 

You can use the skills 
learned in this chapter 

• to express distances and sizes 
of objects in scientific fields 
such as astronomy and biology. 

• to solve problems in math and 
science classes such as Algebra 
and Physics. 



Key 

Vocabulary /Vocabulario 



algebraic expression 


expresion algebraica 


Associative Property 


propiedad asociativa 


Commutative Property 


propiedad 
conmutativa 


Distributive Property 


propiedad distributiva 


equation 


ecuacion 


exponent 


exponente 


numerical expression 


expresion numerica 


order of operations 


orden de las 
operaciones 


term 


termino 



variable 



variable 



Vocabulary Connections 

To become familiar with some of the 
vocabulaiy terms in the chapter, consider 
the following. You may refer to the chapter, 
the glossary, or a dictionary if you like. 

1 . The words equation, equal, and equator all 
begin with the Latin root equa-, meaning 
"level." How can the Latin root word help 
you define equation ? 

2. The word uiunerical means "of numbers." 
How might a numerical expression differ 
from an expression such as "the sum of 
two and five"? 

3. When something is variable, it has the 
ability to change. In mathematics, a 
variable is an algebraic symbol. What 
special property do you think this type 
of symbol has? 



Chapter 1 




Readirijg r 

and Wri^dAta 
Math ^ ^ 



CHAPTER 




Reading Strategy: Use Your Book for Success 

Understanding how your textbook is organized will help you locate and 
use helpful information. 

As you read through an example problem, pay attention to the margin notes , 
such as Helpful Hints, Reading Math notes, and Caution notes. These 
notes will help you understand concepts and avoid common mistakes. 



Ii.i5iil6ijj}''jtl!j']ji| 

Read -4^ as "-4 ti^ 
the 3rd power or 
—4 cubed". 



A repeating decima/ 
can be written with\ 
a bar over the digits 



In Example 1A, 
parentheses are noi^ 
needed because 



Caution! 



An open circle 
means that the 
corresponding valuel 



The glossary is found 
in the back of your 
textbook. Use it to 
find definitions and 
examples of unfamiliar 
words or properties. 



The index is located 
at the end of your 
textbook. Use it to 
find the page where 
a particular concept 
is taught. 



The Skills Bank is 
found in the back of 
your textbook. These 
pages review concepts 
from previous math 
courses. 





Use your textbook for the following problems. 

1 . Use the index to find the page where exponent is defined. 

2. In Lesson 1-8, what does the Remember box, located in the margin of 
page 39, remind you about the perimeter of a figure? 

3. Use the glossary' to find the definition of each term: order of operations, 
numerical expression, equation. 

4. Where can you review how to read and write decimals? 



Algebraic Reasoning 5 



1-1 



Vocabulary 

conjecture 



EXAMP 



Numbers 
and Patterns 



Each year, football teams battle for 
the state championship. The table 
shows the number of teams in 
each round of a division's football 
playoffs. You can look for a pattern 
to find out how many teams are in 
rounds 5 and 6. 




Football Playoffs 


Round 


1 


2 


3 


4 


5 


6 


Number of Teams 


64 


32 


16 


8 







'■? 



Identifying and Extending Number Patterns 

Identify a possible pattern. Use the pattern to write the next 
three numbers. 



A 64,32,16,8, 



64 



16 






4-2 4-2 4-2 4-2 4-2 4-2 
A pattern is to divide each number by 2 to get the next number. 
84-2 = 4 44-2 = 2 24-2=1 

The next three numbers wall be 4, 2, and 1. 

B 51,44,37,30, , , ,... 

51 44 37 30 S S ■ 
-7 -7 -7 -7 -7 -7 
A pattern is to subtract 7 from each number to get the next 
number. 

30 - 7 = 23 23 - 7 = 16 16 - 7 = 9 

The next three numbers will be 23, 16, and 9. 

C 2,3,5,8, 12, ^,fe,^'-- • 

2 3 5 8 12 "'" "' -^ 

-l-l-l-2-(-3-l-4-l-5 -1-6 -1-7 
A pattern is to add one more than you did the time before. 
12 + 5=17 17-1-6 = 23 23-1-7 = 30 

The next three numbers will be 17, 23, and 30. 



Chapter 1 Algebraic Reasoning 



y'l'Jbu Lessor Tutorials OnlinE my.hrw.com 



EXAMPLE [2] Identifying and Extending Geometric Patterns 



Identify a possible pattern. Use the pattern to draw the next three 
figures. 






The pattern is alternating squares and circles with triangles 
between them. 

The next three figures will be 



Helpfuljmji 



For more on 
conjectures, see 
Skills Bank p. SB12. 



The pattern is to shade every other triangle in a clockwise direction. 
The next three figures will be 



You can analyze patterns to make conjectures. A conjecture is a 
statement believed to be true. 



EXAMPLE [bj Using Tables to Identify and Extend Patterns 



Figure 1 Figure 2 



Figure 3 



Make a table that shows 

the number of triangles in 

each figure. Then make a 

conjecture about the 

number of triangles in the 

fifth figure of the pattern. 

Complete the table, and use drawings to justify your answer. 



The pattern is to add 
2 triangles each time. 



+2 +2 +2 +2 
Figure 4 has 6 + 2 = 8 triangles. Figure 5 has 8 + 2 = 10 triangles. 



Figure 


1 


2 


3 


4 


5 


Number of 
Triangles 


2 


4 


6 


8 


10 



AAA 


AAAAA 


Figure 4 



Figure 5 



Thmk and Discuss 


^^ 


^^" 


1. Describe two different number patterns that begin 


vnth 3, 6 


2. Tell when it would be useful to make a 
and extend a pattern. 


table to help 


you identify 



[ ^Mb'j Lesson Tutorials Online my.hrw.com 



1-1 Numbers and Patterns 



1-1 



ZI3. 



/ 



Homework Help Online go.hrw.com, 
[goI 



keyword ■mBiliUBM 
Exercises 1-14, 15, 17, 23 



6UldED PRACTICE 



See Example 1 Identify a possible pattern. Use the pattern to write the next three numbers. 

1. 6, 14, 22, 30, „,__, 2. 1,3,9, 27, ,__,___, ... 

3. 59, 50, 41,32,""%g,B. ••■ 4. 8, 9, 11, 14, S, B- ■> • • • 



See Example 2 Identify a possible pattern. Use the pattern to draw the next three figures. 

=-A A A A ^--0 P Q. d 



See Example 3 



7. Make a table that shows the number of green triangles in each figure. Then 
make a conjecture about the number of green triangles in the fifth figure of 
the pattern. Complete the table, and use drawings to justify your answer. 



Figure 1 Figure 2 



INDEPENDENT PRACTICE 



Figure 3 



See Example 1 Identify a possible pattern. Use the pattern to write the next three numbers. 

8.27,24,21,18, , , ,... 9.4,096,1,024,256,64, , , 



10. 1,3,7, 13,21, 



11. 14,37,60,83, 



"» ^^^'^^^j C=*^» ' 



See Example 2 Identify a possible pattern. Use the pattern to draw the next three figures. 

12. n A O n A ii 13. 







See Example 3 14. Make a table tliat shows the number of dots in each figure. Then make a 

conjecture about the numer of dots in the sixth figure of the pattern. Complete 
the table, and use drawings to justify your answer. 



Figure 1 



« e 
Figure 2 



Figure 3 



9 « • 

« e e 
Figure 4 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EP2. 



Use the rule to write the first five numbers in each pattern. 

15. Start with 7; add 16 to each number to get the next number. 

16. Start with 96; divide each number by 2 to get the next number. 

17. Start wdth 50; subtract 2, then 4, then 6, and so on, to get the next number. 

18. Critical Thinl<ing Suppose the pattern 3, 6, 9, 12, 15 ... is continued 
forever. Will the number 100 appear in the pattern? Why or why not? 



8 Chapter 1 Algebraic Reasoning 



Identify a possible pattern. Use the pattern to find the missing numbers. 

19.3,12, ,192, 768, _,_,... 20.61,55, ,43,^,^,25,.. 

21. ™, ,19,27,35, ,51,... 22.2, ,8, 



32, 64, 



23. Health The table shows the target heart rate 
during exercise for athletes of different ages. 
Assuming the pattern continues, what is the 
target heart rate for a 40-year-old athlete? a 
65-year-old athlete? 

Draw the next three figures in each pattern. 
24 ^ 



Target Heart Rate 


Age 


Heart Rate 
(beats per minute) 


20 


150 


25 


146 


30 


142 


35 


138 



25. ©.[i].A,®.H.A.®, ... 

26. Social Studies In the ancient Mayan civilization, people used a number 
system based on bars and dots. Several numbers are shown below. Look 
for a pattern and write the number 18 in the Mayan system. 



3 5 8 10 13 15 

^^ 27. What's the Error? A student was asked to write the next three numbers 
in the pattern 96, 48, 24, 12, . . . .The student's response was 6, 2, 1. Describe 
and correct the student's error. 

28. Write About It A school chess club meets every Tuesday during the 

month of March. March 1 falls on a Sunday. Explain how to use a number 
pattern to find all the dates when the club meets. 

^ 29. Challenge Find the 83rd number in the pattern 5, 10, 15, 20, 25 



i 



Test Prep and Spiral Review 



30. Multiple Choice Which is the missing number in the pattern 
2, 6, , 54, 162, . . . ? 

c£) 10 a:) 18 CD 30 cd> 43 

31. Gridded Response Find the next number in the pattern 9, 11, 15, 21, 29, 39, . . 

Round each number to the nearest ten. (Previous course) 

32. 61 33. 88 34. 105 35. 2,019 36. 11,403 

Round each number to the nearest hundred. (Previous course) 

37. 91 38. 543 39. 952 40. 4,050 41. 23,093 



7-7 Numbers and Patterns 



^ 7.1.2 Recognize and compute whole number powers of whole numbers 

A DNA molecule makes a copy 
of itself by splitting in half. Each 
half becomes a molecule that is 
identical to the original. The 
molecules continue to split so 
that the two become four, the 
four become eight, and so on. 



Vocabulary 

power 

exponent 

base 




Each time DNA copies itself, the 
number of molecules doubles. 
After four copies, the number of 
molecules is 2 • 2 • 2 • 2 = 16. 

This multiplication can also be 
written as a power, using a base 
and an exponent. The exponent 
tells how many times to use the 
base as a factor. 




Read 2" as "the 
fourth power of 2" 
or "2 to the fourth 
power." 



2*2*2*2 = 



j EXAMPLE 



9 



Base 



Evaluating Powers 

Find each value. 



Interactivities Online ► A 5 

I 5- = 5 • 5 

= 25 







The structure of DNA can be 
compared to a twisted ladder. 




Exponent 



= 16 



Use 5 as a factor 2 times. 



B 2" 



2" = 2 • 2 • 2 • 2 • 2 • 2 



= 64 



C 25' 



25' = 25 



Use 2 as a factor 6 times. 



Any number to tlie first power is equal 
to that number. 



19° =1 



Any number to the zero power, except zero, is equal to 1. 

6° = 1 10" = 1 

Zero to the zero power is undefined, meaning that it does not exist. 

10 Chapter 1 Algebraic Reasoning 



y]'^■i'J Lesson Tutorials OnlinE mv.hrw.com 



To express a whole number as a power, write the number as the 
product of equal factors. Then wTite the product using the base and 
an exponent. For example, 10,000 = 10 ■ 10 • 10 • 10 = 10\ 



EXAMPLE 




L 



Expressing Whole Numbers as Powers 

Write each number using an exponent and the given base. 

A 49, base 7 

49 = 7 • 7 7 is used as a factor 2 times. 

= r- 



B 81, base 3 

81 =3 • 3-3 -3 
= 3^ 



3 is used as a factor 4 times. 



EXAMPLE [T] Earth Science Application 







An earthquake 
measuring 7.2 on the 
Richter scale struck 
Duzce, Turkey, on 
November 12, 1999- 



Earthquake Strength 


Category 


Magnitude 


Moderate 


5 


Large 


6 


Major 


7 


Great 


8 



The Richter scale measures an 
earthquake's strength, or magnitude. 
Each category in the table is 10 times 
stronger than the next lower category. 
For example, a large earthquake is 
10 times stronger than a moderate 
earthquake. How many times stronger 
is a great earthquake than a moderate 
one? 

An earthquake with a magnitude of 6 is 10 times stronger than one 
with a magnitude of 5. 

An earthquake with a magnitude of 7 is 10 times stronger than one 
with a magnitude of 6. 

An earthquake with a magnitude of 8 is 10 times stronger than one 
with a magnitude of 7. 

10 • 10 • 10 = 10' = 1,000 

A great earthquake is 1,000 times stronger than a moderate one. 



Think artd Discuss 

1. Describe a relationship between 3^' and 3^. 

2. Tell which power of 8 is equal to 2*'. Explain. 

3. Explain why any number to the first power is equal to 
that number. 



I yjilBD Lesson Tutorials Online mv.hrw.com 



1-2 Exponents 11 



1-2 




-lifi^jjii 



GUIDED PRACTICE 



See Example 1 Find each value. 

u 1. 2^ 2. 3^ 



3. 6^ 



4. 9' 



keyword ■mbiiwbjM ® 
Exercises 1-30, 37, 39, 41, 45, 
49,51,55 



5. 10'^ 



See Example 2 Write each number using an exponent and the given base. 

L 6. 25, base 5 7. 16, base 4 8. 27, base 3 



9. 100, base 10 



See Example 3 10. Earth Science On the Richter scale, a great earthquake is 10 times 

stronger than a major one, and a major one is 10 times stronger than a large 
one. How many times stronger is a great earthqua]<;e than a large one? 



INDEPENDENT PRACTICE 



See Example 1 Find each value. 

11. 11- 

16. 2^ 



12. 3^ 
17. 5' 



13. 


8^ 


14. 


43 


15. 


3^ 


18. 


2' 


19. 


5^ 


20. 


30' 



See Example 2 Write each number using an exponent and the given base. 

, 21. 81, base 9 22. 4, base 4 23. 64, base 4 



24. 1, base 7 

27. 1,600, base 40 



25. 32, base 2 
28. 2,500, base 50 



26. 128, base 2 

29. 100,000, base 10 



See Example 3 30. In a game, a contestant liad a starting score of one point. He tripled his 

I score even,' turn for four turns. Write his score after four turns as a power. 

' Then find his score. 



Extra Practice 



See page EP2. 



PRACTICE AND PROBLEM SOLVING 



Give two ways to represent each number using powers. 



31. 81 32. 16 

Compare. Write <, >, or 



36. 4- 15 
40. 10,000 



10^ 



37. 2^ 
41. 6^ 



33. 64 

3- 
3.000 



34. 



38. 64 
42. 9^ 



43 
3« 



35. 625 

39. 8^ 
43. 5^ 



17" 



44. To find the volume of a cube, find the third power of the length of an edge 
of the cube. What is the volume of a cube that is 6 inches long on an edge? 

45. Patterns Domingo decided to save $0.03 the first day and to triple the 
amount he saves each day. How much will he save on the seventh day? 

46. Life Science A newborn panda cub weighs an average of 4 ounces. How 
many ounces might a one-year-old panda weigh if its weight increases by 
the power of 5 in one year? 



12 Chapter 1 Algebraic Reasoning 



City 


Population (2004) 


Yuma, AZ 


86,070 


Phoenix, AZ 


1,421,298 



47. Social Studies If the populations of the 
cities in the table double ever\' 10 years, 
what will their populations be in 2034? 

48. Critical Thinking Explain why 6^ ^ 3*^. 

49. Hobbies Malia is making a quilt with a pattern of rings. In the center ring, 
she uses four stars. In each of the next three rings, she uses three times as 
many stars as in the one before. How many stars does she use in the fourth 
ring? Write the answer using a power and find its value. 



Order each set of numbers from least to greatest. 



50. 29, 2l 6-, 16", 3^^ 
53. 2, 1^3^ 16", 



51. 4\33, 6", 5^ 10' 
54. 5-, 21, 11-, 13', 1^ 



52. 7-. 2\80. 10-, 1*^ 



55. 2^ 3^ 9, 5-, 8' 



56. Two weeks before Jackie's birthday her parents gave her one penny. They 
plan to double the amount of pennies she receives each day until her 
birthday. Use exponents to write a pattern that represents the number of 
pennies Jackie receives the first 5 days. Then use the pattern to predict 
how many pennies she will receive on her birthday. 

57. Life Science The cells of some kinds of bacteria divide 
ever\' 30 minutes. If you begin with a single cell, how 
many cells will there be after 1 hour? 2 hours? 3 hours? 

© 58. What's the Error? A student wrote 64 as 8 • 2. 
What was the student's error? 

'^ 59. Write About It Is 2^ greater than or less than 3^? 
Explain your answer. 

Bacteria divide by pincliing in 
two. This process is called binary 
fission. 




[^60. Challenge What is the length of the edge of a cube 
if its volume is 1,000 cubic meters? 



m 



Test Prep and Spiral Review 



61. Multiple Choice Wliat is the value of 4''? 

CS) 24 CD 1,024 (Cj 4,096 CD 16,384 

62. Multiple Choice Which of the following is NOT equal to 64? 

CD 6"* CD 4^ CE) 2^ CD 8^ 

63. Gridded Response Simplify 2^ + 3-. 



Simplify. (Previous course) 

64. 15 + 27 + 5 + 3 + 11 + 16 + 7 + 4 



65. 2 + 6 + 5 + 7+100+1+75 



Identify a possible pattern. Use the pattern to write the next three numbers. (Lesson 1-1) 
66. 100, 91, 82, 73, 64, . . . 67. 17, 19, 22, 26, 31, . . . 68. 2, 6, 18, 54, 162, . . . 



1-2 Exponents 13 







Vocabulary 

scientific notation 
Interactivities Online ► 




7.1.1 Read, write, compare and solve problems using whole numbers in scientific notation 

The distance from Venus to the Sun is 
greater than 100,000,000 kilometers. You 
can write this number as a power of ten 
by using a base often and an exponent. 

10- 10 • 10 • 10 • 10- 10- 10- 10 = 10" 
Power of ten ^ 



The table shows several powers often 



Power of 10 


Meaning 


Value 


10^ 


10 


10 


10^ 


10- 10 


100 


10^ 


10- 10- 10 


1,000 


10^ 


10- 10- 10- 10 


10,000 



You can find the product of a number and a power of ten by 
multiplying or by moving the decimal point of the number. For powers 
of ten v\nth positive exponents, move the decimal point to the right. 



EXAMPLE 



? 



A factor is a number 
that is multiplied by 
another number to 
get a product. 
See Skills Bank p. SB5. 



Multiplying by Powers of Ten 

Multiply 137 • 10^ 

A Method 1: Evaluate the power. 

137 • 10^^ = 137 • (10 • 10 • 10) 
= 137 • 1,000 
= 137,000 

Method 2: Use mental math. 
137 • 10^ = 137.000^ 

= 137,000^^3 places 



B 



Multiply 10 by itself 3 times. 
Multiply. 



Move the decimal point 3 places. 
(You will need to add 3 zeros.) 



Scientific notation is a kind of shorthand that can be used to write 
numbers. Numbers expressed in scientific notation are written as the 
product of two factors. 



14 Chapter 1 Algebraic Reasoning 



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Writing Math 



In scientific notation, 17,900,000 is wTitten as 



In scientific notation, 
it is customary to use 
a multiplication cross 
(x) instead of a dot. 



A number greater 
than or equal to 1 
but less than 10 -^ 



1.79 X 



^ssaaS* 



\ 



A power of W 



EXAMPLE [T] Writing Numbers in Scientific Notation 



Write 9,580,000 in scientific notation. 

9,580,000 = 9,580,000. iviove the decimal point to get 

a number between 7 and 10. 



= 9.58 X 10'^ 



The exponent is equal to the 
number of places the decimal 
point is moved. 



EXAMPLE [bJ Writing Numbers in Standard Form 

Pluto is about 3.7 x 10^ miles from the Sun. Write this distance in 
standard form. 



3.7 X 10' 



3.700000000 



= 3,700,000,000 
Pluto is about 3,700,000,000 miles from the Sun. 



Since the exponent is 9, move the 
decimal point 9 places to the right. 



EXAMPLE 



3 



Comparing Numbers in Scientific Notation 

Mercury is 9.17 x 10^ kilometers from Earth. Jupiter is 6.287 x 10^ 
kilometers from Earth. Which planet is closer to Earth? 

To compare numbers written in scientific notation, first compare the 
exponents. If the exponents are equal, then compare the decimal 
portion of the numbers. 

Mercur\': 9.17 x lO" km 

„ Compare the exponents. 

Jupiter: 6.287 x 10** km 

Notice that 7 < 8. So 9.17 x 10' < 6.287 x \0^. 

Mercury is closer to Earth than Jupiter. 



flH^^H^^^^^^^^^^^^^^^^Hi^^^^^Bli 


Think and Discuss 










1. Tell whether 15 x 


10^ is 


in 


scientific notation 


Explain. 


2. Compare 4 x 10 


and 3 


X 


10^ 


Explain how you know which | 


is greater. 










1 



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1-3 Scientific Notation 



15 




[•JllllK 

^ Homework Help Online go.hrw.com, 



keyword MMtllBcM 
Exercises 1-28, 29, 31, 33, 39, 41 



GUIDED PRACTICE 



See Example 1 Multiply. 

I 1. 15 • 10' 



2. 12 • 10** 



3. 208 • 10^ 



4. 113- 10^ 



See Example 2 Write each number in scientific notation. 

L 5. 3,600,000 6. 214,000 7. 8,000,000,000 8. 42,000 

See Example 3 9. A drop of water contains about 2.0 x 10"' molecules. Write this number in 
L standard form. 

See Example 4 10. Astronomy The diameter of Neptune is 4.9528 x 10' meters. The diameter 
L of Mars is 6.7868 x 10*^ meters. Which planet has the larger diameter? 



INDEPENDENT PRACTICE 



See Example 1 Multiply. 

11. 21 • 10" 

15. 268- 10-^ 



12. 8 • 10* 
16. 550 • 10' 



13. 25 • 10'^ 
17. 2,115 • 10' 



14. 40 • 10"* 
18. 70,030- 10' 



See Example 2 Write each number in scientific notation. 

19. 428,000 20. 1,610,000 21. 3,000,000,000 22. 60,100 

23. 52.000 24. 29.8 • 10' 25. 8,900,000 26. 500 • 10^ 

See Example 3 27. History Ancient Egyptians hammered gold into sheets so thin that it 
took 3.67 X 10^ sheets to make a pile 2.5 centimeters high. Write the 
number of sheets in standard form. 

See Example 4 28. Astronomy Mars is 7.83 x 10' kilometers from Earth. Venus is 
L 4.14 X 10' kilometers from Earth. Which planet is closer to Earth? 



Extra Practice 



See page EP2. 



PRACTICE AND PROBLEM SOLVING 



Find the missing number or numbers. 

29. 24,500 = 2.45 x 10 30. 16,800 = x lO' 

32. 280,000 = 2.8 x 10 33. 5.4 x lO" = 1 



31. =3.40 X 10" 

34. 60,000,000 = X 10 



Tell whether each number is written in scientific notation. Then order the 
numbers from least to greatest. 

35. 43.7 X 10'' 36. 1 x 10' 37. 2.9 x 10' 38. 305 x lO'^ 

39. Physical Science In a vacuum, light travels at a speed of about nine 
hundred and eighty million feet per second. Write this speed in scientific 
notation. 



16 Chapter 1 Algebraic Reasoning 



i* * 



Earth Science 



40. 



41, 



42. 



43. 



The earliest rocks native to 
Eartln formed during tlie 
Archean eon. Calculate the 
length of this eon. Write your 
answer in scientific notation. 

Dinosaurs lived during the 
Mesozoic era. Calculate 
the length of the Mesozoic 
era. Write your answer in 
scientific notation. 

Tropites were prehistoric 
marine animals whose fossil 
remains can be used to date 
the rock formations in which 
they are found. Such fossUs 
are known as index fossils. 
Tropites lived between 
2.08 X 10*^ and 2.30 x 10** years 
ago. During what geologic 
time period did they live? 

(jj Write About It Explain 
why scientific notation is 
especially useful in earth science. 



Geologic Time Scale 



Eon 



Phanerozoic 

(540 mya*-present) 




Era 



Cenozoic 

(65 mya-present) 



Mesozoic 

(248 mya-55 mya) 



Paleozoic 

(540 mya-248 mya) 





Period 



Quaternary (1.8 mya-present) 
Holocene epoch 

(1 1,000 yrs ago-present) 
Pleistocene epoch 
(l,8mya-11,000yrsago) 
Tertiary (65 mya -1.8 mya) 

Pliocene epoch {5.3 mya-1 .8 mya} 
Miocene epoch (23.8 mya-5.3 mya) 
Oligocene epoch (33.7 mya-23.8 mya) 
Eocene epoch (54.8 mya-33.7 mya) 
Paleocene epoch (65 mya-54.8 mya) 



Cretaceous (144 mya-65 mya) 
Jurassic (206 mya -144 mya) 
Triassic (248 mya-206 mya) 



Permian (290 mya-248 mya) 
Pennsylvanian (323 mya-290 mya) 
Mississippian (354 mya-323 mya) 
Devonian (41 7 mya-354 mya) 
Silurian (443 mya-417 mya) 
Ordovician (490 mya-443 mya) 
Cambrian (540 mya -490 mya) 



Proterozoic (2,500 mya-540 mya) 



Archean (3,800 mya-2,500 mya) 



Hadean (4,600 mya-3,800 mya) 



*mya = million years ago 



44. \^ Challenge We live in the Holocene epoch. Write the age of 
this epoch in scientific notation. 



Test Prep and Spiral Review 



45. Multiple Choice Kaylee wrote in her dinosaur report that the Jurassic 
period was 1.75 x lO'^ years ago. According to Kaylee's report, how many 
years ago was the Jurassic period? 

C£) 1,750,000 CX> 17,500,000 CD 175,000,000 CE) 17,500,000,000 

46. Multiple Choice What is 2,430,000 in scientific notation? 



CT) 243 x 10^ 



CS) 24.3 X 10" 



CS) 2 A3 X 10' 



CD 2.43 X 10'' 



Identify a possible pattern. Use the pattern to write the next three numbers. (Lesson 1-1) 
47. 19, 16, 13. 10, _, ,_, _„ . . . 48. 5, 15, 45, 135, 



J* :>i:^.iil \^^^t 



Write each number using an exponent and the given base. (Lesson 1 -2) 

49. 625, base 5 50. 512, base 8 51. 512, base 2 



1-3 Scientific Notation 17 



;. Scientific Notation witli 

LAB/\ a Calculator 



Use with Lesson 1-3 



Scientists often have to work with very large numbers. 
For example, the Andromeda Galaxy contains over 
200,000,000,000 stars. Scientific notation is a compact 
way of expressing large numbers such as this. 



£f. 



Learn It Online 

Lab Resources Online go.hrw.com, 



MSjOLablKGo, 



Activity 



O Show 200,000,000,000 in scientific notation. 

Enter 200,000,000,000 on your graphing 
calculator. Then press 



2 E 11 on the calculator display means 2 x lo", which 
is 200,000,000,000 in scientific notation. Your calculator 
automatically puts very large numbers into scientific notation. 

You can use the EE function to enter 2 x lo" directly into 

EE 

the calculator. Enter 2 X lo" by pressing 2 |m| 11 



200000000000 



>e11 



V ^WWWMMH 



O Simplify 2.31 x 10"* ^ 525. 

Enter 2.31 x 10^ into your calculator in scientific notation, 
and then divide by 525. To do this, 

EE 

press 2.31 HI 4 I fBi 525 



Your answer should be 44. 




Think and Discuss 



1. Explain how scientific notation and calculator notation are similar. 
What could the "E" possibly stand for in calculator notation? 



Try Til is 



Use the calculator to write each number in scientific notation. 

1. 6,500,000 2. 15,000,000 3. 360,000,000,000 

Simplify each expression, and express your answer in scientific notation. 

4. 8.4 X 10'^ ^ 300 5. 9 X lO'^ - 900 6. 2.5 x 10^ x 10 

7. 3 X 10- + 6000 8. 2.85 x lO" -H 95 9. 1.5 x 10' -H 150 



18 Chapter 1 Algebraic Reasoning 



u 



B 



1-4 



Order of Operations 



WV: 



TAJ Solve problems that involve multiplication and division with integers, fractions, decimals 
and combinations of the four operations. 

To assemble the correct product, directions 
must be followed in the correct order, 
hi mathematics, some tasks must also be 
done in a certain order. 



Vocabulary 

numerical expression 
order of operations 



A numerical expression is made up 
of numbers and operations. When 
simplifying a numerical expression, rules 
must be followed so that everyone gets the 
same answer. That is why mathematicians 
have agreed upon the order of operations . 




Interactivities Online ► 



ORDER OF OPERATIONS 



1. Perform operations within grouping symbols. 

2. Evaluate powers. 

3. Multiply and divide in order from left to right. 

4. Add and subtract in order from left to right. 



EXAMPLE Ml n Using the Order of Operations 

Simplify each expression. Use the order of operations to justify 
your answer. 



A 27 - 18 -H 6 
27- 18 H- 6 
27-3 
24 

B 36-18H-2-3 + 8 

36 - 18 H- 2 -3 + 8 

36-9-3 + 8 

36-27 + 8 

9 + 8 

17 

C 5 + 6^-10 

5 + 6" • 10 

5 + 36-10 

5 + 360 

365 



Divide. 
Subtract. 



Divide and multiply from left to right. 
Subtract and add from left to right. 



Evaluate the power. 

Multiply. 

Add. 



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1-4 Order of Operations 19 



EXAMPLE 



Using the Order of Operations with Grouping Symbols 

Simplify each expression. 

A 36 - (2 • 6) H- 3 

36 — (2 • 6) -j- 3 Perform the operation in parentheses. 



36 - 12 H- 3 

36-4 

32 



Divide. 
Subtract. 



When an expression 
has a set of grouping 
symbols within 
a second set of 
grouping symbols, 
begin with the 
innermost set. 



B [(4 + 12 -=- 4) - 2]^ 

[(4 + 12 -=- 4) - 21^ 

1(4 + 3) - 2)3 

17 - 2f 

5' 

125 



The parentheses are inside the bracl<ets, 
so perform the operations inside the 
parentheses first. 



EXAMPLE Q Career Application 



Maria works part-time in a law office, 
where she earns S20 per hour. The 
table shows the number of hours 
she worked last week. Simplify the 
expression (6 + 5 • 3) • 20 to find out 
how much money Maria earned 
last week. 



Day 


Hours 


Monday 


6 


Tuesday 


5 


Wednesday 


5 


Thursday 


5 



(6 + 5 -3) -20 

(6 + 15) • 20 

21 -20 

420 

Maria earned $420 last week. 



Perform the operations in parentheses. 

Add. 

Multiply. 



Think and Discuss 

1. Apply the order of operations to determine if the expressions 
3 + 4~ and (3 + 4)" have the same value. 

2. Give the correct order of operations for simplifying 
(5 + 3 • 20) H- 13 + 3^. 

3. Determine where grouping symbols should be inserted in the 
expression 3 + 9 - 4 • 2 so that its value is 13. 



20 Chapter 1 Algebraic Reasoning 



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I a:iQ j(3fe3£ 



„.„^ 



tJ 








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keyword MJlhiMBM 
Exercises1-18, 21, 23,27,29, 
33,35,37 



GUrDED PRACTICE 



See Example 1 Simplify each expression. Use the order of operations to justify your answer. 

3. 25 - 4- H- 8 

6. 32 + 6(4 - 2^) + 



1. 43 + 16 ^ 4 
See Example 2 4. 26 - (7 • 3) + 2 



2. 28 -4-3^6 + 4 
5. (3- + 11) ^5 



See Example 3 7. Career Caleb earns $10 per hour. He worked 4 hours on Monday, Wednesday, 
and Friday. He worked 8 hours on Tuesday and Thursday. Simplify the 
expression (3 • 4 + 2 • 8) • 10 to find out how much Caleb earned in all. 



L 



INDEPENg|NjLPRACTICE 



See Example 1 Simplify each expression. Use the order of operations to justify your answer. 

L 8. 3 + 7-5-1 9. 5-9-3 10. 3-2 + 6-2" 



See Example 2 11.(3-3-3)^ + 3 + 3 12. 2'' - (4 - 5 + 3) 



13. (3 + 3) + 3 • (3-^- 3) 



L 



14. 4^ + 8-2 



15. (8 - Zy - (8 - 1)- + 3 16. 9,234 + [3 - 3(1 + 8^] 



See Example 3 17. Consumer Math Maki paid a $14 basic fee plus $25 a day to rent a car. 

Simplif\' the expression 14 + 5 - 25 to find out how much it cost her to rent 
the car for 5 days. 

18. Consumer Math Enrico spent $20 per square yard for carpet and $35 for a 
carpet pad. Simplif\' the expression 35 + 20(12*^ + 9) to find out how much 
Enrico spent to carpet a 12 ft by 12 ft room. 



Extra Practice 



See page EP3. 



PRACTICE AND PROBLEM SOLVING 



Simplify each expression. 

19.90-36x2 20.16+14 + 2-7 21. 64 + 2" + 4 



22. (4.5 X 10-) + (6 + 3) 23. (9 - 4)" - 12 x 2 



24. [1 + (2 + 5)-] X 



Compare. Write <, >, or =. 

25. 8 - 3 - 2 8 - (3 - 2) 



26. (6 + 10) + 2 6+10 + 2 
28. 18 + 6-2 18 + (6 - 2) 
30. (18 - 14) + (2 + 2) 18-14 + 2 + 2 



27. 12 + 3 -4 12 + (3 - 4) 
29. [6(8-3) +2] 6(8-3) +2 

Critical Thinking Insert grouping symbols to make each statement true. 

31.4-8-3=20 32.5 + 9-3+2 = 8 33. 12 - 2" + 5 = 20 

34. 4-2 + 6= 32 35. 4 + 6-3+7=1 36. 9-8-6 + 3 = 6 

37. Bertha earned $8.00 per hour for 4 hours babysitfing and $10.00 per hour 
for 5 hours painting a room. Simplify the expression 8 - 4 + 10 - 5 to find 
out how much Bertha earned in all. 



1-4 Order of Operations 21 



38. Consumer Math Mike bought a painting for $512. He sold it at an antique 
auction for 4 times the amount that he paid for it, and then he purchased 
another painting with half of the profit that he made. Simplify the expression 
(512 • 4 - 512) -^ 2 to find how much Mike paid for the second painting. 

39. Multi-Step Anelise bought four shirts 
and two pairs of jeans. She paid $6 in 
sales tax. 

a. Write an expression that shows how 
much she spent on shirts. 

b. Write an expression that shows how 
much she spent on jeans. 

c. Write and evaluate an expression to 
show how much she spent on clothes, 
including sales tax. 

P 40. Choose a Strategy There are four children in a family. The sum of the 

squares of the ages of the three youngest children equals the square of the 
age of the oldest child. How old are the children? 




CA) 1,4,8,9 



(X' 1.3,6, 12 



CD 4,5,8, 10 



CE) 2,3,8, 16 



41. Write About It Describe the order in which you would perform the 
operations to find the correct value of [(2 + 4)- - 2 • 3] -^ 6. 

^ 42. Challenge Use the numbers 3, 5, 6, 2, 54, and 5 in that order to write an 
expression that has a value of 100. 



r 



Test Prep and Spiral Review 



43. Multiple Choice Wliich operation should be performed first to simplify 
the expression 18-1-9-^3 + 8? 



CS) Addition C15 Subtraction <X) Multiplication CE> Division 

44. Multiple Choice Which expression does NOT simplify to 81? 

CD 9 • (4 + 5) CG> 7 + 16 • 4 + 10 CH:' 3 • 25 + 2 CD lO' - 4 • 5 

45. Multiple Choice Quinton bought 2 pairs of jeans for $30 each and 3 pairs 
of socks for $5 each. Which expression can be simplified to determine the 
total amount Quinton paid for the jeans and socks? 



+ 1 



CS) 2 -SOO + 5) 



CD (2 + 3) • (30 + 5) 'X) 2 • (30 + 5) • 3 CD 2 • 30 + 3 • 5 



Find each value. (Lesson 1-2) 
46. 8'' 47. 9^ 

Multiply. (Lesson 1-3) 

51. 612 • 10^ 52. 43.8 • 10^^ 



48. 4^ 



53. 590 • 10-' 



49. 3^ 



54. 3.1 • 10' 



50. 7' 



55. 1.91 • 10- 



I 



22 Chapter 1 Algebraic Reasoning 



Explore Order of 



Use with Lesson 1-4 



REMEMBER 

The order of operations 

1. Perform operations within grouping symbols. 

2. Evaluate powers. 

3. Multiply and divide in order from left to right. 

4. Add and subtract in order from left to right. 



<c?. 



Learn It Online 

Lab Resources Online go.hrw.com. 



Many calculators have an 



key that allows you to find the square of 



a number. On calculators that do not have this key, or to use exponents 

other than 2, you can use the caret key, WSM - 

For example, to evaluate 3", press 3 MSMi 5, and then press ; 



Activity 



O Simplify 4 • 2^ using paper and pencil. Then check your answer with 
a calculator. 



First simplify the expression using paper and pencil: 
4 . 2^ = 4 • 8 = 32. 

Then simplify 4 • 2"^ using your calculator. 

Notice that the calculator automatically evaluates the 
power first. If you want to perform the multiplication 
first, you must put that operation inside parentheses. 



O Use a calculator to simplify — — ^ 



Thinic and Discuss 



1. Is 2 + 5 • 4 ' + 4- equivalent to (2 + 5 • 4'^) + 4"? Explain. 



<4*2)^3 



f 



32 
512 



< 2+5*4) ^3/4-^2 

665.5 



Try This 



Simplify each expression with pencil and paper. Check your answers with 
a calculator. 



1. 3-2' + 5 



2. 3 • (2^ + 5) 



3. (3 • 2)' 



4. 3 • 2- 



5. 2 



Use a calculator to simplify each expression. Round your answers to the 
nearest hundredth. 



6. (2.1 + 5.6 •4-^) ^6^ 



7. [(2.1 + 5.6) -4^] -=- 6^ 



8. [(8.6- 1.5) -^ 21 H- 5 



(3- 2) 



3i 



1-4 Technology Lab 23 






^ 



Vocabulary 

Commutative 
Property 

Associative 
Property 

Identity 
Property 

Distributive 
Property 



,,,^c!ijijjj jJjjJj 



For more on 
properties, see Skills 
Bank p. SB2. 



■ v'«*VJ«>™*.'^'NC 



In Lesson 1-4 you learned how to use the order of operations to 
simplify numerical expressions. The follovdng properties of numbers 
are also useful when you simplify expressions. 





Commutative Property 




Words 


Numbers 


Algebra 


You can add numbers in 
any order and multiply 
numbers in any order. 


3+8=8+3 
5 • 7 = 7 • 5 


a + b = b + a 
ab = ba 



Associative Property 


Words 


Numbers 


Algebra 


When you add or 






multiply, you can group 


(4 + 5) + 1 =4 + (5 + 1) 


{a + b) + c = a + {b + c) 


the numbers together 


(9 • 2) • 6 = 9 • (2 • 6) 


{a • b) ' c = a ■ (b- c) 


in any combination. 







Identity Property 


Words 


Numbers 


Algebra 


The sum of and any 






number is the number. 


4 + = 4 


a + = a 


The product of 1 and any 


8-1=8 


a • 1 = a 


number is the number. 







EXAMPLE |1| Identifying Properties of Addition and Multiplication 

Tell which property is represented. 

A 2 + (7 + 8) = (2 + 7) + 8 

2 + (7 + 8) = (2 + 7) + 8 The numbers are regrouped. 
Associative Property 



B 25 • I = 25 

25 • 1 = 25 
Identity Property 



One of the factors is 7. 



C xy = yx 

xy = yx The order of the variables is switched. 

Commutative Property 



24 Chapter 1 Algebraic Reasoning 



l/jdai) Lesson Tutorials Onlins mv.hrw.com 



You can use properties and mental math to rearrange or regroup 
numbers into combinations that are easier to work with. 



EXAMPLE [2] Using Properties to Simplify Expressions 

Simplify each expression. Justify each step. 



12 + 19 + 18 

12 + 19 + 18 = 19 + 12 + 18 

= 19 + (12 + 18) 
= 19 + 30 
= 49 

25 • 13 • 4 

25 • 13 • 4 = 25 • 4 • 13 

= (25- 4) • 13 
= 100 • 13 
= 1,300 



Commutative Property 
Associative Property 
Add. 



Commutative Property 
Associative Property 
Multiply. 



Multiplication can be 
written as a(b + c) or 
a- (b + c). 



You can use the Distributive Property' to multiply numbers mentally by 
breaking apart one of the numbers and writing it as a sum or difference. 



Distributive Property 


Numbers 


6(9 + 14) = 6 • 9 + 6 • 14 


8(5 - 2) = 8 • 5 - 8 • 2 


Algebra 


a{b + c) = ab + ac 


a{b - c) = ab - ac 



iXAMPLE [3 J Using the Distributive Property to Multiply Mentally 

Use the Distributive Property to find 7(29). 



Method 1 Method 2 

7(29) = 7(20 + 9) Rewrite 29. 7(29) = 7(30 - 1) 

= (7 • 20)+ (7 • 9) Use the Distributive = (7 • 30) - (7 • 1) 

Property. 
= 140 + 63 Multiply. =210-7 

= 203 Simplify. = 203 



Think and Discuss 

1. Describe two different ways to simplify the expression 7 • (3 + 9). 

2. Explain how the Distributive Property can help you find 6 • 102 
using mental math. 



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1-5 Properties of Numbers 25 



1-5 



.iiij'3}^3^ 



HomeworkHelpOnlinego.hrw.com, j 



keyword ■BHIiBBiM ® 
Exercises1-36, 41,47,49,51,53 



GUIDED PRACTICE 



See Example 1 Tell which property is represented. 

1. 1+ (6 + 7) = (1 + 6) +7 2. 1-10=10 



4. 6 + = 6 



3. 3 • 5 = 5 • 3 

5. 4 • (4 ■ 2 ) = (4 • 4) • 2 6. x + y = y + x 



See Example 2 SimpUfy each expression. Justify each step. 

7. 8 + 23 + 2 8. 2 • (17 • 5) 



10. 17 + 29 + 3 



11. 16 + (17 + 14) 



See Example 3 Use the Distributive Property to find each product. 
13. 2(19) 14. 5(31) 



16. (13)6 



17. 8(26) 



9. (25- 11) -4 

12. 5 • 19 • 20 

15. (22)2 

18. (34)6 



INDEPENDENT PRACTICE 



See Example 1 Tell which property is represented. 

19. 1 + = 1 20. xyz = X- qc) 

22. 11 +25 = 25 + 11 23. 7- 1 = 7 

See Example 2 Simplify each expression. Justify each step. 

25. 50 • 16 • 2 26. 9 + 34 + 1 



28. 27 + 28 + 3 



29. 20 + (63 + 80) 



See Example 3 Use the Distributive Property to find each product. 
31. 9(15) 32. (14)5 



34. 10(42) 



35. (23)4 



21. 9 + (9 + 0) = (9 + 9) + 

24. 16 • 4 = 4 • 16 

27. 4 • (25 • 9) 

30. 25 + 17 + 75 

33. 3(58) 

36. (16)5 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EPS. 



Write an example of each property using whole numbers. 

37. Commutative Property 38. Identity Property 

39. Associative Property 40. Distributive Property 



41. Architecture The figure siiows the floor 
plan for a studio loft. To find the area of the 
loft, the architect multiplies the length and 
the width: (14 + 8) • 10. Use the Distributive 
Property to find the area of the loft. 





• • 

• • 



T 

10ft 

1 



14 ft- 



■8ft^ 



Simplify each expression. Justify each step. 

42. 32 + 26 + 43 43. 50 • 45 • Z" 44. 



5+16 + 5" 45. 35 • 25 • 20 



26 Chapter 1 Algebraic Reasoning 



Complete each- equation. Then tell which property is represented. 



46. 5 + 16 = 16 + ... 
48. • (4 + 7) = 3 • 4 + 3 • 7 
50. 2 • • 9 = (2 • 13) • 9 
52. 2 • (6+ 1) = 2 • +2-1 



47. 15 • 1 =s§ 

49. 20 + =20 

51. 8 + ( + 4) = (8 + 8) +4 

53. (12 - 9) • = 12 • 2 - 9 • 2 



54. Sports Janice wants to know the total 
number of games won by the Denver 
Nuggets basketball team over the three 
seasons shov«i in the table. What 
expression should she simplify? Explain 
how she can use mental math and the 
properties of this lesson to simplify 
the expression. 

© 55. What's the Error? A student simplified 
the expression 6 • (9 + 12) as shown. 
What is the student's error? 

i ■ 56. Write About It Do you think there is a 

Commutative Property of Subtraction? Give 
an example to justify your answer. 



Denver Nuggets 


Season 


Won 


Lost 


2001-02 


27 


55 


2002-03 


17 


65 


2003-04 


43 


39 



6 • (9 + 1Z) = 6-9 + IZ 
= 5H + IZ 
= 66 




f0 57. Challenge Use the Distributive Property' to simplify i • (36 + ^). 



i 



Test Prep and Spiral Review 



58. Multiple Choice Which is an example of the Associative Property? 
C£) 4 + = 4 (X) 5 + 7 = 7 + 5 

Ci:)9 + 8 + 2 = 9+(8 + 2) CDS- (12 + 3) = 5- 12 + 5-3 



59. Multiple Choice Which property is 2 • (3 + 7) = (2 • 3) + (2 • 7) an 
example of? 



CE> Associative 



:Sj Commutative (S) Distributive 



(X-> Identity 



60. Short Response Show how to use the Distributive Property to simplify the 
expression 8(27). 

Write each number using an exponent and the given base. (Lesson 1 2) 

61. 36, base 6 62. 64, base 2 63. 9, base 3 64. 1,000, base 10 



Simplify each expression. (Lesson 1-4) 
65. 25 + 5 - (6- - 7) 66. 3^ - (6 + 3) 



67. (4-^ + 5) ^ 7 



68. (5-3)-^ (3^-7) 



1-5 Properties of Numbers 



27 



CHAPTER 




Ready To Go On? 



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^^ ResourcesOnlinego.hrw.com, 

1!HW!B | M s 1 n RTfi0 1 a kgo; 



SECTION 1A 



Quiz for Lessons 1-1 Through 1-5 

^f 1-1 ] Numbers and Patterns 



Identify a possible pattern. Use the pattern to write the next three numbers 
or figures. 



1. 

4. 



8, 


15, 


22,29, . . 






2. 


7£ 


, 66. 53, 40, 










C) 






( ) 






^ 























3. 21,36,51,66,... 



5. Make a table that shows the number of squares in each figure. Then 
make a conjecture about the number of squares in the fifth figure of the 
pattern. Complete tlie table, and use drawings to justify your answer. 



Figure 1 



Figure 2 



Figure 3 



er 







1-2 ] Exponents 

Find each vakie. 

6. 8' 7. 7^ 8. 4'^ 9. 6' 

10. The number of bacteria in a sample doubles every hour. How many bacteria cells 
will there be after 8 hours if there is one cell at the beginning? Write your answer 
as a power. 

1-3 ] Scientific Notation 
Multiply. 



11. 456- 10' 12. 9.3 • 10- 

Write each number in scientific notation. 
14. 8,400,000 15. 521,000,000 



13. 0.36 • 10** 



16. 29,000 



17. In May 2005, the world's population was over 6,446,000,000 and was increasing 
by 140 people each minute! Write this population in scientific notation. 











1-4 ] Order of Operations 

Simplify each expression. 

18. 8- 14 -^ (9 -2) 19.54-6-3 + 4" 20. 4 - 24 H- 2'' 

1-5 j Properties of Numbers 
Simplify each expression. Justify each step. 

22.29 + 50 + 21 23. 5 • 18 • 20 24.34 + 62 + 36 



21. 4(3 + 2)- -9 



25. 3 • 11 -20 



28 Chapter 7 Algebraic Reasoning 



v^ 





Focus on Problem Solving 





Solve 

• Choose an operation: multiplication or division 

To solve a word problem, you must determine which mathematical 
operation you can use to find the answer. One way of doing this is 
to determine the action the problem is asking you to take. If you 
are putting equal parts together, then you need to multiply. If you 
are separating something into equal parts, then you need to divide. 




Decide what action each problem is asking you to take, and tell 
whether you must multiply or divide. Then explain your decision. 



O Judy plays the flute in the band. She 
practices for 3 hours every week, ludy 
practices only half as long as Angle, who 
plays the clarinet. How long does Angle 
practice playing the clarinet each week? 



Each year, members of the band and choir 
are invited to join the bell ensemble for the 
winter performance. There are 18 bells in the 
bell ensemble. This year, each student has 3 
bells to play. How many students are in the 
bell ensemble this year? 




For every percussion instrument in the 
band, there are 4 wind instruments. If 
there are 48 wind instruments in the band, 
how many percussion instruments are 
there? 

O A group of 4 people singing together in 
harmony is called a quartet. At a state 
competition for high school choir students, 
7 quartets from different schools competed. 
How many students competed in the quartet 
competition? j^ ^:.;^ 




Focus on Problem Solving 




1-6 



1 



Variabli 
Expressions 



^aia^-. ^'~1*JB' 







7.2.3 Evaluate numerical expressions and simplify algebraic expressions involving 
rational and irrational numbers. 

Harrison Ford was born in 1942. 
You can find out what year Harrison 
turned 18 by adding 18 to the year 
he was born. 



Vocabulary 

variable 

constant 

algebraic expression 

evaluate 



EXAMPLE 



1942 + 18 

In algebra, letters are often used 
to represent numbers. You can 
use a letter such as a to 
represent Harrison Ford's age. 
When he turns a years old, 
the year will be 

1942 + a. 

The letter a has a value that can 
change, or vary. When a letter 
represents a number that can vary, 
it is called a variable . The year 
1942 is a constant because the 
number cannot change. 

An algebraic expression consists 
of one or more variables. It usually 
contains constants and operations. 
For example, 1942 + n is an 
algebraic expression for the year 
Harrison Ford turns a certain age. 

To evaluate an algebraic expression, substitute a number for the 
variable. 




Age 


Year born + age = year at age 


18 


1942 + 18 


1960 


25 


1942 + 25 


1967 


36 
63 


1942 + 36 


1978 


1942 + 63 


2005 


a 


1942 + a 





1? 



Evaluating Algebraic Expressions 

Evaluate n + 7 for each value of n. 



Interactivities Online ► 



A « = 3 n + 1 
3+7 
10 



Substitute 3 for n. 
Add. 



B n= 5 n + 7 

5+7 Substitute 5 for n. 

12 Add. 



30 Chapter 7 Algebraic Reasoning 



l/jilau Lesson Tutorials Onlins mv.hrw.com 



EXAMPLE 



Multiplication and division 
of variables can be written in 
several ways, as shov«i in the 
table. 

When evaluating expressions, 
use the order of operations. 



Multiplication 


Division 


It 1 • t 
lit) 7 X t 


q 

2 q/2 
q-2 


ab a • b 
a{b) a X b 


s 

r sir 
s - r 



[ 2 1 Evaluating Algebraic Expressions Involving Order of 
!f Operations 

Evaluate each expression for the given value of the variable. 



A 3.1: - 2 for X = 5 

3(5) - 2 Substitute 5 for x. 

15-2 Multiply. 

13 Subtract 

B H ^ 2 + » for /; = 4 

4 -=- 2 + 4 Substitute 4 for n. 

2 + 4 Divide. 

6 Add. 

C 6y- + 2yfory = 2 

6(2)^ + 2(2) Substitute 2 for y. 

6 (4) + 2 (2) Evaluate the power. 

24 + 4 Multiply. 

28 Add. 



EXAMPLE fs J Evaluating Algebraic Expressions with Two Variables 



Evaluate | + 2m for n = 3 and m - A. 



+ 2iu 
^ + 2(4) 

1 + 8 
9 



Substitute 3 for n and 4 for m. 

Divide and multiply from left to right. 
Add. 



Think and Discuss 

1. Write each expression another way. a. 12a' b. j, c. ^ 

2. Explain the difference between a variable and a constant. 



3xy 



VJiJaii Lesson Tutorials Online mv.hrw.com 



1-6 Variables and Algebraic Expressions 31 



GUIDED PRACTICE 



See Example 1 Evaluate n + 9 for each value of n. 

L. 1. n = 3 2. 11 = 2 



3. n = II 



See Example 2 Evaluate each expression for the given value of the variable. 

4. 2.V - 3 for A- = 4 5. ii ^ 3 + n for /; = 6 6. S.y- + 3.v for y = 2 

See Example 3 Evaluate each expression for the given values of the variables. 



L 7. 77 + 3;» for /; = 2 and m = 5 



8. 5a-3b + 5 for rt = 4 and Z? = 3 



INDEPENDENJ^BBACTICE 



See Example 1 Evaluate /; + 5 for each value of ». 

9. ;/ = 17 10. /; = 9 



11. ;? = 



See Example 2 Evaluate each expression for the given value of the variable. 

I 12. 5j'- 1 fori' = 3 13. 10^- 9 for Z; = 2 14. p h- 7 + p for p = 14 

i 

; 15. ;; H- 5 + /; for ii = 20 16. 3.r + 2.v for x = 10 17. 3r" - 5r for r = 3 

See Example 3 Evaluate each expression for the given values of the variables. 



Extra Practice 



18. ^ + 7»; for n = 6 and »? = 4 



20. 9 - ^ + 20.V for .v = 4 and y = 5 



PRACTICE AND PROBLEM SOLVING 



19. 7p - 2r + 3 for p = 6 and T = 2 
21. /•- + I5k for ;• = 15 and k = 5 



See page EP3 



Evaluate each expression for the given values of the variables. 

23. Ad" - 3d ioT d = 2 



22. 20.V- 10for.v = 4 

24. 22p H- 1 1 + p for p = 3 

26. ^ + 7li for k = 8 and /; = 2 

28. 3f H- 3 + fforf = 13 

30. 108 - 12/+ /forj = 9 



25. q + q- + q ^ 2 for q = 4: 

27. />3 +/for/ = 18 

29. 9 + 3p- 5r + 3 for p = 2 and r = 1 

31. 3nr^ + ^ for m = 2 and v = 35 

5 



32. The expression 60iu gives the number of seconds in /// minutes. Evaluate 
60ni for ni = 7. How many seconds are there in 7 minutes? 

33. Money Betsy has /; quarters. You can use the expression 0.25/; to find the 
total value of her coins in dollars. Wliat is the value of 18 quarters? 

34. Physical Science A color TV has a power rating of 200 watts. The 
expression 200r gives the power used by t color TV sets. Evaluate 200f for 
t = 13. How much power is used by 13 TV sets? 



32 Chapter 1 Algebraic Reasoning 



35. Physical Science The expression 1.8r + 32 can be used to convert 
a temperature in degrees Celsius f to degrees Fahrenheit. What is the 
temperature in degrees Fahrenheit if the temperature is 30 °C? 

36. Physical Science The graph shows the changes of state for water. 

a. What is the boiling point of water in degrees Celsius? 

b. Use the expression 1.8c + 32 to find the boiling point of water in 
degrees Fahrenheit. 



50 °C 

25 °C 

O'C 

-25 'C 



Changes of State for Water 



ISCC 
125 ^T 
100 °C 
75 °C -: 






Melting point 

\ Ice and water 




Boiling point Steam 

Water and steam ^^^(gas) 



solid) 



Energy added 






^ 37. What's the Error? A student was asked to identify the variable in the 

expression 72.v + 8. The student answered 72.v. Wliat was the student's error? 

,'-1, 

V 38. Write About It Explain why letters such as .v, p, and n used in algebraic 

expressions are called variables. Use examples to illustrate your response. 
39. Challenge Evaluate the expression " _ '^ for x = 6 and y = 8. 



i 



Test Prep and Spiral Review 



40. Multiple Choice Which expression does NOT equal 15? 

(S) 3r for r = 5 CD 3 + r for r = 12 CD r h- 3 for r = 60 ^ r - 10 for r = 25 

41 . Multiple Choice A group of 1 1 students go rock climbing at a local 
gym. It costs $12 per student plus $4 for each shoe rental. If only 8 
students rent shoes, what is the total cost for the group to go climbing? 
Use the expression 12.v + 4v, where x represents the total number of 
students and y represents the number of students who rent shoes. 



CD $132 



CS) $140 



CH) $164 



Write each number in scientific notation. (Lesson 1-3) 

42. 102.45 43. 62,100,000 44. 769,000 

Use the Distributive Property to find each product. (Lesson 1 -5) 

46. 5(16) 47. (17)4 48. 7(23) 



CD $176 



45. 800,000 



49. (29)3 



1-6 Variables and Algebraic Expressions 33 



& 




7.2.1 Use variables and appropriate operations to write an expression, equation or 
inequality that represents a verbal description 

Although they are closely related, a 
Great Dane weighs about 40 times 
as much as a Chihuahua. An 
expression for the weight of the 
Great Dane could be 40f, where c 
is the weight of the Chihuahua. 

When solving real-world problems, 
you will need to translate words, or 
verbal expressions, into algebraic 
expressions. 



Interactivities Online ► 




Operation 


Verbal Expressions 


Algebraic Expression 


+ 




' add 3 to a number 

• a number plus 3 

• the sum of a number and 3 

• 3 more than a number 

' a number increased by 3 


n + 3 


- 




' subtract 12 from a number 

« a number minus 12 

' the difference of a number and 12 

' 12 less than a number 

' a number decreased by 12 

' take away 12 from a number 

> a number less 12 


X - 12 


H 




' 2 times a number 

• 2 multiplied by a number 

> the product of 2 and a number 


2m or 2 • m 


• 

• 




• 6 divided into a number 

> a number divided by 6 

' the quotient of a number and 6 


a H- 6 or 1 

b 



EXAMPLE [lj Translating Verbal Expressions into Algebraic Expressions 

Write each phrase as an algebraic expression. 



A the product of 20 and t B 

product means "mtiltiply" 

20t 



24 less than a number 
less than means "subtract from" 
/; - 24 



34 Chapter 1 Algebraic Reasoning 



yidau Lesson Tutorials OnlinE mv.hrw.com 



Write each phrase as an algebraic expression. 

C 4 times the sum of a number and 2 
4 times the sum of a number and 2 
4 • » + 2 

4(77 + 2) 

D the sum of 4 times a number and 2 
the sum of 4 times a number and 2 

4-77 +2 

477 +2 



When solving real-world problems, you may need to determine the 
action to know which operation to use. 



Action 


Operation 


Put parts together 


Add 


Put equal parts together 


Multiply 


Find how much more or less 


Subtract 


Separate into equal parts 


Divide 



EXAMPLE [Vj Translating Real-World Problems into Algebraic Express 



ions 



Jed reads p pages each day of a 200-page book. Write an algebraic 
expression for how many days it will take Jed to read the book. 

You need to separate the total number of pages 777fo equal parts. 
This involves division. 

total number of pages _ 2OO 
pages read each day P 

To rent a certain car for a day costs $84 plus $0.29 for every mile 
the car is driven. Write an algebraic expression to show how 
much it costs to rent the car for a day. 

The cost includes $0.29 per mile. Use 777 for the number of miles. 
Multiply to put equal parts togetlier: 0.29777 

In addition to the fee per mile, tlie cost includes a flat fee of $84. 
Add to put parts togetlier: 84 -I- 0.29777 



Think and Discuss 

1. Write three different verbal expressions that can be represented 
by2-y. 

2. Explain how you would determine which operation to use to find 
the number of chairs in 6 rows of 100 chairs each. 



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1-7 Translating Words into Math 35 




:i3.^s:^^3^ 



keyword MteMMM ® 
Exercises 1-13, 15, 17, 19, 21, 
23,25,31 



GUIDED PRACTICE 



See Example 1 Write each phrase as an algebraic expression. 



See Example 2 



1 . the product of 7 and p 
3. 12 divided into a number 



2. 3 less than a number 

4. 3 times the sum of a number and 5 



5. Carly spends $5 for u notebooks. Write an algebraic expression to 
represent the cost of one notebook. 

6. A company charges $46 for cable TV installation and $21 per month for 
basic cable service. Write an algebraic expression to represent the total 
cost of /;; months of basic cable service, including installation. 



INDEPENDENT PRACTICE 



See Example 1 Write each phrase as an algebraic expression. 

7. the sum of 5 and a number 8. 2 less than a number 

9. the quotient of a number and 8 10. 9 times a number 

11. 10 less than the product of a number and 3 

See Example 2 12. Video Express sells used tapes. Marta bought c tapes for $45. Write an 
algebraic expression for the average cost of each tape. 

13. A 5-foot pine tree was planted and grew 2 feet each year. Write an 
algebraic expression for the height of the tree after t years. 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EP3. 



Write each phrase as an algebraic expression. 

14. /» plus the product of 6 and /i 15. the quotient of 23 and » minus f 

16. 14 less than the quantity /c times 6 17. 2 times the sum of y and 5 

18. the quotient of 100 and the quantity 6 plus w 

19. 35 multiplied by the quantity /• less 45 

20. Multi-Step An ice machine can produce 17 pounds of ice in one hour. 

a. Write an algebraic expression to describe the number of pounds of ice 
produced in /; hours. 

b. How many pounds of ice can the machine produce in 4 hours? 

21. Career Karen earns $65,000 a year as an optometrist. She received a 
bonus of b dollars last year and expects to get double that amount as a 
bonus this year. Write an algebraic expression to show the total amount 
Karen expects to earn this year. 



36 Chapter 7 Algebraic Reasoning 



. Q* • . Write a verbal expression for each algebraic expression. 

\mni^ 22. /7 + 3 23. 90 -r y 24. 5 - 405 



Life Scienciiiii^ 



26. 5(^7-8) 



27. 4p- 10 



28. (r+ 1) ^ 14 



25. 
29. 



I6r 




15 



+ 3 



■^2 Life Science Tiny and harmless, follicle mites live in our eyebrows and 
^ eyelashes. They are relatives of spiders and like spiders, they have eight 
legs. Write an algebraic expression for the number of legs in m mites. 



Reddish-brown 
spots appear on 
the leaves and 
fruit of plants 
infested by rust 
mites. 



Nutrition The table shows the estimated 
number of grams of carbohydrates 
commonly found in various types of foods. 

31 . Write an algebraic expression for the 
number of grams of carbohydrates in 
y pieces of fruit and 1 cup of skim milk. 

32. How many grams of carbohydrates are 
in a sandwich made from t ounces of 
lean meat and 2 slices of bread? 



Food 


Carbohydrates 


1 c skim milk 


12 g 


1 piece of fruit 


15g 


1 slice of bread 


15g 


1 oz lean meat 


Og 




m 



e> 



33. What's the Question? Al has twice 
as many baseball cards as Frank and 
four times as many football cards as loe. The expression 2.v + 4y can be 
used to show the total number of baseball and football cards Al has. If the 
answer is y, then what is the question? 

34. Write About It If you are asked to compare two numbers, what two 
operations might you use? Wliy? 

35. Challenge In 2006, one U.S. dollar was equivalent, on average, to 
$1,134 in Canadian dollars. Write an algebraic expression for the 
number of U.S. dollars you could get for /; Canadian dollars. 



m 



Test Prep and Spiral Review 



36. Multiple Choice Which verbal expression does NOT represent 9 - a? 
CS) X less than nine Cc;' subtract x from nine 



CX> X decreased by nine 



CS) the difference of nine and x 



37. Short Response A room at the Oak Creek Inn costs $104 per night for 
two people. There is a $19 charge for each extra person. Write an algebraic 
expression that shows the cost per night for a family of four staying at the 
inn. Then evaluate your expression for 3 nights. 



Simplify each expression. (Lesson 1-4) 

38. 6 + 4 H- 2 39. 9 • 1 - 4 40. 5^ - 3 

42. Evaluate b - a- for a = 2 and i) = 9. (Lesson 1-6) 



41. 24-^3 + 3 



3 



1-7 Translating Words into Matli 37 



1-8 



Simplifying Algebr 
Expressions 







7.2.3 Evaluate numerical expressions and simplify algebraic expressions involving 
rational and irrational numbers. 

Individual skits at the talent show can 
last up to X minutes each, and group 
skits can last up to y minutes each. 
Intermission will be 15 minutes. The 
expression 7x + 9y + 15 represents 
the maximum length of the talent 
show if 7 individuals and 9 groups 
perform. 



Vocabulary 

term 
coefficient 



Caution! 



////// 

A variable by itself, 
such as y, has a 
coefficient of 1. 
So y = ly. 




In the expression 7.y + 9y + 15, 7.y, 9_y, 
and 15 are terms. A term can be a 
number, a variable, or a product of 
numbers and variables. Terms in an 
expression are separated by plus or 
minus signs. 

In the term 7.v, 7 is called the 
coefficient. A coefficient is a 
number that is multiplied by a 
variable in an algebraic expression. 

Like terms are terms with the same variables raised to the same 
exponents. The coefficients do not have to be the same. Constants, 
like 5, \, and 3.2, are also like terms. 



Coefficient 




Variable 



Like Terms 


3xand2x w and ^ 5 and 1.8 


Unlike Terms 


5x'' and 2x 
The exponents 
are different. 


6a and 6fa 

Ttie variables 
are different. 


3.2 and n 

Only one terni 
contains a variable. 



EXAMPLE [1J Identifying Like Terms 

Identify like terms in the list. 

So I 3y- It X- 4z k 



HMH 



Use different shapes 
or colors to indicate 
sets of like terms. 



Look for like variables with like powers. 




3y 



7t 



x^ 



4z 



Like terms: 5a and |a |, 7t, and 2t 



4.5y- 2t ffl 




4.5y 



2f 



3y^ and 4.5y^ 



38 Cliapter 7 Algebraic Reasoning 



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To simplify an algebraic expression that contains like terms, combine 
the terms. Combining like terms is like grouping similar objects. 









^ -.■mi!?.-."-. j«.iiiircv 






X 


X 




X X 




X X X X 






+ 


fl. 


= 


^ 


X 


X 




t.jc:;, ■ X X 




X X X X 


.. 






'- — _:^'j '- 




.- -^.__ - ■~^^._.. 



4x 



+ 



5jc 



9x 



To combine like terms that have variables, add or subtract the coefficients. 



EXAMPLE [2] Simplifying Algebraic Expressions 



EXAMPLE 



CS 



To find the perimeter 
of a figure, add the 
lengths of the sides. 



Simplify. Justify your steps using the Commutative, Associative, 
and Distributive Properties when necessary. 



7x and 2x are like terms. 
Add the coefficients. 



Identify like terms. 
Commutative Property 
Associative Property 
Add or subtract the coefficients. 



A 


7a- + 2x 




7x + 2x 




9x 


1 

B 


Sx^ + 33/ + 7x^ - 2y - 4x2 




5x^ + 3y + 7x-^ - 2y - 4x- 




5x^ + 7x^ + 3,y - 2y - 4x- 




(5x^ + 7x-^) + (3y - 2y) - 4.x- 




\2.x^ +y-4x- 




C 2(fl + 2a'') + 2b 
2(a + 2a") + 2h 
2a + 4a- + 2b 
There are no like terms to combine. 



Geometry Application 

Write an expression for the perimeter 
of the rectangle. Then simplify 
the expression. 



Distributive Property 



b + h + b + h 
{b + b) + (/z + h) 
2b + 2h 



Write an expression using the side lengths. 
Identify and group like terms. 
Add the coefficients. 



Think and Discuss 

1. Explain whether 5x, 5x", and 5x' are like terms. 

2. Explain how you know when an expression cannot be simplified. 



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1-8 Simplifying Algebraic Expressions 39 



1 



(iik^^J>iUiKl£liii£>^i^^ 





Homework Help Online go.hrw.com, 



keyword ■BEiMBJ ® 
Exercises 1-17, 19, 21, 23, 25, 29 



See Example 1 Identify like terms in each list. 



L ^. 6b 5.V- 4x' 



x' 2e 



2. 12a^ 4x^ b 4a- 3.5x^ |t» 

b 



See Example 2 Simplify. Justify your steps using the Commutative, Associative, and 
J Distributive Properties when necessary. 

L 3. 5.V + 3a- 4. 6a" - a^ + 16 5. 4o- + 5a + I4b 



See Example 3 6. Geometry Write an expression for the perimeter 
L of the rectangle. Then simplify the expression. 



Sn 



66 



5n 



INDEPENDENT PRACTICE 



See Example 1 Identify like terms in each list. 

7. 2b b^' b x' 3^'' 2x- 



L 9. lO/c- 111 3^ ^ 2m 2 



8. 6 2i! 3ir 6nf 



10. 6^ v^ 3v^ 6- V 5v^ 



6b 



See Example 2 Simplify. Justify your steps using the Commutative, Associative, and 
Distributive Properties when necessary. 

' 11.3(7 + 2/7 + 5(7 U. 5b + 7b +10 



14. j'+ 4 + 2.V + 3y 



15. q- + 2q + 2q- 



13. (7 + 2/; + 2(7 + /; + 2(: 
16. 18 + 2d^ + d + Zd 



See Example 3 17. Geometry Write an expression for the perimeter 
L of the given figure. Then simplify tlie expression. 



3a 



3a 



See page EP4. 



2n 



2n 



5a 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



23. 7x + 8a- - 3y 

26. 2((7- + 2b + 2(7^) + /; + 2c 



Simplify each expression. 

18. 4a + 5a 19. 32)' - 5.v 20. 4r + 5f + 2c 

21. 5c/- -3,d' + d 22. 5/- + 2/+ /' 

24. 3(p + 9c/ - 2 + 9) + 14p 25. 6b + 6b- + 4/?^ 



27. Geometry Write an 
expression for the 
perimeter of the given 
triangle. Then evaluate 
the perimeter when n 
is 1.2, 3, 4, and 5. 



n 


1 


2 3 4 5 


Perimeter 










40 Chapter 1 Algebraic Reasoning 




Hours Brad Worked 


Week 


Hours 


1 


21.5 


2 


23 


3 


15.5 


4 


19 



The winner of each 
year's National 
Best Bagger 
Competition gets a 
bag-shaped trophy 
and a cash prize. 



28. Critical Thinking Determine whether the expression 9nr + k is equal to 

7m- + 2{2k - nf) + 5k. Use properties to justify your answer. 

29. Multi-Step Brad makes d dollars per hour as a coolc at 
a deli. The table shows the number of hours he worked 
each week in June. 

Write and simplify an expression for the amount 
of money Brad earned in June. 

Evaluate your expression from part a for d - $9.50. 

Wliat does your answer to part b represent? 

Business Ashley earns $8 per hour working at a grocery store. Last week 
she worked /; hours bagging groceries and twice as many hours stocking 
shelves. Write and simplify an expression for the amount Ashley earned. 

31. Critical Thinking The terms 3.v, 23a", 6y-, 2x, y- and one other term 
can be written in an expression which, when simplified, equals 5.v + 7y^. 
Identify the term missing from the list and write the expression. 

i^ 32. What's the Question? At one store, a pair of jeans costs $29 and a shirt 

costs $25. At another store, the same kind of jeans costs $26 and the same 

kind of shirt costs $20. The answer is 29; - 26/ + 25s - 20s - 3/ + 5s. 

What is the question? 

/^* 

._ 33. Write About It Describe the steps for simplifying the expression 

2x + 3 + 5A-- 15. 

^ 34. Challenge A rectangle has a width of x + 2 and a length of 3.v + 1. 
Write and simplify an expression for the perimeter of the rectangle. 




Test Prep and Spiral Review 



35. Multiple Choice Translate "six times the sum of x and y" and "five less 
than y." Which algebraic expression represents the sum of these two 
verbal expressions? 



i-S) 6x + 5 



(Jj 6x + 2y - 5 



<X) 6x + 5)' + 5 



CD 6x + 7v - 5 



36. Multiple Choice The side length of a square is 2x + 3. Which expression 
represents the perimeter of the square? 



CD 2x+ 12 



(Gj 4x + 6 



CH) 6x + 7 



CD 8x + 12 



37. The budget for the 2006 movie Supennan Returns was about two 
hundred and sixty-eight million dollars. Write this amount in scientific 
notation. (Lesson 1-3) 

Evaluate the expression 9y — 3 for each given value of the variable. (Lesson 1-6) 

38. y = 2 39. y=6 40. y=10 41. y=18 



1-8 Simplifying Algebraic Expressions 41 



1-9 



itTtaaODQiLfiig^ 



Their Solutions 







7.2.1 Use variables and appropriate operations to write an expression, equation or 
inequality that represents a verbal description. 

Ella has 22 songs on her MPS player. 
This is 9 more than Kay has. 

This situation can be written as an equation. 
An equation is a mathematical statement 
that two expressions are equal in value. 

Vocabulary An equation is like a balanced scale. 

equation 

solution 

Interactivities Online ► 




Number of 
songs Ella has 

22 



is equal 
to 



9 more than 
Kay has. 

J + 9 



Left expression 



^ 



Right expression 



Just as the weights on both sides of a balanced scale are exactly the 
same, the expressions on both sides of an equation represent exactly 
the same value. 

J ,.,. _ When an equation contains a variable, a value of the variable that 

'liB^ihjMjyl'JlS - / J^l makes the statement true is called a solution of the equation. 

The symbol ^ means 22 = j + 9 j = 13 is a solution because 22 = 13 + 9. 

"is not equal to." I 

r^J^ 22 = J + 9 7=15 is not a solution because 22 ?^ 15 + 9. 



I EXAMPLE ilj Determining WKether a Number Is a Solution of an Equation 

Determine whether the given value of the variable is a solution. 
A 18 = s- 7;s= 11 



/ 



18 = 5 
18= 11 -7 
18 = 4X 



Substitute n for s. 



1 1 is not a solution of 18 = s — 7. 



B H'+ 17= 23; w= 6 



w + 17 = 23 
6+ 17 = 23 
23 = 23 • 



Substitute 6 for w. 



6 is a solution of u> + 17 = 23. 



42 Chapter 1 Algebraic Reasoning 



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EXAMPLE 2 



Writing an Equation to Determine Whetlier a Number is a 
Solution 

Tyler wants to buy a new skateboard. He has S57, which is $38 
less than he needs. Does the skateboard cost S90 or $95? 

You can write an equation to find the price of the skateboard. 
If 5 represents the price of the skateboard, then s - 38 = 57. 

$90 





5 - 38 = 57 
90 - 38 = 57 Substitute 90 for s 
52 = 57X 




$95 

5 - 38 = 57 
95 - 38 = 57 Substitute 95 for s 
57 = 57 • 


1, 


The skateboard costs $95. 




EXAMPLE [3] Deriving a Real-World Situation from an Equation 

Which problem situation best matches the equation 3x + 4 = 22? 

Situation A: 

Harvey spent $22 at the gas station. He paid $4 per gallon for gas 

and $3 for snacks. How many gallons of gas did Harvey buy? 

The variable .v represents the number of gallons of gas that Harvey 
bought. 

$4 per gallon >- 4.t 

Since 4.r is not a term in the given equation, Situation A does not 
match the equation. 



Situation B: 

Harvey spent $22 at the gas station. He paid $3 per gallon for gas 

and $4 for snacks. How many gallons of gas did Harvey buy? 

$3 per gallon >- 3.x 

$4 on snacks *- + 4 

Harvey spent $22 in all, so 3.v + 4 = 22. Situation B matches the 
equation. 



Think and Discuss 

1. Compare equations with expressions. 

2. Give an example of an equation whose solution is 5. 



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1-9 Equations and Their Solutions 43 



1-9 



.im^i£& 



y 



keyword MBtaHMBiM ® 
Exercises 1-13, 15, 17, 19, 21, 
23,25 



GUIDED PRACTICE 



See Example 1 Determine whether the given value of the variable is a solution. 

L 1. 19 = x+4;.v = 23 2. 6» = 78; « = 13 3. k ^ 3= 14; lc= 42 



See Example 
See Example 



4. Mavis wants to buy a book. She has $25, which is $9 less than she needs. 
Does the book cost $34 or $38? 

5. Which problem situation best matches the equation 10 + 2.v = 16? 

Situation A: Angle bought peaches for $2 per pound and laundry detergent 
for $10. She spent a total of $16. How many pounds of peaches did Angle buy? 

Situation B: Angle bought peaches for $10 per pound and laundry detergent 
for $2. She spent a total of $16. How many pounds of peaches did Angle buy? 



INDEPENDENT PRACTICE 



See Example 1 Determine whether the given value of the variable is a solution. 

i 6. ?■- 12 = 25; r = 37 7. 39 h- .v = 13; .v = 4 8. 21 = ni + 9; m = II 



9. 



18 



= 7:a= 126 



10. 16/= 48;/= 3 



11. 71 - y = 26; v = 47 



See Example 2 

L 



12. Curtis wants to buy a new snowboard. He has $119, which is $56 less than 
he needs. Does the snowboard cost $165 or $175? 



See Example 3 13. Wliich problem situation best matches the equation 2/?; + 10 = 18? 

Situation A: A taxi service charges a $2 fee, plus $18 per mile, leremy paid 
the driver $10. How many miles did leremy ride In the taxi? 

Situation B: A taxi service charges a $10 fee, plus $2 per mile, leremy paid 
the driver $18. How many miles did leremy ride in the taxi? 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EP4, 



Determine whether the given value of the variable is a solution. 

14. /■= 6 for 15 -; = 21 

16. /;; = 18 for 16 = 34 - w 

18. y= 8for9_y+2 = 74 

20. <7= 13 for 5^ + 7 -^7= 51 

22. / = 12 for 3(50 - n - lOf = 104 



15. .Y = 36 for 48 = x + 12 

17. k = 23 for 17 + ^- = 40 

19. c= 12 for 100 - 2c = 86 

21. w = 15 for I3w - 2 - 6w = 103 

23. ;• = 21 for 4r - 8 + 9;- - 1 = 264 



24. Hobbies Monlque has a collection of stamps from 6 different countries. 
Jeremy has stamps from 3 fewer countries than Monique does. Write an 
equation shovdng this, using y as the number of countries from which 
Jeremy has stamps. 



44 Chapter 1 Algebraic Reasoning 



Earth Science 



25. 



26 



The diagram shows 

approximate elevations 

for different climate 

zones in the Colorado 

Rockies. Use the 

diagram to write an 

equation that shows the 

vertical distance d from 

the summit of Mount 

Evans (14,264 ft) to the tree line, which marks 

the beginning of the alpine tundra zone. 



-41— West 


East^^- 


Tree line 


Alpine tundra, above 10,500 ft 




Subalpine, 9,000-10,500 ft 


Pinon-Juniper, 7,000-9,000 ft 


Montane forest, 7,500-9,000 ft 


Semidesert, 5,500-7,000 ft 


Foothills, 5,500-7,500 ft 


'■■"'' — ; •: 


Great Plains, 3,000-5,500 ft 



Source. Colorado Mall 



The top wind speed of an F5 tornado, the strongest known 
kind of tornado, is 246 mi/h faster than the top wind speed 
of an Fl tornado, the weakest kind of tornado. The top 
wind speed of an Fl tornado is 72 mi/h. Is the top wind 
speed of an F5 tornado 174 mi/h, 218 mi/h, or 318 mi/h? 

27. \£) Write a Problem The mean surface temperature of 
Earth increased about 1 °F from 1861 to 1998. In 1998, the 
mean surface temperature was about 60 °F. Use these data 
to write a problem involving an equation with a variable. 



28. 



^ Challenge In the 1980s, about 9.3 x lO^* acres of 
tropical forests were destroyed each year due to 
deforestation. About how many acres of tropical 
forests were destroyed during the 1980s? 




Maroon take and Maroon Bells in the 
Colorado Rockies 




Test Prep and Spiral Review 



29. Multiple Choice lack's rectangular bedroom has a length of 10 feet. He 
used tlie formula A— 10»' to find the area of his room. He found that his 
bedroom had an area of 150 square feet. Wliat was the width of his bedroom? 



(X* 15 feet 



CS) 25 feet 



CD 30 feet 



CD 15,000 feet 



30. Multiple Choice The number of seventh-graders at Pecos Middle School 
is 316. This is 27 more than the number of eighth-graders. How many 
eighth-graders are enrolled? 



CE' 289 



CD 291 



CH) 299 



Write each number in scientific notation. (Lesson 1-3) 

31. 10,850,000 32. 627,000 

Tell which property is represented. (Lesson 1-5) 
34. (7 + 5) + 3 = 7 + (5 + 3) 35. 181 + = 181 



CD 343 



33. 9,040,000 



36. be = cb 



1-9 Equations and Their Solutions 45 




Model Solving Equations 



Use with Lessons 1-10 and 1-11 



KEY 




REMEMBER 


m-iD^ 


= variable 


• In an equation, the expressions on both 
sides of the equal sign are equivalent. 


OR 




• A variable can have any value that 


s = 1 i_^_- 


= variable 


makes the equation true. 



X?, 



Learn It Online 

Lab Resources Online go.hrw.com 



You can use balance scales and algebra tiles to model and solve equations. 



Activity 



Q Use a balance scale to model and solve the equation 3 + x = 11. 



a. On the left side of the scale, place 
3 unit weights and one variable weight. 
On the right side, place 11 unit weights. 
This models 3 + .v = 1 1 . 



m 



jTE 



3 + 



X = 



1 


1 


1 




1 


1 


1 


1 


1 


1 


1 


1 



11 



b. Remove 3 of the unit weights from 
each side of the scale to leave 
the variable weight by itself on 
one side. 











^ 


2,2,^ 


1^1 


1 


1 1 1 


li^l 


J_ 


1 1 1 








■-=—-- 


■^_ 


z^ 




*- 


i + 

i 


X 


= 




11 
-3 



Count the remaining unit weights 
on the right side of the scale. This 
number represents the solution 
of the equation. 



The model shows that if 3 + x = 11, then x = 8. 



1 


1 


1 


1 


1 


1 


1 


1 





46 Chapter 7 Algebraic Reasoning 



Q Use algebra tiles to model and solve the equadon 3y = 15. 

a. On the left side of the mat, 
place 3 variable tiles. On 
the right side, place 15 
unit tiles. This models 
3_v = 15. 




b. Since there are 3 variable 
tiles, divide the tiles on 
each side of the mat into 3 
equal groups. 





IL.^^-^-^-^ 


: F5 g ^ gt5 gj 


f[ 


L_ Li L„ L: lal 


(L_._,^. 


:. L L„ L-_, b: L;.] 


1 



3y 
3 



15 
3 



c. Count the number of unit 
tiles in one of the groups. 
This number represents the 
solution of the equation. 



The model shows that if 3v = 15, then y = 5. 



To check your solutions, substitute the variable in each equation with your 
solution. If the resulting equation is true, your solution is correct. 



3 + .Y =11 
3 + 8 = 11 
11 = !!• 



3.y = 15 
3- 5 = 15 



15 = 15v/ 



Think and Discuss 



1. What operation did you use to solve the equation 3 + .v = 1 1 in Q'? 
What operation did you use to solve 3y = 15 in©? 

2. Compare using a balance scale and weights with using a mat and 
algebra tiles. Which method of modeling equations is more helpful 
to you? Explain. 



Try Tliis 



Use a balance scale or algebra tiles to model and solve each equation. 

1. 4.\-=16 2.3 + 5 = 11 3. 5r=15 4. ;; + 7 = 12 

5. y + 6 = 13 6. 8 = 2r 7. 9 = 7 + w 8. 18 = 6p 



7-70 Hands-On Lab 47 







1-1 



jj Solving Equations by Adding 
9 or Subtracting 



7.2.1 Use variables and appropriate operations to write an expression, equation or 
inequality tliat represents a verbal description. 

To solve an equation means to find a solution to the equation. To do 
this, isolate the variable — that is, get the variable alone on one side of 
the equal sign. 



Vocabulary 

Addition Property 
of Equality 

inverse operations 

Subtraction Property 
of Equality 



Interactivities Online ► 



X = 8 - 5 
7-3=y 

The variables are isolated. 



X + 5 = 8 

7 = 3+y 

The variables are not isolated. 



Recall that an equation is like a balanced scale. If you increase or 
decrease the weights by the same amount on both sides, the scale will 
remain balanced. 



ADDITION PROPERTY OF EQUALITY 








Words 


Numbers 


Algebra 


You can add the 


2 + 3 = 5 


X = y 


same amount to both 


+ 4 +4 


+ z + z 


sides of an equation, 
and the statement 


2 + 7 = 9 


X + z = y + z 


will still be true. 










^ 



EXAMPLE 



Use inverse operations when isolating a variable. Addition and subtraction 
are inverse operations, which means that they "undo" each other. 



2+5=7 



-/ 



-5=2 



^ 



Using the Addition Property of Equality 

Solve the equation x - 8 = 17. Check your answer. 



X - 8 = 17 

+ 8 +8 

X = 25 

Check 

X - 8 = 17 
25 - 8= 17 
17= 17i/ 



Think: 8 is subtracted from x, so 
add 8 to botli sides to isolate x. 



Substitute 25 for x. 
25 is a solution. 



48 Chapter 1 Algebraic Reasoning 



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^SUBTRACTION PROPERTY OF EQUALITY 



Words 



You can subtract the same 
amount from both sides 
of an equation, and the 
statement will still be true. 



Numbers 



4 + 7 = 11 
-3 -3 

4 + 4 = 8 



Algebra 



X = y 

- z — z 

X — z = y — z 



EXAMPLE 



C3 



Using the Subtraction Property of Equality 

Solve the equation a + 5 = 11. Check your answer. 

(7 + 5 = 11 Think: 5 is added to a, s 

— 5 — 5 subtract 5 from both sides to isolate a. 

a =6 

Check 



a + 5= n 
6 + 5=11 
11 = It*/ 



Substitute 6 for a. 
6 is a solution. 



EXAMPLE [T] Sports Application 



Michael Jordan's highest point total for a single game was 69. The 
entire team scored 117 points in that game. How many points did 
his teammates score? 

Let p represent the points scored by the rest of the team. 

Jordan's points + Teammates' points = Final score 

69 + p = 117 

69 + p= 117 
- 69 — 69 Subtract 69 from both sides to isolate p. 

p= 48 

His teammates scored 48 points. 



Think and Discuss 

1. Explain how to decide which operation to use in order to isolate 
the variable in an equation. 

2. Describe what would happen if a number were added or 
subtracted on one side of an equation but not on the other side. 



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7-70 Solving Equations by Adding or Subtracting 49 



1-10 



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'; J 



□^ 



Ci3,J^3-03: 





GUIDED PRACTICE 



See Example 1 Solve each equation. Check your answer. 



1. r- 77 = 99 
See Example 2 4.^+83 = 92 



2. 102 = r- 66 
5. 45 = 36 + / 



3. A- - 22 = 66 
6. 987 = 16 + m 



See Example 3 7. After a gain of 9 yards, your team has gained a total of 23 yards. How 
L many yards had your team gained before the 9-yard gain? 



INDEPENDENT PRACTICE 



See Example 1 Solve each equation. Check your answer. 

9. r - 28 = 54 



See Example 2 



8. /; -36 = 17 

11. /;- 41 = 26 

14. A-l- 15 = 43 

17. 110 = 5+ 65 

20. 97 = /■ + 45 



12. Ill - 51 = 23 

15. /('+ 19 = 62 

18. X + 47 = 82 

21. ^+ 13 = 112 



10. p- 56 = 12 
13. k- 22 = 101 
16. c? + 14 = 38 
19. 18 +7 = 94 
22. 44 = 16 + n 



See Example 3 



23. Hank is on a field trip. He has to travel 56 miles to reach his destination. 
He has traveled 18 miles so far. How much farther does he have to travel? 

24. Sandy read 8 books in April. If her book club requires her to read 6 books 
each month, how many more books did she read than what was required? 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EP4. 



Solve each equation. Check your answer. Tell which property you used. 



25. ^-7 = 3 
28. 356 = y- 219 
31. 651 + r = 800 
34. 16 = /; - 125 
37. 63 + X = 902 



26. /; + 17 = 98 
29. 105 = <7 + 60 
32. /-63 = 937 
35. .s + 841 = 1,000 
38. ;:- 712 = 54 



27. 23 + Z? = 75 

30. g- 720 = 159 

33. 59 + m = 258 

36. 711 =^7- 800 

39. 120 = ((' + 41 



40. Physical Science An object weighs less when it is in water. This is 
because water exerts a buoyant force on the object. The weight of an object 
out of water is equal to the object's weight in water plus the buoyant force 
of the water. Suppose an object weighs 103 pounds out of water and 55 
pounds in water. Write and solve an equation to find the buoyant force of 
the water. 

41 . Banking After Lana deposited a check for $65, her new account balance 
was $315. Write and solve an equation to find the amount that was in 
Lana's account before the deposit. 



50 Chapter 1 Algebraic Reasoning 



42. 



©43. 



Music Jason wants to buy the tmmpet 
advertised in tlie classified ads. He lias 
saved $156. Using tlie information from 
the ad, write and solve an equation to 
find how much more money he needs 
to buy the trumpet. 

What's the Error? Describe 

and correct the error. 

.Y = 50 for (8 + 4)2 + x = 26 



TICKerS, Fridayl0/5, Ampitheaire, 7 30 p.m. 
Good seats. SlOO/both. Will deliver lick- 
etsloyou! Jason. 123-4 S6- 7852. 

TICKETS, Rafael MencJoza in concert. 

EscamihQl^^w*^^BHCT'^gWw^^i]M. two 

fsea^, row 17. $75 123-567' 




J,ES. 2 . Traiect 6 7^ 

protessibnar mlWf^^^^Rush M-8S8 
great condition. Must sell last, 
SlOOO/best First buyer takes all. Chad, 
321-321-3211 
KCV. Ultrasonic 16-33-45-78 rpm, S100. 
" ■ * '- Rhure V-15, Mesto, Lanii-'' 



vertrsea . 

FITNESS 

sell, tradt 
weights, L 
Again Spt 
0222 La 
playitagamsd 

FREE DIVE 

the best g< 
We (ully = 
toys, Liq' 
Flexifoil b' 
toys, D- 
suits. F 
graphiii 



44. Write About It Explain how you know whether to add or subtract to 
^45. 



solve an equation. 



Challenge Kwan keeps a 
record of his football team's 
gains and losses on each 
play of the game. The 
record is shown in the 
table. Find the missing 
information by writing and 
solving an equation. 



Play 


Play Gain/Loss 


Overall Gain/Loss 


1st down 


Gain of 2 yards 


Gain of 2 yards 


2nd down 


Loss of 5 yards 


Loss of 3 yards 


3rd down 


Gain of 7 yards 


Gain of 4 yards 


4th down 




Loss of 7 yards 




Test Prep and Spiral Review 



itrwwfwrwwiiwtnt 



46. Gridded Response Morgan has read 78 pages of Treasure Island. The book 
has 203 pages. How many pages of the book does Morgan have left to read? 

47. Multiple Choice Wliich problem situation best represents the equation 

42 - .V = 7? 

CS) Craig is 42 years old. His brother is 7 years older than he is. How old is 
Craig's brother? 

CE> Dylan has 42 days to finish his science fair project. How many weeks 
does he have left to finish his project? 

CD The total lunch bill for a group of 7 friends is $42. If the friends split 
the cost of the meal evenly, how much should each person pay? 

CS) Each student in the Anderson Junior High Spanish Club has paid for a 
club T-shirt. If there are 42 students in the club and only 7 shirts are left 
to be picked up, how many students have already picked up their shirts? 

Write each phrase as an algebraic expression. ( Lesson 1-7) 

48. the product of 16 and n 49. 17 decreased by A; 50. 8 times the sum of x and 4 



Simplify each expression. (Lesson 1-8) 

51. 6(2 -I- 2/7) -I- 3» 52. 4x - 7v + x 



53. 8-l-3r-l-2(4f) 



7-70 Solving Equations by Adding or Subtracting 51 







Solving Equations 
Multiplying or Dividing 



7.2.1 Use variables and appropriate operations to write an expression, equation or 
inequality tinat represents a verbal description. 

Like addition and subtraction, 
multiplication and division 
are inverse operations. 
They "undo" each other. 



2|:S|=io 

10 



ffi 







Words 


Numbers 


Algebra 


You can multiply both 
sides of an equation by 
the same number, and the 
statement will still be true. 


3-4 = 12 

2-3 -4 = 2-12 

6 • 4 = 24 


X = y 
zx = zy 



Vocabulary 

Multiplication 
Property of Equality 

Division Property 
of Equality 

Interactivi ties Online ► 



If a variable is divided by a number, you can often use multiplication to 
isolate the variable. Multiply both sides of the equation by the number. 

EXAMPLE [ij Using the Multiplication Property of Equality 

Solve the equation ^ = 20. Check your answer. 

4 = 20 



(7)f = 20(7) 


Think: x is divided by 7, so multiply both 


x= 140 


sides by 7 to isolate x. 


Check 




1 = 20 




If = 20 


Substitute 140 for x. 


20 = 20 • 


140 is a solution. 



.,.,^-.«to«&&^d&!l 


i. :- 


in^iMiiH 


m 


Words 


Numbers 


Algebra 


You can divide both sides 
of an equation by the same 
nonzero number, and the 
statement will still be true. 


5 • 6 = 30 

5 • 6 _ 30 
3 3 

5. 1=10 
5-2 = 10 


.X' = y 

X _ y 

z z 

Z9t 





52 Chapter 1 Algebraic Reasoning 



yidaD Lesson Tutorials Online mv.hrw.com 



If a variable is multiplied by a number, you can often use division to 
isolate the variable. Divide both sides of the equation by the number. 

EXAMPLE [2] Using the Division Property of Equality 

Solve the equation 240 = 42. Check your answer. 

240 = 4;: .. _ ,. _, .. 



240 _ 4z 


aiviae ootn siaes b 


4 4 




60 = z 




Check 




240 = 4z 




240 = 4 (60) 


Substitute 60 for z 


240 = 240 • 


60 is a solution. 



EXAMPLE 



[3 




1 /)! I / ■ 



In 2005, tance 
Armstrong won his 
seventh consecutive 
Tour de France. He is 
the first person to win 
the 2,051 -mile bicycle 
race more than five 
years in a row. 



Health Application 

If you count your heartbeats for 10 seconds and multiply that 
number by 6, you can find your heart rate in beats per minute. 
Lance Armstrong, who won the Tour de France seven years in a 
row, from 1999 to 2005, has a resting heart rate of 30 beats per 
minute. How many times does his heart beat in 10 seconds? 

Use the given information to write an equation, where b is the 
number of heartbeats in 10 seconds. 

Beats in 10 s • 6 = beats per minute 
b • 6 = 30 



6b = 30 

6b _ 30 
6 6 

b = 5 



Tliir ultiplied by 6, so 

divide botii sides by 6 to isolate b. 



Lance Armstrong's heart beats 5 times in 10 seconds. 



Think and Discuss 

1. Explain how to check your solution to an equation. 

2. Describe how to solve 13x = 91. 

3. When you solve 5p = 35, will p be greater than 35 or less than 35? 
Explain your answer. 

4. When you solve ^ = 35, will p be greater than 35 or less than 35? 
Explain your answer. 



VJiliD L€SSon Tutorials OnlinE 



7-7 7 Solving Equations by Multiplying or Dividing 53 




keyword ■mBiwiiHiiM @ 

Exercises 1-20, 21, 27, 31, 33, 
35,41,43 



GUIDED PRACTICE 



See Example 1 Solve each equation. Check your answer. 



L 

See Example 2 
See Example 3 



1. — = 11 

77 

4. 72 = 8x 



2. ti ^ 25 = 4 
5. 3c = 96 



3. J' ^ 8 = 5 
6. X- 18= n 



7. On Friday nights, a local bowling alley charges $5 per person to bowl 
all night. If Carol and her friends paid a total of $45 to bowl, how many 
people were in their group? 



INDEPENDENT PRACTICE 



See Example 1 Solve each equation. Check your answer. 

8. 12 = .s- ^ 4 



9. 4 = 72 



18 



11. ^ = 35 
See Example 2 14. 17.v = 85 
L 17. 97(7 = 194 



12. 



= 22 



15. 63 = 3p 
18. 9(7 = 108 



10. 13=1 

13. 17 = 11^ 18 

16. 6;/ = 222 

19. 495 = 11(^ 



See Example 3 20. It costs $6 per ticket for groups often or more people to see a minor league 
baseball game. If Albert's group paid a total of $162 for game tickets, how 
many people were in the group? 



Extra Practice 



See page EP4. 



PRACTICE AND PROBLEM SOLVING 



Solve each equation. Check your answer. 



21. 9=g-3 

24. 7/- = 84 

27. /; + 33 = 95 

30. 504 = f-212 

33. 2l=d^2 



22. 150 = 3; 
25. 5.V = 35 
28.^ = 6 
31. Ha = 288 
34. ^ = 83 



23. 68 = 7?? - 42 

29. 12/= 240 

32. 157 + ^ = 269 

35. r-92 = 215 



Multi-Step Translate each sentence into an equation. Then solve the equation. 

36. A number d divided by 4 equals 3. 

37. The sum of 7 and a number /? is 15. 

38. The product of a number b and 5 is 250. 

39. Twelve is the difference of a number q and 8. 

40. Consumer Math Nine weeks from now Susan hopes to buy a bicycle 
that costs $180. How much money must she save per week? 



54 Chapter 1 Algebraic Reasoning 



41. School A school club is collecting toys for a chUdren's charity. There are 18 
students in the club. The goal is to collect 216 toys. Each member will collect 
the same number of toys. How many toys should each member collect? 

42. Travel Lissa drove from Los Angeles to New York Cit}' and averaged 45 miles 
per hour. Her driving time totaled 62 hours. Write and solve an equation to 
find the distance Lissa traveled. 



43. 



44. 



©45. 
#46. 



47. 



Business A store rents space in a building at a cost of $19 per square foot. If 
the store is 700 square feet, how much is the rent? 



Favorite Fruits 




Ms. Ryan asked her students to name their 
favorite fruit. If 6 times as many people like 
bananas as like peaches, how many people like 
peaches? 

What's the Error? For the equation 7.v = 56, a 
student found the value of .v to be 392. What was 
the student's error? 

Write About It How do you know whether to 
use multiplication or division to solve an equation? 

Challenge In a survey, 8,690,000 college students were asked about their 
electronic equipment usage. The results are as follows: 7,299,600 use a 
TV, 6,604,400 use a DVD, 3,389,100 use a video game system, 3,041,500 
use a VCR, and x students use an MP3 player. If you multiply the number 
of students who use MPS players by 5 and divide by 3, you get the 
total number of students represented in the survey. Write and solve an 
equation to find the number of students who use MP3 players. 




Test Prep and Spiral Review 



48. Multiple Choice Mr. Tomkins borrowed $1,200 to buy a computer. He wants 
to repay the loan in 8 equal payments. How much vdll each payment be? 



CA) $80 CD $100 (c:> $150 

49. Multiple Choice Solve the equation 16x = 208. 

CD A- = 11 CS>.v=12 CE)x=13 



CD $200 



CD x= 14 



50. Extended Response It costs $18 per ticket for groups of 20 or more 
people to enter an amusement park. If Celia's group paid a total of $414 
to enter, how many people were in her group? 

Determine whether the given value of the variable is a solution. ' Lesson 19) 

51. .T + 34 = 48;x= 14 52. d - 87 = 77; cf = 10 



Solve each equation. Lesson 1-10) 

53. 76 + H = 115 54. ; - 97 = 145 



55. t- 123 = 455 



56. f? + 39 = 86 



7-7 7 Solving Equations by Multiplying or Dividing 55 




To Go On? 



.^^ Learn It Online 

^J* ResourcesOnlinego.hrw.com, 



Quiz for Lessons 1-6 Through 1-11 

(^ 1-6 ] Variables and Algebraic Expressions 

Evaluate each expression for the given values of the variable. 

1. 7(.v + 4)forA = 5 2. 11 - 7? H-3for/; = 6 3. /7 + 6r forp = 11 and r = 3 

^ 1-7 ] Translating Words into Math 

Write each phrase as an algebraic expression. 

4. the quotient of a number and 15 5. a number decreased by 13 

6. 10 times the difference of /J and 2 7. 3 plus the product of a number and 8 

8. A long-distance phone company charges a $2.95 monthly fee plus $0.14 for each 
minute. Write an algebraic expression to show the cost of calling for t minutes in 
one month. 

V^ 1-8 ] Simplifying Algebraic Expressions 

Simplify each expression. Justify your steps. 

9. 2y+ 5y'- 2y' 



10. .v + 4 + 7.V + 9 



12. Write an expression for the perimeter of the given 
figure. Then simplify the expression. 



11. 10 + 9t> - 6(1 - /; 



4fa 



7a 



Q^ 1-9 ) Equations and Their Solutions 



la 



Ab 



Determine whether the given value of the variable is a solution. 

13. 22 -.v= 7;.v= 15 14. 



56 



= 8; 1=9 



15. /;/ + 19 = 47;/;? = 28 



16. Last month Sue spent $147 on groceries. This month slie spent $29 more on 
groceries than last month. Did Sue spend $118 or $176 on groceries this month? 



Qy 1-10] Solving Equations by Adding or Subtracting 

Solve each equation. 

17. g- 4 =13 18.20 = 7 + ^ 19. r- 18 = 6 

Qy 1-11] Solving Equations by Multiplying or Dividing 



20. J71 + 34 = 53 



Solve each equation. 

21. 1 = 7 



22. 3b = 39 



23. ;; ^ 16 = 7 



24. 330 = 22x 



25. A water jug holds 128 fluid ounces. How many 8-ounce servings of water does the 
jug hold? 



56 Chapter 1 Algebraic Reasoning 



CONNECTIONS 



Sears Tower when it was completed in 1973, the Sears Tower 
in Chicago became the tallest building in the United States. The 
tower's Skydeck on the 103rd floor offers an incredible view that 
attracts 1.3 million visitors per year. The express elevators to the 
Skydeck are among the fastest in the world. 

For 1-4, use the table. 

1. The table shows the distance the Sk\'deck 
elevators travel in seconds. Describe the 
pattern in the table. 

2. Find the distance an elevator can travel 
in 7 seconds. Explain how you found the 
distance. 

3. Write an expression that gives the distance 
an elevator travels in 5 seconds. 

4. The Skydeck is 1,353 feet above ground. 
Write and solve an equation to find out 
about how long it takes an elevator to go 
from the ground to the Skydeck. 

5. The Sears Tower has 1.61 x lO"* windows. The Empire 
State Building in New York has 6.5 x 10'^ windows. 
Which building has more windows? How many more 
windows does it have? 

6. Approximately 2.5 x 10'* people enter the Sears Tower 
each day. About how many people enter the building 
during a typical work week from Monday to Friday? 




irfiiii 




Real-World Connections T 57 




Jumping Beans 



You will need a grid that is 4 squares by 6 squares. 
Each square must be large enough to contain a bean. 
Mark off a 3-square by 3-square section of the grid. 
Place nine beans in the nine spaces, as shown below. 

You must move all nine beans to the nine marked-off 
squares in the fewest number of moves. 



Follow the rules below to move the beans. 

You may move to any empty square in any direction. 

Q You may jump over another bean in any direction to an empty square. 

Q You may jump over other beans as many times as you like. 




e • # 

























f 




1 




# 


t 


^ 


% 




» 






« 


1 










„ 







Moving all the beans in ten moves is not too difticult. but can you 
do it in nine moves? 



Trading Spaces 



The purpose of the game is to replace the red 
counters with the yellow counters, and the 
yellow counters with the red counters, in the 
fewest moves possible. The counters must be 
moved one at a time in an L-shape. No two 
counters may occupy the same square. 

A complete copy of the rules and a game board are 
available online. 




Learn It Online 

Game Time Extra go.hrw.com, 

■PEIi.lMSlDi.aT^^TlG? 



^ 



58 Chapter 1 Algebraic Reasoning 



/^^ mfJi 




Materials 

• I full sheet of 
decorative paper 

• 3 smaller pieces of 
decorative paper 

• stapler 

• scissors 

• markers 
■ pencil 




PROJECT 



Step-by-Step 
Algebra 



This "step book" is a great place to record sample 
algebra problems. 

Directions 

O Lay the ll^-by-7| inch sheet of paper in front 
of you. Fold it down 2:^ inches from the top and 
make a crease. Figure A 

Q Slide the 7^-by-7^-inch sheet of paper under 
the flap of the first piece. Do the same with the 
5^-by-7|-inch and 3|-by-7|-inch sheets of paper 
to make a step book. Staple all of the sheets 
together at the top. Figure B 

Q Use a pencil to divide the three middle sheets 
into thirds. Then cut up from the bottom along 
the lines you drew to make slits in these three 
sheets. Figure C 

Q On the top step of your booklet, write the 
number and tide of the chapter. 

Taking Note of the Math 

Label each of the steps in your booklet 
with important concepts from the chapter: 
"Using Exponents," "Expressing Numbers 
in Scientific Notation," and so on. On the 
bottom sheet, write "Solving Equations." 
Write sample problems from the chapter 
on the appropriate steps. 








2 




59 



CHAPTER I 



i 



Study Guide: 



Vocabulary 

Addition Property 
of Equality 



48 



algebraic expression 30 

Associative Property 24 

base 10 

coefficient 38 

Commutative 

Property 24 

conjecture 7 

constant 30 



Distributive Property ... 25 



Division Property 
of Equality 



52 



equation 42 

evaluate 30 

exponent 10 

Identity Property 24 

inverse operations 48 



Multiplication 
Property of Equality 



, . 52 



numerical 

expression 19 

order of operations 19 

power 10 

scientific notation 14 

solution 42 

Subtraction Property 

of Equality 49 

term 38 

variable 30 



Complete the sentences below with vocabulary words from the list above. 

1. The ? tells how many times to use the ? as a factor. 

2. A (n) ? is a mathematical phrase made up of numbers and 



operations. 



is a mathematical statement that two expressions are equal 



3. A(n) ? 
in value. 

4. A(n) ? consists of constants, variables, and operations. 



EXAMPLES 



1-1 } Numbers and Patterns (pp. 6-9) 

■ Identify a possible pattern. Use the 
pattern to write the next three numbers. 

2,8, 14,20,.. . 

2+6 = 8 8 + 6= 14 14 + 6 = 20 
A possible pattern is to add 6 each time. 
20 + 6 = 26 26 + 6 = 32 32 + 6 = 38 



1-23 Exponents (pp. 10-13) 

■ Find the value of 4^. 

4 • 4 = 64 



43 = 4 



EXERCISES 



Identify a possible pattern. Use the pattern 
to write the next three numbers. 



5. 6, 10, 14, 18, . . . 
7. 7, 14,21,28, . . . 
9. 41,37,33,29, . . 



Find each value. 
11. 9^ 12. 10' 



6. 15,35,55,75,... 

8. 8,40,200, 1,000, 

10. 68,61,54,47, .. 



13. 2' 14. r 15. IV 



8=, 



60 Chapter 1 Algebraic Reasoning 



EXAMPLES 



EXERCISES 



^1-3] Scientific Notation (pp. 14-17) 

j ■ Multiply 157 • 10^ 

! 157 • 10^ = 1570000 
= 1,570,000 



Multiply. 

16. 144 • 10- 



17. 1.32 • 10' 18. 



10' 



Write each number in scientific notation. 
19. 48,000 20. 7,020,000 21. 149,000 

22. In 2006 the population of Switzerland 
was about 7.507 x 10''. Write this 
population in standard form. 



1-4] Order of Operations (pp 19-22) 

■ Simplify 150 - (18 + 6) • 5. 

:: 150 -(18 + 6) -5 /erro. rr: ■;\.^ ^,^^. 

In parentheses. 

150-24-5 Multiply. 

150 - 120 Subtract. 

30 



Simplify each expression. 

23. 2 + (9 - 6) ^ 3 24. 12 • 3" - 5 

25. 1 1 + 2 • 5 - (9 + 7) 26. 75 ^ 5- + 8" 

27. Lola decides to join a 15 mile 

walk-a-thon. Her parents give her 
$3 for each mile walked and her brother 
gives her $10. Simplify the expression 
3 • 15 + 10 to find out how much 
monev she raised. 



1-5j Properties of Numbers (pp 24 27) 

■ Tell which property is represented. 

(10- 13) -28 = 10- (13 -28) 
Associative Property 



Tell which property is represented. 

28. 42 + 17 = 17 + 42 

29. /;/ + = 111 

30. 6 • (.Y - 5) = 6 • A- - 6 • 5 

Simplify each expression. Justify each step. 

31. 28 + 15 + 22 32. 20 • 23 • 5 



1-6] Variables and Algebraic Expressions (pp. 30-33) 
■ Evaluate 5a — 6b + 7 for a = 4 and b = 3. 



5a -6b + 7 
5(4) -6(3) + 7 
20-18 + 7 
9 



Evaluate each expression for the 
given values of the variables. 

33. 4.V - 5 for x = 6 

34. 83'^ + 3yfory = 4 

35. ^ + 6ni - 3 for « = 5 and in - 2 



Tidb'j Lesson Tutorials Online mv.hrw.com 



Study Guide: Review 61 



EXAMPLES 



EXERCISES 



1-7 J Translating Words into Math (pp 34-37) 

■ Write as an algebraic expression. Write as an algebraic expression. 

36. 4 divided by the sum of a number and 12 

37. 2 times the difference of t and 1 1 

38. Missy spent $32 on s shirts. Write an 
algebraic expression to represent the 
cost of one shirt. 



5 times the sum of a number and 6 
Sin + 6) 



1-8^ Simplifying Algebraic Expressions (pp. 38-41) 



Simplify the expression. 

4a-^ + 5y + 8a-^ - 4y - 5x^ 
Ax^ + 5y + 8x^ - 4y- 5x' 



IZx-" + V - 5a-^ 



Simplify each expression. 

39. 7b' + 8 + 3/7- 

40. 12rt' + 4 + 3(7- - 2 

41. A- + x^ + A-"* + 5x- 



1-9) Equations and Their Solutions (pp. 42 45) 



Determine whether 22 is a solution. 



24 = 5 - 13 
24 = 22 - 
24 = 9X 



13 



22 is not a solution. 



Determine whether the given value of the 
variable is a solution. 

42. A = 3; ^ = 48 

43. 36 = ;; - 12; ii = 48 

44. 9x= 117; x= 12 



1-10j Solving Equations by Adding or Subtracting (pp. 48-51) 

■ Solve the equation. Then check. Solve each equation. Then check. 



b+ 12= 16 
- 12 - 12 

b^ 4 



/?+ 12 = 16 
4 + 12= 16 
16= \6t/ 



45. 8 + /? = 16 
47. 27 + r = 45 



46. 20 = ;? - 12 
48. f - 68 = 44 



1-11j Solving Equations by Multiplying or Dividing (pp. 52-55) 



Solve the equation. Then check. 

2r= 12 2r= 12 



Solve each equation. Then check. 

49. /? + 12 = 6 50. 3p=27 



2r _ 12 
2 2 

/■=6 



2(6)= 12 



12= 12t/ 



51. 



14 



= 7 



52. 6x = 78 



53. Lee charges $8 per hour to baby-sit. Last 
montli she earned $136. How many 
hours did Lee baby-sit last month? 



62 Chapter 1 Algebraic Reasoning 



Chapter Test 



Identify a possible pattern. Use the pattern to write the next three numbers. 




1. 24,32,40,48, 

Find each value. 
5. 6^ 

Multiply. 

9. 148 • 10- 



2. 6, 18,54, 162, 



6. r 



3. 64,58,52,46,.. 



7. 



4. 13, 30, 47, 64, . 



8. 3^ 



10. 56.3 • 10^ 



Write each number in scientific notation. 
13. 406,000,000 14. 1,905,000 



11. 6.89 • 10-* 



15. 22,400 



12. 7.5 • 10"* 



16. 500,000 



17. The deepest point in the Atlantic Ocean is the Milwaukee Depth lying at a depth of 
2 J493 X lO'* feet. Write this depth in standard form. 

Simplify each expression. 

18. 18 -3 ^3-^ 19.36+16-50 20. 149 - (2^ -200) 21. (4 -=- 2) • 9 + 11 



24. 84 • 3 = 3 • 84 



Tell which property is represented. 

22. + 45 = 45 23. (r + s) + t- r+ is + t) 

Evaluate each expression for the given values of the variables. 

25. 4rt + 6Z; + 7 for (? = 2 and i; = 3 26. 7y " + 7y for y = 3 

Write each phrase as an algebraic expression. 

27. a number increased by 12 28. the quotient of a number and 7 

29. 5 less than the product of 7 and 5 30. the difference between 3 times .v and 4 



Simplify each expression. Justify your steps. 

31. /7 + 2 + 5Z7 32. 16 + 5i) + 3Z; + 9 



33. 5rt + 6f + 9 + 2a 



34. To join the gym Halle must pay a $75 enrollment fee and $32 per month. Write an 
algebraic expression to represent the total cost of m months at the gym, including the 
enrollment fee. 



Solve each equation. 

35. .v + 9= 19 36. 21 = y- 20 



37. m - 54 = 72 



39. 16 = 



y 



40. 102 = 17y 



41. 



= 1,400 



38. 136=.y+ 114 
42. 6.V = 42 



43. A caterer charged $15 per person to prepare a meal for a banquet. If the total catering 
charge for the banquet was $1,530, how many people attended? 



Chapter 1 Test 63 




^ . Test Tackier 

STANDARDIZED TEST STRATEGIES 




Multiple Choice: Eliminate Answer Choices 

With some multiple-choice test items, you can use mental math or 
number sense to quickly eliminate some of the answer choices before 
you begin solving the problem. 



EXAMPLE 



Which is the solution to the equation x + 7 = 15? 

'...- ■.._,' 09 Q-V O O O '^J 

>-^ W' = v_.'' •■~^j O '^ <» 

WW <^ <^ <!& v_, :,_/ 




®x = 22 CE).v=15 CD x = 8 CE)a- = 7 

READ the question. Then try to eliminate some of the answer choices. 

Use number sense: 

When you add, you get a greater number than what you started with. Since 
X + 7 = 15, 15 must be greater than x, or x must be less than 15. Since 22 
and 15 are not less than 15, you can eliminate answer choices A and B. 

The correct answer choice is C. 



EXAMPLE 



What is the value of the expression 18x + 6 for x = 5? 

CD 90 eg) 96 CS) 191 CD 198 

LOOK at the choices. Then try to eliminate some of the answer choices. 

Use mental math: 

Estimate the value of the expression. Round 18 to 20 to make the 
multiplication easier. 



20x + 6 

20(5) + 6 

106 



Substitute 5 for x. 
Multiply. Then add. 



Because you rounded up, the value of the expression should be less than 
106. You can eliminate choices H and J because they are too large. 

The correct answer choice is G. 



64 Chapter 1 Algebraic Reasoning 




Before you work a test question, use 
mental math to help you decide if 
there are answer choices that you can 
eliminate right away. 



Read each test item and answer the 
questions that follow. 

Item A 

During the August back-to-school sale, 
2 pairs of shoes cost $34, a shirt costs 
$15, and a pair of pants costs $27. Janet 
bought 2 pairs of shoes, 4 shirts, and 4 
pairs of pants and then paid an 
additional $7 for tax. Which expression 
shows the total that Janet spent? 

(S) 34 -1-4(15 -I- 27) 

® 34 + 4(15 + 27) + 7 

CD 4(34+ 15 + 27) + 7 

CD 34+15 + 4-27 

1. Can any of the answer choices be 
eliminated immediately? If so, which 
choices and why? 

2. Describe how you can determine the 
correct answer from the remaining 
choices. 

Item B 

Anthony saved $1 from his first 
paycheck, $2 from his second 
paycheck, then $4, $8, and so on. How 
much money did Anthony save from 
his tenth paycheck? 



CD $10 
CD $16 



CE) $512 
CD $1,023 



3. Are there any answer choices you 
can eliminate immediately? If so, 
which choices and why? 

4. What common error was made in 
finding answer choice F? 



Item C 

Craig has three weeks to read an 
850-page book. Which equation can 
be used to find the number of pages 
Craig has to read each day? 

CD 3a- = 850 



CS) 1 = 850 
CD 2Lv = 850 



CS) 



21 



= 850 



5. Describe how to use number sense 
to eliminate at least one answer 
choice. 

6. What common error was made in 
finding answer choice D? 

Item D 

What value of t makes the following 



equation 


true? 










22f 


= 132 




CD 


6 




® 


154 


CD' 


110 




CD 


2,904 



7. Which choices can be eliminated 
by using number sense? Explain. 

8. What common error was made in 
finding answer choice J? 

9. Describe how you could check 
your answer to this problem. 



Item E 

What is 

(1+2)2 


the value of the 
+ 14-4-2 + 5? 


expression 


® 







CD 


17 


(D 


11 




CD 


21 



10. Use mental math to quickly 
eliminate one answer choice. 
Explain your choice. 

11. What common error was made 
in finding answer choice B? 

12. What common error was made 
in finding answer choice C? 



Test Tackier 65 



CHAPTER 



1 



ISTEP+ 




.£*9 Learn It Online 

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Applied Skills Assessment 

Constructed Response 

1. Luke can swim 25 laps in one hour. 
Write an algebraic expression to show 
how many laps Luke can swim in h 
hours. How many hours will it take 
Luke to swim 100 laps? 

2. An aerobics instructor teaches a 
45-minute class at 9:30 a.m., three times 
a week. She dedicates 12 minutes 
during each class to stretching. The rest 
of the class consists of aerobic dance. 
How many minutes of each class does 
the instructor spend teaching aerobic 
dance? Write and solve an equation to 
explain how you found your answer. 

3. Ike and Joe ran the same distance but 
took different routes. Ike ran 3 blocks 
east and 7 blocks south. Joe ran 4 blocks 
west and then turned north. How far 
north did Joe run? Show your work. 

Extended Response 

4. The Raiders and the Hornets are buying 
new uniforms for their baseball teams. 
Each team member will receive a new 
cap, a jersey, and a pair of pants. 



Uniform Costs 




Raiders 


Hornets 


Cap 


$15 


$15 


Jersey 


$75 


$70 


Pants 


$50 


$70 



a. Let r represent the number of 
Raiders team members, and let h 
represent the number of Hornets 
team members. For each team, 
write an expression that gives the 
total cost of the team's uniforms. 

b. If the Raiders and the Hornets both 
have 12 team members, how much 
will each team spend on uniforms? 
Which team will spend the most, 
and by how much? Show your work. 

Multiple-Choice Assessment 

5. Which expression has a value of 74 
when X = 10, y = 8, and z = 12? 

A. 4xyz C. 2xz - 3y 

B. X + 5y + 2z D. 6xyz + 8 

6. What is the next number in the pattern? 



3, 3^ 27, 3\ 3' 



A. 729 

B. 3' 



C. 243 

D. 3« 



7. A contractor charges $22 to install 
one miniblind. How much does the 
contractor charge to install m 
miniblinds? 

C. 22 + m 



A. 22m 
^- 22 



D. 



22 

m 



8. Which of the following is an example 
of the Commutative Property? 

A. 20 + 10 = 2(10 + 5) 

B. 20 + 10 = 10 + 20 

C. 5 + (20 + 10) = (5 + 20) + 10 

D. 20 + = 20 



66 Chapter 1 Algebraic Reasoning 



9. Which expression simplifies to 9x + 3 
when you combine like terms? 

A. lOx^ -x^ - 3 

B. 3x + 7 - 4 + 3x 

C. 1 8 + 4x - 1 5 + 5x 

D. 7x^ + 2x + 6 - 3 

10. What is the solution to the equation 
810 = x- 625? 



A. X = 185 

B. x = 215 



C. X = 845 

D. X = 1,435 



11. Tia maps out her jogging route 
as shown in the table. How many 
meters does Tia plan to jog? 



Tia's Jogging Route 


Street 


Meters 


1st to Park 


428 


Park to Windsor 


112 


Windsor to East 


506 


East to Manor 


814 


Manor to Vane 


660 


Vane to 1st 


480 



A. 3 X 10 m 

B. 3 X 10 m 



C. 3 X 10 m 

D. 3 X lo" m 



12. To make a beaded necklace, Kris needs 
88 beads. If Kris has 1,056 beads, how 
many necklaces can she make? 

A. 968 C. 264 

B. 12 D. 8 

13. What are the next two numbers in 
the pattern? 

75, 70, 60, 55, 45, 40 

A. 35, 30 

B. 30, 20 

C. 30, 25 

D. 35, 25 



14. Marc spends $78 for n shirts. Which 
expression can be used to represent 
the cost of one shirt? 
n ^78 



"■78 
B. 78n 



C. 
D. 



n 
78 + n 



15. Which situation best matches the 
expression 0.29x + 2? 

A. A taxi company charges a $2.00 flat 
fee plus $0.29 for every mile. 

B. Jimmy ran 0.29 miles, stopped to 
rest, and then ran 2 more miles. 

C. There are 0.29 grams of calcium in 
2 servings of Hearty Health Cereal. 

D. Amy bought 2 pieces of gum for 
$0.29 each. 

16. Which of the following should be 
performed first to simplify this 
expression? 

1 6 • 2 + (20 H- 5) - 3^" H- 3 + 1 

A. 3^ -H 3 C. 16 • 2 

B. 20 ^ 5 D. 3 + 1 

Gridded Response 

17. If X = 15 and y = 5, what is the value 
of ^ + 3y? 

18. What is the exponent when you write 
the number 23,000,000 in scientific 
notation? 

19. An airplane has seats for 198 
passengers. If each row seats 6 people, 
how many rows are on the plane? 

20. What is the value of the expression 
3^ X (2 + 3 X 4) - 5? 

21. What is the solution to the equation 
10 + s = 42? 

22. What is the sum of 4 and the product 
of 9 and 5? 



Cumulative Assessment, Chapter 1 67 




2A 

2-1 

EXT 
LAB 
2-2 

LAB 

2-3 

LAB 



LAB 
2-5 

2B 

2-6 
2-7 
2-8 

2C 

2-9 



2-11 



Subtraction 

Subtracting Integers 

Model Integer 
Multiplication and 
Division 

/lultiplying and Dividing 
Integers 

Model Integer Equations 

Solving Equations 
Containing Integers 

Factors and Multiples 

Prime Factorization 
Greatest Common Factor 
Least Common Multiple 

Rational Numbers 

Equivalent Fractions and 
lixed Numbers 

Equivalent Fractions and 
Decimals 

Comparing and Ordering 
Rational Numbers 



7.1.6 







Why Learn This? 



Integers are commonly used to describe 
temperatures. In many parts of the world, 
winter temperatures are often negative 
integers, meaning it is colder than 0°. 






' w^^ 

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Chapter Project Online qo.hrw.com, 



keyword tlllWiM ® 



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68 Chapter 2 



■^ £^'i 




;P^%;;.S r^ ; 







Are You Ready? 



7 



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** ResourcesOnlinego.hrw.com, 
■ IMIilli.lijMSIQ-AYRT gG.: 



0^ Vocabulary 



Choose the best term from the list to complete each sentence. 

1. To ? a number on a number line, mark and label 
the point that corresponds to that number. 

2. The expression 1 < 3 < 5 tells the ? of these three 
numbers on a number line. 

3. A(n) ? is a mathematical statement showing two 
things are equal. 

4. Each number in the set 0, 1, 2, 3, 4, 5, 6, 7, . . . is 
a(n) ? ■ 

5. To ? an equation, find a value that makes it true. 



Complete these exercises to review skills you will need for this chapter. 



whole number 

expression 

graph 

solve 

equation 

order 



Q) Order of Operations 

Simplify. 

6. 7 + 9-5-2 
8. 115 - 15-3 + 9(8 - 2) 
10. 300 + 6(5 -3) - 11 



7. 12 -3 - 4 • 5 
9. 20 • 5 • 2 (7 + 1) H- 4 
11. 14-13 + 9-2 



0^ Find Multiples 

Find the first five multiples of each number. 

12. 2 13. 9 14. 15 15. I 

16. 101 17. 54 18. 326 19. 1,024 

Of Find Factors 

List all the factors of each number. 

20. 8 21. 22 22. 36 23. 50 

24. 108 25. 84 26. 256 27. 630 

(y) Use Inverse Operations to Solve Equations 

Solve. 

28.^ + 3 = 10 29. X- 4 =16 30. 9p = 63 31. | = 80 

32. X- 3 =14 33. 1 = 21 34. 9 + r=91 35.15^ = 45 




gers and Rational Numbers 69 




study Guid 



y"»-^Vi| r£|Vi 



(&W® 



'we Been 



Prevooysiy, y©y 

o compared and ordered non- 
negative rational numbers. 

• generated equivalent forms of 
rational numbers including 
v^hole numbers, fractions, and 
decimals. 

® used integers to represent real- 
life situations. 



Key 

Vocabulary /Vocabulario 




In This C 



You will study 

• comparing and ordering 
integers and rational numbers. 

• converting between fractions 
and decimals mentally, on 
paper, and with a calculator. 

• using models to add, subtract, 
multiply, and divide integers. 

• finding the prime factorization, 
greatest common factor, and 
least common multiple. 



Where You're Going 

You can use the skills 
learned in this chapter 

• to express negative numbers 
related to scientific fields 
such as marine biology or 
meteorology. 

• to find equivalent measures. 



equivalent fraction 


fraccion equivalente 


greatest common 
factor (GCF) 


maximo comun 
divisor (MCD) 


integer 


entero 


least common multiple 
(LCIVI) 


minimo comun 
multiplo (MCM) 


prime factorization 


factorizacion prima 


rational number 


numero racional 


relatively prime 


primo relative 


repeating decimal 


decimal periodico 


terminating decimal 


decimal finito 



Vocabulary Connections 

To become familiar with some of the 
vocabulary terms in the chapter, consider 
the following. You may refer to the chapter, 
the glossary, or a dictionary if you like. 

1 . The word commou means "belonging to 
or shared by two things." How can you use 
this definition to explain what the least 
common multiple of two numbers is? 

2. Wlien something is relative, it is "carried 
back" to or compared to certain values. A 
prime number is only divisible by itself and 
one. If two numbers are relatively prime 
and neither are prime numbers, how do 
you think they relate to each other? 

3. A decimal is a number that has digits 
to the right of the decimal point. What 
might you predict about those digits in 
a repeating decimal? 



70 Chapter 2 




.^ 



^ 



Reading 

and WriMaxn 

MathX -^ 




Writing Strategy: Translate Between Words 
and iVIath 

As you read a real-world math problem, look for key words to help you 
translate between the words and the math. 



Example 



At FunZone the cost to play laser tag is $8 per game. The cost to play 
miniature golf is $5 per game. The one-time admission fee to the park 
is $3. Jonna wants to play both laser tag and miniature golf. Write an 
algebraic expression to find the total amount Jonna would pay to play 
i laser tag games and m golf games at FunZone. 



r. 



more than one game, yo 
1 pay multiples om 

Miniature golf costs $5 per game. 
For more than one game, you 
would pay multiples of $5. 



Jonna wants to play both laser 
tag and miniature golt. 





There is a one-time admission 
fee of $3. 



> 

^ 
\ 



The total cost of m miniature golf 
games is 5m. 



/ 



Add the two totals to find the total 
cost of laser tag and mimature golf. 

8€-i-5m 

Add the admission fee to the total 
cost of the games. 

8€ -H 5w -H 3 



r 



IVy This 



Write an algebraic expression tfiat describes the situation. Explain why 
you chose each operation in the expression. 

1 . School supplies are half-price at Bargain Mart this week. The original prices 
were $2 per package of pens and $4 per notebook. CaUy buys 1 package of 
pens and n notebooks. How much does Cally spend? 

2. Fred has /cookies, and Gary has g cookies. Fred and Gary each eat 3 cookies. 
How many total cookies are left? 



Integers and Rational Numbers 71 



2-1 



j 







7.1.6 Identity, write, rename, compare and order rational and common 
irrational numbers and plot them on a number line. 

The opposite, or additive inverse, 
of a number is the same 
distance from on a number 
line as the original number, 
but on the other side of 0. 
Zero is its ow^n opposite. 

-4 and 4 are opposites. 



Vocabulary 

opposite 
additive inverse 
integer 
absolute value 



The whole numbers 
are the natural 
numbers and zero: 
0, 1, 2, 3 . 



I 4 I 

-5 -4 -3 



H — \ — \ — h- 



1 
4 



1 



3 4 



Y I Y 

Negative integers I Positive integers 

is neither positive 
nor negative. 




Dr. Sylvia Earle holds the world 
record for the deepest solo dive. 



The integers are the set of whole numbers and their opposites. By 
using integers, you can express elevations above, below, and at sea 
level. Sea level has an elevation of feet. Sylvia Earle's record dive was 
to an elevation of — 1,250 feet. 

EXAMPLE 1^1 Graphing Integers and Their Opposites on a Number Line 

Graph the integer -3 and its opposite on a number line. 

, 3 units , 3 units , 

H 4< H 

-H — \ — 4 — I — I — I — I — I — ♦ — \ — h* The opposite of -3 is 3. 
-5-4-3-2-1 1 2 3 4 5 



EXAMPLE 



IJJJaT: 



The symbol < means 
"is less than," and 
the symbol > means 
"is greater than." 



You can compare and order integers by graphing them on a number 
line. Integers increase in value as you move to the right along a 
number line. They decrease in value as you move to the left. 



^ 



Comparing Integers Using a Number Line 

Compare the integers. Use < or >. 
A 2 -2 



H — I 4 I — h 



H — h 



-4-3-2-1 1 2 3 4 



2 is farther to the right than -2, so 2 > -2. 



72 Chapter 2 Integers and Rational Numbers 



y]'h 



Lesson Tutorials OnlinE mv.hrw.com 



Compare the integers. Use < or >. 
B -10 -7 



+ 



4- 



+ 



+ 



4- 



+ 



+ 



-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 

- 10 is farther to the left than -7, so - 10 < -7. 



EXAMPLE [3] Ordering Integers Using a Number Line 



Use a number line to order the integers -2, 5, -4, 1, -1, and 
from least to greatest. 

Graph the integers on a number line. Then read them from left to 
right. 



I ♦ I — » » ♦ » I — \ — h-*- 



-5-4-3-2-1 1 2 3 4 5 

The numbers in order from least to greatest are -4, 



■1,0, l,and5. 



jJj?Jj;Jj.JjJlui' 

For more on absolute ^ number's absolute value is its distance from on a number line. 

value, see Skills Bank c- 4- * i *• 1 1 * 1 

^„ ' Since distance can never be negative, absolute values are never 
p. SB16^ ^ 

negative. They are always positive or zero. 



EXAIVIPLE [4] Finding Absolute Value 



Use a number line to find each absolute value. 



ijj.'f'TnjT 






The symbol 1 1 is read 
as "the absolute value 
of." For example, 
|— 3| means "the 
absolute value 
of -3." 



h 



7 units 



H 



*-\ — I — \ — \ — \ — \ — I — \ — h-# — h 
-2-1012345678 

7 is 7 units from 0, so 1 7 1 = 7. 

|-4| 

4 units 



H — h 



-^ — h 



H — \ — h 



-6-5 -4-3-2-1 1 2 3 4 

—4 is 4 units from 0, so I -4 1 = 4. 



Think and Discuss 


1. Tell which number is greater: -4,500 or -10,000. 


2. Name the greatest negative integer and the least nonnegative 


integer. Then compare the absolute values of these integers. 



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2-1 Integers 73 



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keyword ■aaiifcaM ® 
Exercises 1-30, 31, 33, 35, 37, 
39,41,45 



GUDED PRACTICE S 



See Example 1 Graph each integer and its opposite on a number line. 
i 1. 2 2. -9 3. -1 



4. 6 



See Example 2 Compare the integers. Use < or >. 

L 5. 5 -5 6. -9 -18 



7. -21 -17 8. -12 12 



See Example 3 Use a number line to order the integers from least to greatest. 

L 9.6,-3,-1,-5,4 10.8,-2,7,1,-8 11.-6,-4,3,0,1 



See Example 4 Use a number line to find each absolute value. 

I 12. |-2| 13. |8| 14. I 



15. -10 



INDEPENDENT PRACTICE 



See Example 1 Graph each integer and its opposite on a number line. 
L. 16. -4 17. 10 18. -12 



19. 7 



See Example 2 Compare the integers. Use < or >. 

[ 20. -14 -7 21. 9 -9 



22. -12 12 23. -31 -27 



See Example 3 Use a number line to order the integers from least to greatest. 

I. 24.-3,2,-5,-6,5 25.-7,-9,-2,0,-5 26.3,-6,9,-1,-2 

See Example 4 Use a number line to find each absolute value. 

. 27. |-16| 28. |12| 29. |-20| 30. |l5| 



Extra Practice 



See page EPS. 



PRACTICE AND PROBLEM SOLVING 



Compare. Write <, >, or =. 

-'""- '^'^ 32. 18 -55 



31. -^[j z'j 
35. 34 |34| 



36. 64 



33. |-2l| 21 34. -9 -27 

37. |-3| |3| 38. -100 -82 



39. Earth Science The table shows the average temperatures in Vostok, 
Antarctica from March to October. List the months in order from coldest 
to warmest. 



Month 


Mar 


Apr 


May 


Jun 


Jul 


Aug 


Sep 


Oct 


Temperature (°F) 


-72 


-84 


-86 


-85 


-88 


-90 


-87 


-71 



40. What is the opposite of |3 



? ? 



41. What is the opposite of |-29i? 



42. Business A company reported a net loss of $2,000,000 during its first year. In its 
second year it reported a profit of $5,000,000. Write each amount as an integer. 



74 Chapter 2 Integers and Rational Numbers 




In wakeboarding, 
a rider uses the 
waves created by a 
boat, the wake, to 
jump into the air 
and perform tricks 
such as rolls and 
flips. 



43. Critical Thinking Give an example in which a negative number has a greater 
absolute value than a positive number. 

44. Social Studies Lines of latitude are imaginary lines that circle the globe 
in an east-west direction. They measure distances north and south of the 
equator. The equator represents 0° latitude. 

a. What latitude is opposite of 30° north latitude? 

b. How do these latitudes' distances from the equator compare? 



Sports The graph shows how 
participation in several sports 
changed between 1999 and 2000 in 
the United States. 

By about what percent did 
participation in racquetball 
increase or decrease? 

By about what percent did 
participation in wall climbing 
increase or decrease? 



45 



46 



Popular Recreational Sports 



Group sports declined in 1999 as 
Americans turned to individual sports. 



Source USA Today. July 6, 2001 







47. What's the Error? At 9 a.m. the 

outside temperature was -3 °F. 
By noon, the temperature was 
— 12 °E A newscaster said that it 
was getting warmer outside. 
Why is this incorrect? 

48. Write About It Explain how to compare two integers. 

49. Challenge What values can .y have if |.v| = 11? 



Wakeboarding 




■■■■■i 


Snowmobilmg 




■■■■ 


Wall climbing 




.- .-.^ 


Beach volleyball 


■ 


/""% 


Racquetball 
Baseball 


. 


^. 



-10 10 20 

Percent change 



30 



40 



Test Prep and Spiral Review 



50. Multiple Choice Which list shows the integers in order from least to greatest? 

(S) -5,-6,-7,2,3 ex- 2,3,-5,-6,-7 (c:> -7,-6,-5,2,3 CD 3,2,-7,-6,-5 

51. Multiple Choice The table shows the average temperatures 
in Barrow, Alaska, for several months. In which month is the 
average temperature lowest? 



CT) January 
CG> March 



CK' May 
CD July 



Monthly Temperatures 


January 


-12 °F 


March 


-13 °F 


May 


20 "F 


July 


40 °F 



Write each number in scientific notation. (Lesson 1-3) 
52. 400,000 53. 1,802,300 54. 59.7 



lo-" 



Use the Distributive Property to find each product. (Lesson 1 -5) 
56. 3(12) 57. 2(56) 58. (27)6 



55. 800 • 10'^ 



59. (34)5 



2-1 Integers 75 



LESSON 2-1 I Negative 
EXTEiysioN J Exponents 



When a natural number 
has a positive exponent, the 
value of the power is greater 
than or equal to 1. Wlien a 
natural number has a negative 
exponent, the value of the 
power is less than or equal to 1. 
When any natural number has 
a zero exponent, the value of 
the power is equal to 1. 

Notice: The negative exponent 

becomes positive when it is 

moved to the denominator of the fraction. 



Power 


Meaning 


Value 


102 


10- 10 


100 


101 


10 


10 


10° 


1 


1 


10-1 


1 

IQi 


^ or 0.1 


10-2 


10 10 10^ 


100°^ 0-01 


10-3 


1 • 1 • 1 or 1 

10 10 10 103 


1000 °^ 0001 



EXAMPLE fij EvaBuating Negative Exponents 



Evaluate 10 

10-^ = -^ 

1 



10,000 
= 0.0001 



Write the fraction with a positive 
exponent in the denominator. 

Evaluate the power. 
Write the decimal form. 



In Chapter 1, you learned to write large numbers in scientific notation 
using powers often with positive exponents. In the same way, you can 
write very small numbers in scientific notation using powers of ten with 
negative exponents. 



EXAMPLE 



? 



jMiMiSMIi 



Move the decimal 
point to get a 
number that is 
greater than or 
equal to 1 and less 
than 10. 



Writing Small Numbers in Scientific Notation 

Write 0.000065 in scientific notation. 



0.000065 = 0.000065 



Move the decimal point 5 places 
to the right. 



= 6.5 X 0.00001 Write as a product of two factors. 

= 6.5 X 10"^ Write the exponential form. Since the 

decimal point was moved 5 places, 
the exponent is -5. 



76 Chapter 2 Integers and Rational Numbers 



EXAMPLE 



[ 3 J Writing Small Numbers in Standard Form 

' Write 3.4 x 10^ in standard form. 



Bi -. -i . i - v 



3.4 X 10 *^ = 0000003.4 since the exponent is -6, move tlie 



= 0.0000034 



decimal point 6 places to the left. 



When comparing numbers in scientific notation, you may need to 
compare only the powers of ten to see which value is greater. 

XAMPLE [4] Comparing Numbers Using Scientific Notation 

Compare. Write <, >, or =. 
A 3.7 X 10 " 6.1 X 10 1- 

10"''> 10~'- Compare the powers often. 

Since 10'« > lO''^, 3.7 x lO"" > 6.1 x lO-'l 

B 4.9 X 10 "^ 7.3 X 10"5 

10-5 = ^0-5 . n. 

Since the powers often are equal, compare the decimals. 
4.9 < 7.3 /. 

Since 4.9 < 7.3, 4.9 x 10"^ < 7.3 x lO'^. 




Find each value. 
1. 10-» 2. lO-'^ 



3. 10-5 



4. 10 



■10 



Write each number in scientific notation or standard form. 



6. 0.00000021 
10. 0.0009 
14. 5.8 X 10-9 
18. 2.77 X 10-' 



7. 0.00086 

11. 0.0453 

15. 4.5 X 10-5 

19. 9.06 X 10"- 



Compare. Write <, >, or =. 

22. 7.6 X 10-' 7.7 x IQ-' 

24. 2.8 X 10-'^ 2.8 x 10"^ 



8. 0.0000000066 

12. 0.0701 

16. 3.2 X 10 ' 

20. 7 X 10-"^ 

23. 8.2 X 10 ' 

25. 5.5 X 10-' 



5. 10 



-7 



9. 0.007 

13. 0.00003021 

17. 1.4 X 10" 

21. 8 X 10-" 

8.1 X 10-'^ 

2.2 X 10-5 



26. Write About It Explain the effect that a zero exponent has on a power. 

2-1 Extension 77 



LABl^ Model Integer Addition 



Use with Lesson 2-2 



KEY 

+ li = 



REMEMBER 

• Adding or subtracting zero does not 
change the value of an expression. 




You can model integer addition by using integer chips. Yellow chips 
represent positive numbers and red chips represent negative numbers. 



Activity 



Wlien you model adding numbers with the same sign, you can count 
the total number of chips to find the sum. 




The total number of 
positive cliips is 7. 




Tlie total number of 
negative chips is 7. 



3 + 4 = 7 



O Use integer chips to find each sum. 
a. 2 + 4 b. -2 + (-4) 



-3 + (-4) = -7 



c. 6 + 3 



d. -5 + (-4) 



Wlien you model adding numbers with different signs, you cannot count 
the chips to find their sum. 

J + _.! = 2 and ^ + H = -2 

but ) + |B = A red chip and a yellow chip make a neutral pair. 



When you model adding a positive and a negative number, you need to 
remove all of the neutral pairs that you can find — that is, all pairs of 1 red 
chip and 1 yellow chip. These pairs have a value of zero, so they do not 
affect the sum. 



78 Chapter 2 Integers and Rational Numbers 




You cannot just count the colored chips 
to find their sum. 

3+ (-4) = 
Before you count the chips, you need to remove all of the zero pairs. 



When you remove the zero pairs, 
there is one red chio left. 
So the sum of t: s -1. 




3 + (-4) = -l 

Q Use integer chips to find each sum. 
a. 4 + (-6) b. -5 + 2 



Think and Discuss 



c. 7 + (-3) 



-6 + 3 



1. Will 8 + (-3) and -3 + 8 give the same answer? Why or why not? 

2. If you have more red chips than yellow chips in a group, is tiie sum of the 
chips positive or negative? 

3. If you have more yellow chips than red chips in a group, is the sum of the 
chips positive or negative? 

4. Make a Conjecture Make a conjecture for the sign of the answer when 
negative and positive integers are added. Give examples. 



Try This 



Use integer chips to find each sum. 

1.4 + (-7) 2. -5 + (-4) 

Write the addition problems modeled below. 



3. -5 + 1 





4. 6+ (-4) 





2-2 Hands-On Lab 79 



2-2 


\ 


1 ; 

Adding Integer^^^^^^^^l 




/ 






1 


■^■K"""^ 


PP' ^ 


- J. 




Vv_ 




The math team wanted 
to raise money for a trip 
to Washington, D.C. They 
began by estimating their 
income and expenses. 

hicome items are positive, 
and expenses are negative. 
By adding all your income 


■ 


Ml 






i--^ 


SSIHIEimJ 










■1 tSi^S^"" 


'^^^^sHBH 




WK/tm^MSStana^A^^ 


m 




lI3iiJiStS 






Chf> Ledger 




and expenses, you can find 






Estimated Xncome ay\d Expenses 






your total earnmgs or losses. 




'Description \ Pimount 








Car \^csln supplies 


-$P'^nn 






One way to add integers is by 
using a number line. 


Car i^ash earnin^is ■ 


^ J3_0Q,QP 






Ihake sale supplies 


-$?0.00 






fiake sale ecjrr^incis 


f2?0.00 





: EXAMPLE 

Interactivities Online ► 



ModeEing Integer Addition 
Use a number line to find each sum. 
-3 + (-6) 

+ (-6) 



'SCt. 



1^- 

H — \ — \ — \ \ h 



-+• 



H — \ — \ — h 



-9 -8 -7 -6 -5 -4 -3 -2 -1 

-3 + (-6) = -9 



1 



Start at 0. Move left 
3 units. Then move left 
6 more units. 



"B 4 + (-7) 



+ {-7) 



-H — \ — \ — \ — \ — \ — \ — I — \ — \ — h 
-5-4-3-2-1 1 2 3 4 5 

4+ (-7) = -3 



Start at 0. Move right 
4 units. Then move 
left 7 units. 



You can also use absolute value to add integers. 



InteQiis 



To add two integers with the same sign, find the sum of their 
absolute values. Use the sign of the two integers. 

To add two integers with different signs, find the difference of their 
absolute values. Use the sign of the integer with the greater absolute 
value. 



80 Chapter 2 Integers and Rational Numbers 



VjiJaiJ Lesson Tutorials Online mv.hrw.com 



EXAMPLE [2J Adding Integers Using Absolute Values 






:j;J jj JJljii' 



When adding 
integers, thinl<: If the 
signs are the same, 
find the sum. 
If the signs are 
different, find the 
difference. 



Find each sum. 

A -7+ (-4) 

The signs are the same. Find the sum 
of the absolute values. 

-7 + (-4) Thinl<: 7 + 4= 11. 

— 1 1 Use the sign of tlie 

two integers. 

B -8 + 6 

The signs are different. Find the difference 

of the absolute values. 

-8 + 6 r^/n/c; 8-6 = 2. 

—2 Use the sign of the 

integer with the 
greater absolute value. 



J J J -J -J J, 



\ 



EXAMPLE [3J Evaluating Expressions with Integers 



Evaluate a + fa for a = 6 and b = -10. 

a + b 

6 + (-10) Substitute 6 for a and -10 for b. 

The signs are different. Think: 10 - 6 = 4. 
— 4 Use the sign of the integer with the greater 

absolute value (negative). 



EXAMPLE 



C3 



Banking Application 

The math team's income from a car wash was S300, including tips. 
Supply expenses were $25. Use integer addition to find the team's 
total profit or loss. 

300 + (—25) Use negative for the expenses. 

300 — 25 Find the difference of the absolute values. 

275 The answer is positive. 

The team earned $275. 



^^^^^^^^^^^^^^^^^^^^^^^^^^^^^B 




Tftink and Discuss 








1. 


Explain wrhether -7 + 2 is the 


same as 7 + (- 


-2). 




2. 


Use the Commutative Property 
equivalent to 3 + (-5). 


to write an expression 


that is 



'Mb'j Lesson Tutorials Online mv.hrw.com 



2-2 Adding Integers 81 




^festitij^' 




'^Jj^JS3i 




Homework Help Online go.hrw.com, 



keyword MMaiilMM ® 
Exercises 1-32, 33, 37, 39, 43, 
47,49,51 



GUIDED^RRAvCTICE 



See Example 1 Use a number line to find each sum. 

.1. 9 + 3 2.-4 + (-2) 



3. 7 + (-9) 



See Example 2 Find each sum. 

L 5. 7 + 8 



6. -1 + (-12) 



7. -25 + 10 



4. -3 + 6 



8. 31 + (-20) 



See Example 3 Evaluate a + [; for the given values. 

i 9. (7 = 5,/7= -17 ^0.^^8.b=-i 



11. a= -4.b= -16 



See Example 4 12. Sports A football team gains 8 yards on one play and then loses 13 yards 
L on the next. Use integer addition to find the team's total yardage. 



'MPiEitai.^LgMQ:'CE 



See Example 1 Use a number line to find each sum. 

i 13. -16 + 7 14. -5 + (-1) 



[ 17. 10 + (-3) 

See Example 2 Find each sum. 
21. -13 + (-6) 

! 25. -81 + (-7) 



18. 



■20 + 2 



22. 14 + 25 
26. 28 + (-3) 



15. 4 + 9 

19. -12 + (-5) 

23. -22 + 6 

27. -70+ 15 



16. -7 + 8 

20. -9 + 6 

24. 35+ (-50) 

28. -18 + (-62) 



See Example 3 Evaluate c + rf for the given values. 

! 29. r = 6, f/ = -20 30. r = -8, ff = -21 



31. f = -45, ff = 32 



See Example 4 32. The temperature dropped 17 °F in 6 hours. The final temperature was —3 °F. 
I Use integer addition to find the starting temperature. 



Extra Practice 



See page EP5. 



PRACTICE AND PROBLEM SOLVING 



Find each sum. 

33. -8 + (-5) 

36. -22 + (-18) +22 



34. 14+ (-7) 

37. 27+ (-29) + 16 



35. -41 + 15 

38. -30 + 71 + (-70) 



Compare. Write <, >, or =. 

39.-23+18 -41 40. 59 + (-59) 41. 31 + (-20) ^- 9 



42. 



-24+ (-24) 



48 



43. 25 + (-70) -95 



44. 16 + (-40) 



-24 



45. Personal Finance Cody made deposits of $45, $18, and $27 into his 
checking account. He then wrote checks for $21 and $93. Write an 
expression to show tlie change in Cody's account. Then simplify tlie 
expression. 



82 Chapter 2 Integers and Rational Numbers 



-« Q*^ ' . Evaluate each expression for iv - — 12, .v = 10, and y - —7. 

\*ixi^ 46. 7 + y 47. -4 + w 48. w + y 49. .v + y 



50. R' + A- 




The Appalachian 
Trail extends 
about 2,160 
miles from Maine 
to Georgia. It 
takes about 5 to 
7 months to hike 
the entire trail. 



Recreation Hikers along the Appalachian Trail camped overnight at Horns 
Pond, at an elevation of 3,100 ft. Then they hiked along the ridge of the 
Bigelow Mountains to West Peak, which is one of Maine's highest peaks. 
Use the diagram to determine the elevation of West Peak. 



Bigelow Mountain Range, Maine 




52. Multi-Step Hector and Luis are playing a game. In the game, each player 
starts with points, and the player with the most points at the end wins. 
Hector gains 5 points, loses 3, loses 2, and then gains 3. Luis loses 5 points, 
gains 1, gains 5, and then loses 3. Determine the final scores by modeling the 
problem on a number line. Then tell who wins the game and by how much. 

P 53. What's the Question? The temperature was -8 °F at 6 a.m. and rose 15 °F 
by 9 A.M. The answer is 7 °F. Wliat is the question? 

•'_ 54. Write About It Compare the method used to add integers with the same 
sign and the method used to add integers with different signs. 



^ 55. Challenge A business had losses of $225 million, $75 million, and $375 
million and profits of $15 million and $125 million. How much was its 
overall profit or loss? 



i 



Test Prep and Spiral Review 



56. Multiple Choice Which expression is represented by the model? 

CS) -4 + (-1) CS:) -4 + 3 

CD -4 + CE)-4 + 4 

57. Multiple Choice Which expression has the greatest value? 
CD -4 + 8 CD -2 + (-3) CH) 1 + 2 



+ 3 



Simplify each expression. (Lesson 

58. 2 + 5 • 2 - 3 59. 3^ - (6 • 4) + 1 



1-4) 



60. 30 - 5 • (3 + 



Compare. Write <, >, or =. (Lesson 2-1) 

62. -14 -12 63. |-4| 3 



64. -6 



-4 



H \ \ — \ — h 



-5-4-3-2-1 



CD 4+ (-6) 



61. 15 -3 -2- + 1 



65. -9 



■11 



2-2 Adding Integers 83 



\ - ' ■ " 

LABIV^ Model Integer Subtraction 



Use with Lesson 2-3 

,.■■■' 


KEV 


REMEMBER 


!'•) = 1 


• Adding or subtracting zero does not 


# = -1 


change the value of an expression. 


o-«=» 





>^*pLearn It Online 

*** LabResourcesOnllnego.hrw.com, 

■a«M510Lab2 ga 



You can model integer subtraction by using integer chips. 



Activity 



These groups of chips sliow three different ways of modeling 2. 






Q Show two other ways of modeling 2. 



These groups of chips show two different ways of modeling -2. 





Q Show two other ways of modeling -2. 



You can model subtraction problems involving two integers with the 
same sign by taking away chips. 





8-3 = 5 
O Use integer chips to find each difference. 

a. 6- 5 b. -6- (-5) c. 10-7 



-8- (-3) = -5 



d. -7- (-4) 



84 Chapter 2 Integers and Rational Numbers 



To model subtraction problems involving two integers with different 
signs, such as -6 - 3, you will need to add zero pairs before you can 
take chips away. 



Use 6 red chips to represent -6. 




Since you cannot tafee away 

3 yellow ct)ips, ac' ' ~ 

ctiips paired wlti^ _ , ^„ .,..,,, 



OS. 



-6-3 = -9 
O Use integer chips to find each difference. 

a. -6 - 5 b. 5 - (-6) c. 4 - 7 



d. 



(-3) 



Think and Discuss 



1. How could you model the expression — 5? 

2. When you add zero pairs to model subtraction using chips, does it 
matter how many zero pairs you add? 

3. Would 2-3 have the same answer as 3 - 2? Wliy or why not? 

4. Make a Conjecture Make a conjecture for the sign of the answer when 
a positive integer is subtracted from a negative integer. Give examples. 



Try This 



Use integer chips to find each difference. 

1. 4 -2 2. -4 - (-2) 

4. 3 - 4 5. 2 - 3 

7. 5 -3 8. -3 - (-5) 



3. -2- (-3) 
6. - 3 
9. 6 - (-4) 



2-3 Hands-On Lab 85 



Subtracting 
J Integers 



During flight, the space shuttle may 
be exposed to temperatures as low as 
-250 "F and as high as 3,000 °F. 

To find the difference in these 
temperatures, you need to know how 
to subtract integers with different signs. 

You can model tlie difference between 
two integers using a number line. When 
you subtract a positive number, the 
difference is less than the original 
number, so you move to the left. To 
subtract a negative number, move to 
the rigln. 




(EXAMPLE 



[ 1 J Modeling integer Subtraction 

Use a number line to find each difference. 



If the number being 
subtracted is less 
than the number it 
is subtracted from, 
the answer will be 
positive. If the 
number being 
subtracted is greater, 
the answer will be 
negative. 



A 3-8 



-8 



h 



H — \ — \ — I — \ — I- 



H — \ — \ — h 



-6 -5-4-3-2-1 1 2 3 4 

3 - 8 = -5 



B -4-2 

, -2 



-4 



H 



H — \ — h- 



H — \ — \ — \ — \ — I — h 



-6-5-4-3-2-1 12 3 4 

-4 - 2 = -6 



Start at 0. 
Move right 3 units. 
To subtract 8, 
move to the left. 



Start atO. 
Move left 4 units. 
To subtract 2, 
move to the left. 



C 2 - (-3) 



h 



H- 



•(-3) 



H — I — I — \ — \ — \ — \ — h 



-3-2-101234567 

2 - (-3) =5 



Start atO. 
Move right 2 units. 
To subtract -3, 
move to the right. 



Addition and subtraction are inverse operations — they "undo" each 
other. Instead of subtracting a number, you can add its opposite. 



86 Chapter 2 Integers and Rational Numbers 



'Mb'j Lesson Tutorials OnlinE mv.hrw.com 



EXAMPLE [2] Subtracting Integers by Adding the Opposite 



Interactivities Online ► 



Find each difference. 

A 5 -9 

5-9 = 5 + (-9) 

= -4 

B -9 -(-2) 

-9 -(-2) = -9 + 2 

C -4-3 

-4-3= -4 + (-3) 



Add the opposite of 9. 



Add the opposite of -2. 



Add the opposite of 3. 



EXAMPLE [3 1 Evaluating Expressions with Integers 



Evaluate a - b for each set of values. 

A a= -6,b = 7 

a-b 

— 6-7 = — 6 + (-7) Substitute for a and b. Add the opposite 

= -13 of 7. 



B rt = 14, /;= -9 

a-b 

14 - (-9) = 14 + 9 
= 23 



Substitute for a and b. Add the opposite 
of -9. 



EXAMPLE 



[*) 



Temperature Application 

Find the difference between 3,000 °F and -250 °F, the tempera- 
tures the space shuttle must endure. 

3,000- (-250) 

3,000 + 250 = 3,250 Add the opposite of -250 

The difference in temperatures the shuttle must endure is 3,250 °F. 



Think and Discuss 

1. Suppose you subtract one negative integer from another. Will your 
answer be greater than or less than the number you started with? 

2. Tell whether you can reverse the order of integers when 
subtracting and still get the same answer. Why or why not? 



^Mb'j] Lesson Tutorials Online my.hrw.com 



2-3 Subtracting Integers 87 



2-3 




GUIDED PRACTICE 



See Example 1 Use a number line to find each difference. 

L 1.4-7 2.-6-5 3. 2 -(-4) 4. -8 - (-2) 



See Example 2 Find each difference. 

L 5.6-10 6. 



-3 - (-8) 



7. -1 - 



8. -12- (-2) 



See Example 3 Evaluate a - bfor each set of values. 

L 9.a = 5,b=-2 10. <7 = -8, /; = 6 



11. a = 4,b= 18 



See Example 4 12. In 1980, in Great Falls, Montana, the temperature rose from -32 °F to 15 °F 
L in seven minutes. How much did the temperature increase? 



INDEPENDENI.ERACTICE 



See Example 1 Use a number line to find each difference. 

13.7-12 14. -5 -(-9) 15. 2 -(-6) 

^ 17. 9 -(-3) 18. -4-10 19. 8 -(-8) 

See Example 2 Find each difference. 

21. -22 - (-5) 22.-4-21 23.27-19 

L 25. 30 -(-20) 26.-15-15 27. 12 - (-6) 



16. 7- (-8) 

20. -3 - (-3) 

24. -10- (-7) 

28. -31 - 15 



See Example 3 Evaluate a - bfor each set of values. 

29. (7 = 9, ^= -7 30.a=-l\,b = 2 3A.a = -2,b = 3 

32. a ^8. b= 19 33. a = -10. b = 10 34. a = -4, b = -15 

See Example 4 35. In 1918, in Granville, North Dakota, the temperature rose from -33 °F to 
50 °F in 12 hours. How much did the temperature increase? 





. fKAUIK-t ANU fKUIS 


Ltm M 










Simplify. 








[Extra Practice J 




See page EPS. 


36. 2-8 


37. 


-5-9 


38. 


15-12-8 




39. 6+ (-5) -3 


40. 


1 - 8 + (-6) 


41. 


4- (-7) -9 




42. (2 - 3) - (5 - 6) 


43. 


5- (-8) - (-3) 


44. 


10-12 + 2 



Evaluate each expression for m = -5, u = 8, and p = —14. 

45. Ill - 11 + p 46. /; - 111 - p 47. p - m - ii 48. Jii + n - p 

49. Patterns Find the next three numbers in the pattern 7, 3, —1, -5, -9, . . . 
Then describe tlie pattern. 



88 Chapter 2 Integers and Rational Numbers 



QL 



Astronomy 



50. The temperature of Mercury can be as high as 873 °F. 
The temperature of Pluto is about -393 °F. What is the 
difference between these temperatures? 

51. One side of Mercury always faces the Sun. 
The temperature on this side can reach 873 °F. 
The temperature on the other side can be as low as 
—361 °F. What is the difference between the two 
temperatures? 

52. Earth's moon rotates relative to the Sun about once a 
month. The side facing the Sun at a given time can be 
as hot as 224 °F. The side away from the Sun can be 
as cold as -307 °F. What is the difference between 
these temperatures? 

53. The highest recorded temperature on Earth is 
136 °F. The lowest is - 129 °F. What is the 
difference between these temperatures? 

Use the graph for Exercises 54 and 55. 

54. How much deeper is the deepest canyon on 
Mars than the deepest canyon on Venus? 

55. ^^ Challenge Wliat is the difference 
between Earth's highest mountain and its 
deepest ocean canyon? What is the difference 
between Mars' highest mountain and its 
deepest canyon? Which difference is greater? 
How much greater is it? 




Temperatures in the Sun range from about 5,500 °C at 
its surface to more than 1 5 million °C at its core. 



Highest and Lowest Points on 
Venus, Earth, and IVIars 



Q. 



80.000 
70.000 
50.000 
50.000 
40.000 
30.000 
20.000 
10.000 

10,000 
20,000 
30,000 
40,000 
-50,000 



Highest 
Points 



70,000 



35,000 



29,035 



-9,500 



Lowest 
Points 




36,198 
Venus Earth 



26,000 



Mars 




Test Prep and Spiral Review 



56. IVIultiple Choice Wliich expression does NOT have a value of —3? 

C^ -2 - 1 cX) 10 - 13 CD 5 - (-8) CE) -4 - (-1) 

57. Extended Response \f in = -2 and /; = 4, which expression has the 
least absolute value; )ii + n, n - in, or m — /;? Explain your answer. 

Evaluate each expression for the given values of the variables. (Lessors 1 5) 

58. 3.V - 5 for x = 2 59. 2ir + n for n = 1 60. 4y" - 3.y for y = 2 

61. 4a + 7 for a = 3 62. .v^ + 9 for .v = 1 63. 5;: + z'' for ^ = 3 

64. Sports In three plays, a football team gained 10 yards, lost 22 yards, 
and gained 15 yards. Use integer addition to find the team's total yardage 
for the three plays. (Lesson 2-2) 



2-3 Subtracting Integers 89 



Model Integer 
Multiplication and Division 



Use with Lesson 2-4 



KEY 



1 = 1 

;+ H) = 



REMEMBER 

• The Commutative Property states that 
two numbers can be multiplied in any 
order without changing the product. 

• Multiplication is repeated addition. 

• Multiplication and division are Inverse 
operations. 



,^5^ Learn It Online 

*J* LabResourcesOnlinego.hrw.com, 

■yj.ii.ii.i ivMii Lab2 m^o^ 



You can model integer multiplication and division by using integer chips. 



Activity 1 



Use integer chips to model 3 • (-5). 




Think: 3 • ( — 5) means 3 groups of —5. 

Arrange 3 groups of 5 red chips. 
There are a total of 15 red chips. 



3 • (-5) = -15 



O Use integer chips to find each product. 
a. 2 • (-2) b. 3 • (-6) 



c. 5 • (-4) 



d. 6- (-3) 



Use integer chips to model -4 • 2. 

Using the Commutative Property, you can write -4 • 2 as 2 • (-4). 

Thinl<: 2 • (—4) means 2 groups of —4. 

Arrange 2 groups of 4 red chips. 
There are a total of 8 red chips. 

-4-2= -8 




Q Use integer chips to find each product. 

a. -6- 5 b. -4 • 6 c. -3 • 4 



d. -2-3 



90 Chapter 2 Integers and Rational Numbers 



■ "I'.^'tn'^n^W;^ WTF ."■" 



Think and Discuss 



1 . What is the sign of the product when you multiply two positive numbers? 
a negative and a positive number? two negative numbers? 

2. If 12 were the answer to a multiplication problem, list all of the 
possible factors that are integers. 



Try Tiiis 



Use integer chips to find each product. 
1. 4- (-5) 2. -3-2 



3. 1 • (-6) 



4. -5 • 2 



5. On days that Kathy has swimming lessons, she spends $2.00 of her 
allowance on snacks. Last week, Kathy had swimming lessons on 
Monday, Wednesday, and Friday. How much of her allowance did 
Kathy spend on snacks last week? Use integer chips to model the 
situation and solve the problem. 



Activity 2 



Use integer chips to model -15 h- 3. 




Think: -15 is separated into 3 e^ 
Arrange 15 red rh;^^ Jntn ? e^miAi nroups. 
There are 5 reu L////ji m eaLn yiuup. 



-15 H- 3 = -5 

O Use integer chips to find each quotient. 
a. -20 H- 5 b. -18 ^ 6 



c. -12 H- 4 



d. -24 



Thinic and Discuss 



1. What is the sign of the answer when you divide two negative 
integers? a negative integer by a positive integer? a positive integer 
by a negative integer? 

2. How are multiplication and division of integers related? 



Try This 



Use integer chips to find each quotient. 

1. -21 H- 7 2. -12 H- 4 



-:- 9 



5. Ty spent $18 of his allowance at the arcade. He hit baseballs, played 
pinball, and played video games. Each of these activities cost the 
same amount at the arcade. How much did each activity cost? Use 
integer chips to model the situation and solve the problem. 



4. -10 H- 5 



2-4 Hands-On Lab 91 



2-4 



Multiplying and Dividin 
Integers — 



7.1.7 Solve problems that involve multiplication and division with integers, fractions, decimals 
and combinations of the tour operations. 

You can think of multiplication as 
repeated addition. 

3-2 = 2 + 2 + 2 = 6 

3 • (-2) = (-2) + (-2) + (-2) = -6 



EXAMPLE 

Interactivities Online ► 



[ 1 J Multiplying Integers Using Repeated Addition 

Use a number line to find each product. 







Remember' 



Multiplication and 
division are inverse 
operations. They 
"undo" each other. 
Notice how these 
operations undo 
each other in the 
patterns shown. 



A^ 3 • (-3) 

+ (-3) +(-3) +(-3) 

■ I I I I I ■ ^^'"'^- ^^^'^ ^^ 0. 

-10-9 -8 -7 -6 -5 -4 -3 -2-1 1 Add -3 three times. 

3- (-3) = -9 



B -4-2 

-4 -2 = 2 ■ (-4) 



| . +'-4) |. +(-4) 



H — \ — \ — h 



H — I — \ \ \ — h 



Use the Commutative 
Property. 

Thinl<: Start at 0. 



-10 -9 -8 -7 -6 -5 -4 -3 -2-1 1 Add -4 two times. 
-4 -2= -8 



The patterns below suggest that when the signs of two integers are 
different, their product or quotient is negative. The patterns also 
suggest that the product or quotient of two negative integers is positive. 



-3- 


2 = -6 


-3- 


1 = -3 


-3- 


0=0 


-3- 


(-1) = 3 


-3- 


(-2) = 6 



-6 + 




-3) = 


2 


-3 + 




-3) = 


1 


+ 




-3) = 





3 + 




-3) = 


-1 


6 + 




-3) = 


-2 



Multiplying and Dividing Two Inte 



If the signs are: 

the same 
different 



Your answer will be: 

positive 
negative 



92 Chapter 2 Integers and Rational Numbers \ y'ni^u] Lesson Tutorials Online my.hrw.com 



EXAMPLE [?] Multiplying Integ 

Find each product 



ers 



-4- (-2) 




B 


-3-6 


-4- (-2) 
8 


Both signs are 
negative, so the 
product is positive. 




— 3*6 The signs are 

different, so the 
product is negative 



EXAMPLE 



(B 



Dividing Integers 

Find each quotient. 



A 72 ^(-9) 



I he signs are 



B 



72 -=- (-9) different, so the 
— 8 quotient is negative. 



lUU . ( 5) The signs are the 
-100 -=- (-5) same, so the 

20 quotient is positive. 



Zero divided by any number is zero, but you cannot find an answer for 
division by zero. For example, -6 -h 9^: 0, because • ?i: -6. We say 
that division by zero is undefined. 



EXAMPLE [Sj Sports Application 



A football team must move the ball forward at least 10 yards 
from its starting point to make a first down. If the team has 
2 losses of 3 yards each and a gain of 14 yards, does the team 
make a first down? 

Add the total loss to the gain to find how far the ball moved forward. 
2 • (-3) + 14 Multiply -3 by 2 to find the total loss; 

then add the gain of 14. 
— 6 + 14 Use the order of operations. Multiply first. 

8 Then add. 

The team moved the ball forward 8 yards, so it did not make a 
first down. 



Think and Discuss 

1. List at least four different multiplication examples that have 24 as 
their product. Use both positive and negative integers. 

2. Explain why the rules for multiplying integers make sense. 



Mh'j Lesson Tutorials Online mv.hrw.com 



2-4 Multiplying and Dividing Integers 93 



2-4 



[•Li I hi 

Homework Help Online go.hrw.com, 



keyword ■MMMKaM @ 
Exercisesl-34,35,37,39,41, 
43,45,47 



GUIDED PRACTICE 



See Example 1 Use a number line to find each product. 

_ 1. 5- (-3) 2. 5- (-2) 3. 



-3-5 



See Example 2 Find each product. 

L 5.-5- (-3) 



6. -2-5 



See Example 3 Find each quotient. 

9. 32 H- (-4) 10. -18 H-3 



13. -63^ (-9) 



14. -50 ^ 10 



7. 3 • (-5) 

11. -20 H- (-5) 
15. 63 H- 



4. -4-6 

8. -7- (-4) 

12. 49 H- (-7) 

16. -45 -f (-5) 



See Example 4 17. Angelina hiked along a 2,250-foot mountain trail. She stopped 5 times 
along the way to rest, walking the same distance between each stop. 
L How far did Angelina hike before the first stop? 



INDEPENDENT PRACTICE 



See Example 1 Use a number line to find each product. 

18. 2- (-1) 19.-5-2 20.-4-2 21. 3 - (-4) 



See Example 2 Find each product. 

L 22. 4 • (-6) 23. -6- (-8) 



24. -8 - 4 



See Example 3 Find each quotient. 

26. 48 ^(-6) 27. -35 ^(-5) 28.-16-4-4 



30. -42 ^ 



31. 81 ^ (-9) 



32. -77^ 11 



25. -5- (-7) 

29. -64 ^ 8 
33. 27 H- (-3) 



See Example 4 34. A scuba diver descended below the ocean's surface in 35-foot intervals as 
he examined a coral reef. He dove to a total depth of 140 feet. In how 
L many intervals did the diver make his descent? 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EPS. 



Find each product or quotient. 

35.-4-10 36. -3 H- 37. -45 h- 15 38. -3-4- (-1) 



39. 



-500 H- (-10) 40. 5- (-4) - (-2) 41. 225 -^ (-75) 42. -=- (-3) 



Evaluate each expression for a = —5, b = 6, and c = — 12. 

43. -2c -I- b 44. 4a - b 45. ab + c 



46. ac -^ b 



47. Earth Science A scuba diver is swimming at a depth of - 12 feet in the 
Flower Garden Banks National Marine Sanctuaiy. She dives dov^i to a coral 
reef that is at five times this depth. What is the depth of the coral reef? 



94 Chapter 2 Integers and Rational Numbers 



Simplify each expression. Justify your steps using the Commutative, 
Associative, and Distributive Properties wlien necessary. 

48. (-3)2 49. -(-2 + 1) 50. 8 + (-5)' + 7 51. (-!)'• (9 + 3) 

52. 29 -(-7) -3 53. -4- 14- (-25) 54. 25 - (-2) • 4' 55. 8 -(6 +(-2)) 

56. Earth Science The table shows 
the depths of major caves in the 
United States. Approximately how 
many times deeper is Jewel Cave 
than Kartchner (taverns? 

Personal Finance Does each person 
end up with more or less money than 
he started with? By how much? 

57. Kevin spends $24 a day for 3 days. 

58. Devin earns $15 a day for 5 days. 



Depths of Major U.S. 


Caves 


Cave 


Depth (ft) 




"T 1 


Carlsbad Caverns 


-1,022 


11 


Caverns of Sonora 


-150 


11 


Ellison's Cave 


-1,000 




J, J 


Jewel Cave 


-696 KPV 


Kartchner Caverns 


-137 




'r. 


Mammoth Cave 


-379 





Source NSS U S A Long Cave List 



59. Evan spends $20 a day for 3 days. Then he earns $18 a day for 4 days. 

^ 60. What's the Error? A student writes, "The quotient of an integer divided 
by an integer of the opposite sign has the sign of the integer with the 
greater absolute value." Wliat is the student's error? 

61. Write About It Explain how to find the product and the c]uotient of 
two integers. 

• (-1) -4 -2 • (-3) and 



(^ 62. Challenge Use > or < to compare 
-1 + (-2) + 4 + (-25) + (-10). 



i 



Test Prep and Spiral Review 



63. Multiple Choice Which of the expressions are ec]ual to -20? 

1-2-10 II -40 + (-2) III -5 -(-2)' IV -4-2-12 

(X) I only CE) I and II (E;) I. Ill, and IV CE:) I, II, III, IV 

64. Multiple Choice Which expression has a value that is greater than the 
value of -25 + (-5)? 



CD 36 + (-6) 



CD -100 + 10 



CE;. -50 + (-10) 



CD -45 + (-5) 



Write each phrase as an algebraic expression. (Lesson 1 -7) 

65. the sum of a number and 6 66. the product of -3 and a number 

67. 4 less than twice a number 68. 5 more than a number divided by 3 



Find each difference. (Lesson 2-3) 
69. 3 - (-2) 70. -5-6 



71. 6-8 



72. 2- (-7) 



2-4 Multiplying and Dividing Integers 95 



\'\ ' 



Model Integer Equations 



Use with Lesson 2-5 



KEY 

'f *! = 1 

B=-i 






REMEMBER 

• Adding or subtracting zero does not 
change the value of an expression. 






g + O = 

You can use algebra tiles to model and solve equations. 



£?. 



Learn It Online 

Lab Resources Online go.hrw.com, 

IBlijMblULab^mr 



Activity 



To solve the equation .v + 2 = 3, you need to get x alone on one side of the 
equal sign. You can add or remove tiles as long as you add the same amount 
or remove the same amount on both sides. 








X + 2 = 3 



Remove 2 from each side. 



X = 1 



O Use algebra tiles to model and solve each equation. 

a. X + 3 = 5 b. .Y + 4 = 9 c. .v + 5 = 8 

The equation x + 6 = 4 is more difficult to model because there are not 
enough tiles on the right side of the mat to remove 6 from each side. 




X + 6 = 4 



Add -6 to each side. 













1 


'BB 

B B 

_ BB 




:j ,^j B B 

,_J .:! B B 






/BB 






/ 


i 





'J T 

Remove zero pairs from eact) side. 




d. X + 6 = 6 




96 Chapter 2 Integers and Rational Numbers 



Q Use algebra tiles to model and solve each equation. 

a. .V + 5 = 3 b. .V + 4 = 2 c. .v + 7 = -3 

When modeling an equation that involves subtraction, such as .v - 6 = 2, 
you must first rewrite the equation as an addition equation. For example, 
the equation .r — 6 = 2 can be rewritten as .v + (- 6) = 2. 

Modeling equations that involve addition of negative numbers is similar to 
modeling equations that involve addition of positive numbers. 



d. .v + 6 = -2 




X + (-6) = 



^ 









J 


BB 
BB 
BB 











-6+6 = 
Remove zero pairs. 



Add 6 to er 







> 






BB 
BB 
BB 




} 


J 






^ 




4 






Q Use algebra tiles to model and solve each equation, 
a. .V - 4 = 3 b. .V - 2 = 8 c. .v - 5 = 



d. x-7 = 



Think and Discuss 



1. When you remove tiles, what operation are you modeling? When 
you add tiles, what operation are you modeling? 

2. How can you use the original model to check your solution? 

3. To model .v - 6 = 2, you must rewrite the equation as x + ( - 6) = 2. 
Why are you allowed to do this? 



Try This 



Use algebra tiles to model and solve each equation. 

1. .V + 7 = 10 2. .V - 5 = -8 3. x + (- 5) = -4 4. .v - 2 = 1 

5. A- + 4 = 8 6. .V + 3 = -2 7. x + (- 1) = 9 8. x - 7 = -6 



2-5 Hands-On Lab 97 



B 



d-' 



Containing Integers 



4 



7.2.1 Use variables and appropriate operations to write an expression, equation or 
inequality that represents a verbal description 

Recall that the sum of a number and its opposite is 0. This is called tlie 
Inverse Property of Addition. 



Inverse Property of Addition 




Words 


Numbers 


Algebra 


The sum of a number and its 
opposite, or additive inverse, is 0. 


3 + (-3) = 


a + (-a) = 



You can use the Inverse Property of Addition to solve addition and 
subtraction equations that contain integers, such as -3 + i' = -5. 



EXAMPLE M 


11 Solving Addition and Subtraction Equations 


Interactivities Online ► 


Solve each equation. Check 
A -3 + y= -5 


> your answer. 




-3+3'= -5 


Use the Inverse Property of Addition 




+ 3 +3 

y ^ 9 


Add 3 to both sides. 




Check -3 + )' = -5 






-3+ (-2) = -5 


Substitute -2 for y. 




-5 = -5i/ 


True. 




B » + 3 = -10 






n + 3 = -10 


Use the Inverse Property of Addition 




+ (-3) + (-3) 


Add -3 to both sides. 




/; = -13 






Check 7? + 3 = -10 






-13 + 3= -10 


Substitute - 13 for n. 




-10= -10*/ 


True. 




C .V - 8 = -32 






A- - 8 = -32 


Use the Inverse Property of Addition 




+ 8 +8 


Add 8 to both sides. 




X = -24 






Check .V - 8 = -32 






-24- 8= -32 


Substitute -24 for x. 




-32= -32 • 


True. 



98 Chapter 2 Integers and Rational Numbers [VjJaijl Lesson Tutorials Online my.hrw.com 



In Chapter 1, you used inverse operations to solve multiplication 
and division equations. You can also use inverse operations to solve 
multiplication and division equations that contain integers. 



EXAMPLE 



m 



Solving Multiplication and Division Equations 

Solve each equation. Check your answer. 

A 



-3 ^ 

(-3)(^)= (-3)9 

a = -27 
Check ^3 = 9 

-27 ? 



-3 



9=9%/ 



MultifDly both sides by -3. 



Substitute -27 for a. 
True. 



-120 = 6a- 




-120 _ 6.V 
6 6 


Divide both sides by 


-20 = A- 




Check -120 = 6a- 




-120 = 6(-20) 
-120= -120i/ 


Substitute -20 for x 


True. 



EXAMPLE fij Business Application 



A shoe manufacturer made a profit of S800 million. This amount 
is S200 million more than last year's profit. What was last 
year's profit? 

Let p represent last year's profit (in millions of dollars). 

This year's profit is $200 million more than last year's profit. 



800 

800 = 200 + p 
200 - 200 
600= p 



200 



+ 



Last year's profit was $600 million. 



Think and Discuss 

1. Tell what value of ii makes — /z + 32 equal to zero. 

2. Explain why you would or would not multiply both sides of an 
equation by to solve it. 



yidijul Lesson Tutorials OnliriE my.hrw.com 2-5 Solving Equations Containing Integers 99 



2-5 



iJ 



GUIDED PRACTICE 



1 HomeworkHelpOnlinego.hrw.com, 



keyword ■MWllKBiM ® 
Exercises 1-20, 23, 25, 31, 33, 
35,37,43 



See Example 1 
See Example 2 



Solve each equation. Check your answer. 

1. w-6= -2 2. x + 5 = -7 



4. ^ = 2 
-4 



-240 = 8v 



3. A = -18 + 11 
6. -5a = 300 



See Example 3 7. Business Last year, a chain of electronics stores had a loss of $45 million. 
This year the loss is $12 million more than last year's loss. What is this 
year's loss? 



See Example 1 



INDIBEMD 



Solve each equation. Check your answer. 

8. /)- 7 = -16 9. A-+ 6 = 3 



L 11. r+ 14 = 10 

See Example 2 14. 9c = -99 

I 17. -^= -30 

1 — b 



12. r + 8 = -20 

15 - = -4 

18. 200 = -25p 



10. s + 2 = -4 
13. (7 - 25 = -5 
16. -16 = 2;: 
19. ^=12 



See Example 3 20. The temperature in Nome, 7\laska, was -50 "F. This was 18 °F less than 
the temperature in Anchorage, Alaska, on the same day. What was the 
L temperature in Anchorage? 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EP6. 



Solve each equation. Check your answer. 

21. 9j'= 900 

24 



-^=7 
-20 



27. llF= -121 



30. 



= 8 



238 
33. 4.x- = 2 + 14 



22. <^ - 15 = 45 
25. -85 = -5c 
28. ^ = 9 
31. -23 = .v-l-35 
34. c + r + f = 6 



23. ,/ + 56 = -7 
26. ('-39 = -16 
29. /('+ 41 = 
32. 0= -15;)? 
35. r- 3 = 4 + 2 



36. Geometry The three angles of a triangle have equal measures. The sum 
of their measures is 180°. Wliat is the measure of each angle? 

37. Sports Herb has 42 days to prepare for a cross-country race. During his 
training, he will run a total of 126 miles. If Herb runs the same distance 
every day, how many miles will he run each day? 

38. Multi-Step Jared bought one share of stock for $225. 

a. He sold the stock for a profit of $55. What was the selling price of 
the stock? 

b. The price of the stock dropped $40 the day after Jared sold it. At what 
price would Jared have sold it if he had waited until then? 



100 Chapter 2 Integers and Rational Numbers 



Translate each sentence into an equation. Then solve the equation. 

39. The sum of - 13 and a number p is 8. 

40. A number .v divided by 4 is -7. 

41. 9 less than a number t is —22. 

42. Physical Science On the Kelvin temperature scale, pure water boils at 
373 K. The difference between the boiling point and the freezing point of 
water on this scale is 100 K. Wliat is the freezing point of water? 

Recreation The graph shows the 
most popular travel destinations over 
Labor Day weekend. Use the graph 
for Exercises 43 and 44. 

43. Which destination was 5 times 
more popular than theme or 
amusement parks? 

44. According to the graph, the 
mountains were as popular as 
state or national parks and what 
other destination combined? 

45. Choose a Strategy Matthew 
(M) earns $23 less a week than his 
sister Allie (yl). Their combined 




State or 

national parks 

Theme or 

amusement parks 

Other 



Source AAA 

salaries are $93. How much does each of them earn per week? 

C£) A-$35;M;$12 CT) A- $35; M; $58 CD A $58; M; $35 

M 46. Write About It Explain how to isolate a variable in an equation. 

47. Challenge Write an equation that includes the variable p and the 
numbers 5, 3, and 31 so that the solution is p - 16. 



m 



Test Prep and Spiral Review 



48. Multiple Choice Solve -15/?; = 60. 

CK) m = -4 CD '" = 5 CE) m = 45 

49. Multiple Choice For which equation does .v = 2? 

CD -3jc = 6 CS) A- + 3 = -5 CK) X + A- = 4 



<CE) m - 75 



CD f =-8 



Identify a possible pattern. Use the pattern to write the next three numbers. (Lesson 1-1 ) 
50.26,21,16,11,6.... 51.1,2,4,8,16,... 52.1,4,3,6,5,... 



Compare. Write <, >, or 
53. -5 -8 

56. -10 I-IOI 



(Lessons 2-1, 2-2, and 2-3) 
54. 4 |-4| 

57. -7-8 -15 



55. |-7| |-9| 

58. -12 10 -f (-12) 



2-5 Solving Equations Containing Integers 101 




Ready To Go On? 



yi^ Learn It Online 



ResourcesOnlinego.hrw.com, 
■ Mll.lJjMSIimiG02ALG°l 



Quiz for Lessons 2-1 Through 2-5 

Q) 2-1 I Integers 



Compare the integers. Use < or > . 
1. 5 -8 2. -: 



3.-4 *3 



4. Use a number line to order the integers -7, 3, 6, -1,0, 5, -4, and 7 
from least to greatest. 







Use a number Une to find each absolute value. 

5. 1-231 6. |17| 

2-2 j Adding Integers 

Find each sum. 

8.-6 + 3 9. 5+ (-9) 



7. I-IOI 



ST 



Evaluate p + t for the given values. 

11. ;;= 5, r= -18 12. ^= -4, f = -13 

2-3 ] Subtracting Integers 
Find each difference. 

14. -21 - (-7) 15. 9 - (-11) 



10. -7 + (-11) 



13. p= -37, ^ = 39 



16. 6-17 



& 



17. Wlien Cai traveled from New Orleans, Louisiana, to the Ozark 

Mountains in Arkansas, the elevation changed from 7 ft below sea level 
to 2,314 ft above sea level. How much did the elevation increase? 

2-4 j Multiplying and Dividing Integers 
Find each product or quotient. 
18.-7-3 19. 30 H- (-15) 20. -5 -(-9) 



21. After reaching the top of a cliff, a rock climber descended the rock face 
using a 65 ft rope. The distance to the base of the cliff was 585 ft. How 
many rope lengths did it take the climber to complete her descent? 



er 



2-5 ] Solving Equations Containing Integers 
Solve each equation. Check your answer. 

22. 3a- = 30 23. k - 25 = 50 



24. v+ 16 = -i 



25. This year, 72 students completed projects for the science fair. This was 
23 more students than last year. How many students completed projects 
for the science fair last year? 



102 Chapter 2 Integers and Rational Numbers 



Focys on Problem Solving 





• Choose a method of computation 

Wlien you know the operation you must use and \ou know exactly 
which numbers to use, a calculator might be tiie easiest way to 
solve a problem. Sometimes, such as when the numbers are small 
or are multiples of 10, it may be quicker to use mental math. 

Sometimes, you have to write the numbers to see how they relate 
in an equation. When you are working an equation, using a pencil 
and paper is the simplest method to use because you can see each 
step as you go. 



For each problem, tell whether you would use a calculator, mental 
math, or pencil and paper to solve it. Explain your answer. Then 
solve the problem. 



A scouting troop is collecting aluminum 
cans to raise money for charity. Their 
goal is to collect 3,000 cans in 6 months. 
If they set a goal to collect an equal 
number of cans each month, how many 
cans can they expect to collect each 
month? 

The Grand Canyon is 29,000 meters 
wide at its v«dest point. The Empire 
State Building, located in New York City, is 
381 meters tall. Laid end to end, about how 
many Empire State Buildings would fit across 
the Grand Canyon at its widest point? 

On a piano keyboard, all but one of the 
black keys are arranged in groups so that 
there are 7 groups with 2 black keys each and 
7 groups with 3 black keys each. How many 
black keys are there on a piano? 



Some wind chimes are made of rods. 
The rods are usually of different lengths, 
producing different sounds. The frequency 
(which determines the pitch) of the sound 
is measured in hertz (Hz). If one rod on a 
chime has a frequency of 55 Hz and another 
rod has a frequency that is twice that of the 
first rod's, what is the frequency of the 
second rod? 




Focus on Problem Solving 103 



2-6 




1^ 7.1.4 Recognize or use prime and composite numbers to solve problems. 

In the Chinese zodiac, each year is named 
by one of twelve animals. The years are 
named in an established order that 
repeats every twelve years. 201 1 is 
the Year of Rabbit, and 2012 is the Year of 
the Dragon. 201 1 is a prime fuiinber. 2012 
is a composite number. 



-Ar^^^ 



Vocabulary 

prime number 
composite number 
prime factorization 




mp---- 


ss^i- 


«^--^ 


4^=~_.U 


^ftiZ 


li^l^t 


Sifc 


m=--- 


sf---- 


11 ==■--: 


il^r-^ 



A prime number is a whole number 
greater than 1 that has exactly two 
factors, 1 and itself. Three is a prime 
number because its only factors are 
1 and 3. 

A composite number is a whole number that has more than two 
factors. Six is a composite number because it has more than two 
factors — 1, 2, 3, and 6. The number 1 has exactly one factor and is 
neither prime nor composite. 



EXAMPLE [ij Identifying Prime and Composite Numbers 

Tell whether each number is prime or composite. 



For a review of 
factors, see Skills 
Bank p. SB6. 



A 19 

The factors of 19 are 

1 and 19. 

So 19 is prime. 



B 20 

The factors of 20 are 
1,2,4,5, 10, and 20. 
So 20 is composite. 



A composite number can be written as the product of its prime 
factors. This is called the prime factorization of the number. You can 
use a factor tree to find the prime factors of a composite number. 



EXAMPLE 2 



JEMi 



•m 



You can write prime 
factorizations by 
using exponents. The 
exponent tells how 
many times to use 
the base as a factor. 



Using a Factor Tree to Find Prime Factorization 

Write the prime factorization of each number. 
A 36 

36 

4 . g Write 36 as the product of two factors. 

rZ)«r2)«(3)«(3) Continue factoring until all factors are prime. 

The prime factorization of 36 is 2 • 2 • 3 • 3, or 2" • 3"^. 



104 Chapter 2 Integers and Rational Numbers \ VjiJhi;| Lesson Tutorials Online mv.hrw.com 



Write the prime factorization of each number. 

B 280 

280 



Write 280 as the product of two factors. 
Continue factoring until all factors are prime. 



10 • 28 

@-®- 4 .(7 

The prime factorization of 280 is 2 • 2 • 2 • 5 • 7, or 2 ' • 5 • 7. 



You can also use a step diagram to find a prime factorization. At each 
step, divide by a prime factor until the quotient is 1. 

EXAMPLE [3] Using a Step Diagram to Find Prime Factorization 

Write the prime factorization of each number. 
A 252 

2 I 252 Divide 252 by 2. Write the quotient below 252. 

Keep dividing by a prime factor. 



2 126 


3 63 


3 21 


7 7 



1 Stop when the quotient is 1. 

The prime factorization of 252 is 2 • 2 • 3 • 3 • 7, or 2~ • 3" • 7. 
B 495 



3 495 



3 165 



Divide 495 by 3. 

Keep dividing by a prime factor. 



5 55 



11 11 



1 Stop when the quotient Is 1. 

The prime factorization of 495 is 3 • 3 • 5 • 1 1 , or 3" • 5 • 11. 



There is only one prime factorization for 
any given composite number (except for 
different orders of the factors). Example 
3B began by dividing 495 by 3, the smallest 
prime factor of 495. Beginning with any 
prime factor of 495 gives the same result. 



5|495 


11 |495 


3 99 


3 45 


3 |33 


5 1 15 


11 11 


3 3 


1 


1 



Think and Discuss 

1. Explain how to decide whether 47 is prime. 

2. Compare prime numbers and composite numbers. 



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2-6 Prime Factorization 



105 



2-6 




a3ji3d333 -^ 




GUIDED PRACTICE 



See Example 1 Tell whether each number is prime or composite. 



L 



1. 



2. 15 



3. 49 



Write the prime factorization of each number. 



See Example 2 5. 16 




6. 54 




16 




54 




4 • 

/\ 

7.7. 


4 

/\ 

7 . 7 


6 • 9 

7.7.7.7 




^ 9. 18 




10. 26 


See Example ; 


1 13. 250 




14. 190 




. 17. 639 




18. 414 



INDEPENDENT PRACTICE 



See Example 1 Tell whether each number is prime or composite. 
21. 31 22. 18 23. 67 



25. 77 



26. 



27. 9 



Write the prime factorization of each number. 



See Example 2 29. 68 

33. 135 

37. 800 

See Example 3 41. 315 

45. 242 

49. 1,225 



30. 75 

34. 48 

38. 310 

42. 728 

46. 700 

50. 288 



31. 120 

35. 154 

39. 625 

43. 189 

47. 187 

51. 360 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



4. 12 



7. 81 


8. 105 


81 

9 • ? 

/\ /\ 
7.7.7.7 


105 

/\ 
5 • ? 

/ /\ 
7.7.7 


11. 45 


12. 80 


15. 100 


16. 360 


19. 1,000 


20. 140 



24. 8 

28. 113 

32. 150 

36. 210 

40. 2,000 

44. 396 

48. 884 

52. 1,152 



See page EP6. 



Complete the prime factorization for each composite number. 

53. 180 = 2" • -5 54. 462 = 2 • 3 • 7 • 55. 1,575 = 3" 



56. 117 = 3- 



57. 144 = 



58. 13,000 = 2^ 



59. Critical Thinking One way to factor 64 is 1 • 64. 

a. What other ways can 64 be written as the product of two factors? 

b. How many prime factorizations of 64 are tliere? 

60. Critical Thinl<ing If the prime factors of a number are all the prime 
numbers less than 10 and no factor is repeated, what is the number? 



13 



106 Chapter 2 Integers and Rational Numbers 



61. A number ii is a prime factor of 28 and 63. Wliat is the number? 



62. If you were born in one of the years hsted in the table, was your birth 
year a composite number? List five composite numbers in the table. 



Chinese Zodiac LiJPr 


Animal Sign 


Years 


Animal Sign 


Years '^ 


Horse 


1990, 2002 


Rat 


1996, 2008 *'^ 


Ram 


1991, 2003 


Ox 


1997, 2009 


Monkey 


1992, 2004 


Tiger 


1998, 2010 


Rooster 


1993, 2005 


Rabbit 


1999, 2011 


Dog 


1994, 2006 


Dragon 


2000, 2012 


Boar 


1995, 2007 


Snake 


2001, 2013 



63. Business Eric is catering a part^' for 152 people. He wants to seat the same 
number of people at each table. He also wants more than 2 people but 
fewer than 10 people at a table. How many people can he seat at each table? 



64. Write a Problem Using the information 
in the table, write a problem using prime 
factorization that includes the number of 
calories per serving of the melons. 



@65. 



Write About It Describe how to use 
factor trees to find a prime factorization. 

§^66. Challenge Find the smallest number that 
is divisible bv 2, 3, 4, 5, 6, 7, 8, 9, and 10. 







Cantaloupe 


^^ 


Watermelon 


'^•Si^l 


Honeydew 


42 ^ 



Test Prep and Spiral Review 



67. Multiple Choice Which is the prime factorization of 75? 

(3) 3" • 5 CD 3 • 5' Cc:> 3- • 5" CD) 3 • S'' 

68. Multiple Choice Write the composite number for 2 • 3* • 5". 

CD 84 CD 180 CH) 450 CD 1,350 

69. Short Response Create two different factor trees for 120. Then write the 
prime factorization for 120. 



Multiply. (Lesson 1-3) 

70. 2.45- 10^ 71. 58.7- lO' 



72. 200 • 10^ 



Solve each equation. Check your answer. (Lesson 2-5) 

74. 3.Y = -6 75. V - 4 = -3 76. z -I- 4 = 3 - 5 



73. 1,480 • lO'' 



77. = -4x 



2-6 Prime Factorization 107 




Vocabulary 

greatest common 
factor (GCF) 



EXAMPLE 




Wlien getting ready for his birthday 
party, David used the greatest common 
factor to make matching favor bags. 
The greatest common factor (GCF) 
of two or more whole numbers is the 
greatest whole number that divides 
evenly into each number. 



^i ?\ I ^:^\i?:' p!7V7 






One way to find the GCF of two or 

more numbers is to list all the factors of each number. 

The GCF is the greatest factor that appears in all the lists 




^ 



Using a List to Find the GCF 

Find the greatest common factor (GCF) of 24, 36, and 48. 

Factors of 24: 1, 2, 3, 4, 6, 8,(12), 24 List all the factors of 

Factors of 36: 1, 2, 3, 4, 6, 9,(12), 18, 36 ^^<^^ number 

Factors of 48: 1, 2, 3, 4. 6, 8,(12), 16, 24, 48 ^''''^ '^^ ^'^''^'' ^^'^°' 

^^ that is in all the lists. 
The GCF is 12. 



A second way to find the GCF is to use prime factorization. 



EXAMPLE 



53 



Using Prime Factorization to Find the GCF 

Find the greatest common factor (GCF). 

A 60, 45 

60==2-2-(3)-(5) 



45 =(3)- 3 •© 

3-5 = 15 
The GCF is 15. 

B 504,132,96,60 



504 = 

132 = 

96 = 

60 = 

9.0. 



11 

2 • 
5 



3= 12 
The GCF is 12. 



Write the prime factorization of each 
number and circle the prime factors common 
to all the numbers. 
Multiply the common prime factors. 



3 • 7 Write the prime factorization of 

each number and circle the prime 
2 '(3) factors common to all the numbers. 

Multiply the common prime factors. 



108 Chapter 2 Integers and Rational Numbers Ifi'Jh'j] Lesson Tutorials Online mv.hrw.com 



EXAMPLE 



'-r— 







PROBLEM 
SOLVING 





PROBLEM SOLVING APPLICATION 

David is mal<ing favor bags 
for his birthday party. 
He has 50 confetti eggs 
and 30 noisemakers. What is the 
greatest number of matching 
favor bags he can mal<e 
using all of the confetti eggs 
and noisemakers? 

n» Understand the Problem 

Rewrite the question as a statement. 

• Find the greatest number of favor bags David can make. 
List the important information: 

• There are 50 confetti eggs. 

• There are 30 noisemakers. 

• Each favor bag must have the same number of eggs 

and the same number of noisemakers. 
The answer will be the GCF of 50 and 30. 




Make a Plan 

You can write the prime factorizations of 50 and 30 to find the GCF. 

%] Solve 

50 =(2)-(5> 5 

30 =(2)- 3 -(5) Multiply the prime factors that are 
2-5=10 common to both 50 and 30. 

David can make 10 favor bags. 



Q Look Back 

If David makes 10 favor bags, each one wall have 

5 confetti eggs and 3 noisemakers, with nothing left over. 



Think and Discuss 

1. Tell what the letters GCF stand for and explain what the GCF 
of two numbers is. 

2. Discuss whether the GCF of two numbers could be a 
prime number. 

3. Explain whether every factor of the GCF of two numbers 
is also a factor of each number. Give an example. 



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2-7 Greatest Common Factor 109 



2-7 



i 



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keyword ■miaiiKBiM ® 

Exercises 1-20, 23, 25, 29, 33, 
35,37,39 



See Example 1 
See Example 2 

See Example 3 



•i^H^THiJii^ 



Find the greatest common factor (GCF). 



1. 30,42 
4. 60,231 



2. 36,45 
5. 12,28 



3. 24,36,60,84 
6. 20, 40, 50, 120 



7. The Math Club members are preparing identical welcome kits for the 

SLXth-graders. They have 60 pencils and 48 memo pads. Wliat is the greatest 
number of kits they can prepare using all of the pencils and memo pads? 



INDEPENDENT PRACTICE 



Find the greatest common factor (GCF). 
See Example i 8. 60,126 9. 12,36 

L 11. 22, 121 12. 28,42 

See Example 2 14.28,60 15.54,80 

L 17. 26,52 18. 11,44,77 



10. 75,90 

13. 38,76 

16. 30,45,60, 105 

19. 18,27,36,48 



See Example 3 20. Hetty is making identical gift baskets for the Senior Citizens Center. She has 
39 small soap bars and 26 small bottles of lotion. What is the greatest number 
of baskets she can make using all of the soap bars and bottles of lotion? 



Extra Practice 



See page EPS 



PRACTICE AND PROBLEM SOLVING 



Find the greatest common factor (GCF). 
21. 5,7 22. 12, 15 



24. 9, 11 
27. 80, 120 
30. 4,6, 10,22 
33. 6, 15,33,48 



25. 22,44,66 
28. 20,28 
31. 14,21,35,70 
34. 18,45,63,81 



23. 4, 6 

26. 77, 121 
29. 2,3,4,5,7 
32. 6, 10, 11, 14 
35. 13,39,52,78 



36. Critical Thinking Which pair of numbers has a GCF that is a prime 
number, 48 and 90 or 105 and 56? 

37. Museum employees are preparing an exliibit of ancient coins. They have 
49 copper coins and 35 silver coins to arrange on shelves. Each shelf will 
have the same number of copper coins and the same number of silver 
coins. How many shelves will the employees need for this exliibit? 

38. Multi-Step Todd and FJizabeth are making treat bags for the hospital 
volunteers. They have baked 56 shortbread cookies and 84 lemon bars. 
What is the greatest number of bags they can make if all volunteers 
receive identical treat bags? How many cookies and how many lemon 
bars will each bag contain? 



110 Chapter 2 Integers and Rational Numbers 




The sculpture 
Balloon Dog by U.S. 
artist Jeff Koons 
was featured in an 
exhibit in Bregenz, 
Austria. 



39. School Some of the students in the 
Math Club signed up to bring food 
and drinks to a party. 

a. If each club member gets the same 
amount of each item at the party, 
how many students are in the 
Math Club? 

b. How many carrots, pizza slices, 
cans of juice, and apples can each 
club member have at the party? 



Food and Drink Sign-up Sheet 


Student 


Item 


Amount 


Macy 


Apples 


14 


Paul 


Pizza slices 


21 


Christie 


Juice boxes 


7 


Peter 


Carrot sticks 


35 



-^2 Art A gallerv' is displaying a collection of 12 sculptiu'es and 20 paintings by 
^ local artists. The e.xliibit is arranged into as many sections as possible so 
that each section has the same number of sculptures and the same number 
of paintings. How many sections are in the exliibit? 



?j 41. What's the Error? A student used these factor trees to find the GCF of 50 
and 70. The student decided that the GCF is 5. Explain the student's error 
and give the correct GCF. 



50 

25 <2 



70 



@- 10 



5)- (5 



0^2. Write About It The GCF of 1,274 and 1,365 is 91, or 7 • 13. Are 7, 13, and 

91 factors of both 1,274 and 1,365? Explain. 



^ 43. Challenge Find three coDiposlte numbers that have a GCF of 1. 



Test Prep and Spiral Review 



44. Multiple Choice Which pair of numbers has a greatest common factor 
that is NOT a prime number? 

Ca:) 15, 20 CD 18, 30 CC) 24, 75 CS:) 6, 10 

45. Gridded Response What is the greatest common factor of 28 and 91? 
Find each value. (Lesson 1-2) 

46. 10^ 47. 13' 48. 6^ 49. 3^ 

Use a number line to find each sum or difference. (Lessons 2-2 and 2-3) 

50. -5-^(-3) 51.2-7 52. 4 + (-8) 53. -3 - (-5) 

Complete the prime factorization for each composite number. (Lesson 2-6) 

54.100= -5- 55. 147 = 3 • isBi 56. 270 = 2 • 3^ • 57.140= •5-7 



2-7 Greatest Common Factor 111 



2-8 



, lanraittiwiTU •; 



Vocabulary 

multiple 

least common 
multiple (LCM) 



St Common Multiple 



The maintenance schedule on 
Kendra's pickup truck shows that 
the tires should be rotated every 
7,500 miles and that the oil filter 
should be replaced every 5,000 
miles. Wliat is the lowest mileage 
at which both services are due at 
the same time? To find the answer, 
you can use least common Duiltiples. 

A multiple of a number is tlie 
product of that number and a 
nonzero whole number. Some 
multiples of 7,500 and 5,000 are 
as follows: 

7,500: 7,500, 15,000, 22,500, 30,000, 37,500, 45,000, . . . 
5,000: 5,000, 10,000, 15,000, 20,000, 25,000, 30,000 

A common multiple of two or more numbers is a number that 

is a multiple of each of the given numbers. So 15,000 and 30,000 are 

common multiples of 7,500 and 5,000. 

The least common multiple (LCM) of two or more numbers is the 
common multiple with the least value. The LCM of 7,500 and 5,000 
is 15,000. This is the lowest mileage at which both services are due at 
the same time. 




EXAMPLE 



[T] Using a List to Find the LCM 

Find the least common multiple (LCM). 

A 3,5 

Multiples of 3: 3, 6, 9, 12,(15), 18 
Multiples of 5: 5, 10,(15), 20, 25 
The LCM is 15. 



List multiples of each number. 
Find the least value that 
is in both lists. 



B 4, 6, 12 

Multiples of 4: 4, 8,(12), 16, 20, 24, 28 List multiples of each number. 
Multiples of 6: 6,(12), 18, 24, 30 Find the least value that 

Multiples of 12:@), 24, 36, 48 
The LCM is 12. 



is in all the lists. 



112 Chapter 2 Integers and Rational Numbers [VJiJ-juj Lessod Tutorials OnlinE mv.hrw.com 



EXAMPLE 



Sometimes, listing the multiples of numbers is not the easiest way to 
find the LCM. For example, the LCM of 78 and 110 is 4,290. You would 
have to list 55 multiples of 78 and 39 multiples of 1 10 to reach 4,290! 

[ 2 1 Using Prime Factorization to Find the LCM 

Find the least common multiple (LCM). 



A 78,110 



110 = 



13 
11 



(2)3, 13.5, 11 

2 -3 -5 • 11 • 13 
The LCM is 4,290. 



6, 27, 45 

6=2- 



27 = 
45 = _ 
2,(3)(3)3, 5 



3 
5 



2 • 3' • 5 

The LCM is 270. 



Write the prime factorization of eacli number. 

Circle any common prime factors. 

List tiie prime factors of the numbers. Use each 

circled factor only once. 

Multiply the factors in the list. 



Write the prime factorization of each number. 

Circle any prime factors that are common to at 

least 2 numbers. 

List the prime factors of the numbers. Use each 

circled factor only once. 

Multiply the factors in the list. 



EXAMPLE 



[3 



Recreation Application 

Charia and her little brother are walking laps on a track. Charla 
walks one lap every 4 minutes, and her brother walks one lap 
every 6 minutes. They start together. In how many minutes will 
they be together at the starting line again? 

Find the LCM of 4 and 6. 



4 = 
6 = 



TheLCMis@-2 • 3 = 12. 

They will be together at the starting line in 12 minutes. 



Think and Discuss 

1. Tell what the letters LCM stand for and explain what the LCM of 
two numbers is. 

2. Describe a way to remember the difference between GCF and LCM. 



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2-8 Least Comnnon Multiple 113 



2-8 



See Example 1 
See Example 2 

See Example 3 



L 




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keyword MBteinBiB;« ® 

Exercises 1-21, 23, 25, 27, 29, 
31,33,37 



GUIDED PRACTICE 



Find the least common multiple (LCM). 
1. 4,7 2. 14,21,28 



4. 30,48 



5. 3,9, 15 



4,8, 12, 16 
10,40,50 



7. Jeriy and his dad are walking aiound the track. Jerry completes one lap every 
8 minutes. His dad completes one lap every 6 minutes. They start together. 
In liow many minutes will they be together at the starting line again? 



See Example 1 



See Example 2 



See Example 3 



INDEPENDENT PRACTICE 



Find the least common multiple (LCM). 
8. 6,9 9. 8, 12 



11. 6, 14 
14. 6,27 
17. 10, 15, 18,20 



12. 18,27 
15. 16,20 
18. 11,22,44 



10. 15,20 
13. 8, 10, 12 
16. 12, 15,22 
19. 8, 12, 18,20 



20. Recreation On her bicycle, Anna circles the block every 4 minutes. Her 
brotlier, on his scooter, circles the block eveiy 10 minutes. They start out 
together. In how many minutes will they meet again at the starting point? 

21. Rod helped his mom plant a vegetable garden. Rod planted a row every 
30 minutes, and his mom planted a row every 20 minutes. If they started 
together, how long will it be before they both finish a row at the same time? 



Extra Practice 



See page EP7. 



PRACTICE AND PROBLEM SOLVING 



Find the least common multiple (LCM). 
22. 3,7 23. 4,6 



25. 22, 44, 66 
28. 3,5,7 
31. 24,36,48 



26. 80, 120 
29. 3,6, 12 
32. 2,3,4,5 



24. 9, 12 

27. 10, 18 

30. 5,7,9 

33. 14,21,35,70 



34. Jack mows the lawn every three weeks and washes the car every two weeks. 
If he does both today, how many days will pass before he does them both 
on the same day again? 

35. Critical Thinking Is it possible for two numbers to have the same LCM 
andGCF? Explain. 

36. Multi-Step Milli jogs every day, bikes every 3 days, and swims once a 
week. She does all three activities on October 3. On what date will she 
next perform all three activities? 



114 Chapter 2 Integers and Rational Numbers 



Q 



<»•» 




Social Studies 



.-WB*^ 



The Mayan, the Chinese, and the standard western calendar 
are all based on cycles. 

37. The Mayan ceremonial calendar, or tzolkin, was 260 days 
long. It was composed of two independent cycles, a 13-day 
cycle and a 20-day cycle. At the beginning of the calendar, 
both cycles are at day 1. Will both cycles be at day 1 at the 
same time again before the 260 days are over? If so, when? 

38. The Chinese calendar has 12 months of 30 days each and 
6-day weeks. The Chinese New Year begins on the first day of a 
month and the first day of a week. Will the first day of a month 
and the first day of a week occur again at the same time before 
the 360-day year is over? If so, when? Explain your answer. 



39. 




*V-, 



Write About It The lulian Date calendar assigns each 
day a unique number. It begins on day and adds 1 for each 
new day. So ID 2266296, or October 12, 1492, is 2,266,296 days 
from the beginning of the calendar. What are some advantages 
of using the lulian Date calendar? What are some advantages 
of using calendars that are based on cycles? 




40.® 



^ Challenge The Mayan Long Count calendar 
used the naming system at right. Assuming the 
calendar began on ID 584285, express ID 2266296 
in terms of the Mayan Long Count calendar. Start 
by finding the number of pictun that had passed 
up to that date. 









1 Pictun = 


20 Baktun = 


2,880,000 days 


1 Baktun 


= 20 Katun = 


= 144,000 days 


1 Katun = 20 Tun = 


7,200 days 


1 Tun 


= 18 Winal = 


= 360 days 


1 Winal = 20 Kin 


= 20 days 




1 Kin = 1 d 


ay 



i 



Test Prep and Spiral Review 



41. Multiple Choice Which is the least common multiple of 4 and 10? 

(X) 2 d:' 10 CD 20 CE) 40 

42. Multiple Choice Wliich pair of numbers has a least common multiple 
of 150? 



CD 10, 15 



CS) 150,300 



Simplify each expression. (Lesson 1-8) 
43. 3c -I- 2f - 2 44. 5.r + 3x^ - 2x 



(E) 2,300 



45. 7u + 3i'- 4 



CD 15,50 



46. ;?; -I- 1 - 6;?; 



Find the greatest common factor (GCF). (Lesson 2-7) 

47. 12,28 48. 16,24 49. 15.75 



50. 28, 70 



2-8 Least Common Multiple 115 




To Go On? 



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ResourcesOnlinego.hrw.com, 
|gBaMSI0RTGO2B|51 







Quiz for Lessons 2-6 Through 2-8 

2-6 ] Prime Factorization 
Complete each factor tree to find the prime factorization. 
1. 24 2. 140 3. 45 



6 

/\ 

7.7 



4 

/\ 
7.7 



14 



10 



/ /\ 



42 

/\ 

7.7 

/\ \ 

3 • 7 • ? 



I 



Write tlie prime factorization of each number. 
5. 96 6. 125 

8. 105 9. 324 

2-7 ) Greatest Common Factor 
Find tlie greatest common factor (GCF). 
11. 66,96 12. 18,27,45 

14. 14,28,56 15. 85, 102 

17. 52,91, 104 18. 30, 75,90 



7. 99 
10. 500 



13. 16,28,44 
16. 76,95 
19. 118, 116 



20. Yasmin and Jon have volunteered to prepare snacks for the first-grade 
field trip. They have 63 carrot sticks and 105 strawberries. What is the 
greatest number of identical snacks they can prepare using all of the 
carrot sticks and strawberries? 



er 



2-8 ] Least Common Multiple 
Find the least common multiple (LCM). 

21. 35,40 22. 8,25 23. 64,72 

24. 12,20 25. 21,33 26. 6,30 

27. 20,42 28. 9, 13 29. 14, 18 

30. Eddie goes jogging every other day, lifts weights eveiy third day, and swims 
every fourth day. If Eddie begins all three activities on Monday, how many 
days will it be before he does all three activities on the same day again? 

31. Sean and his mom start running around a 1-mile track at the same time. 
Sean runs 1 mile every 8 minutes. His mom runs 1 mile every 10 minutes. 
In how many minutes will they be together at the starting line again? 



116 Chapter 2 Integers and Rational Numbers 



ik^M^. 




Focus on Problem Soliring 




tLook Back 
• Check that your answer is reasonable 

In some situations, such as when you are looking for an estimate 
or completing a multiple-choice question, check to see whether 
a solution or answer is reasonably accurate. One way to do this 
is by rounding the numbers to the nearest multiple of 10 or 100, 
depending on how large the numbers are. Sometimes it is useful 
to round one number up and another down. 



t 



Read each problem, and determine whether the given solution is too 
high, is too low, or appears to be correct. Explain your answer. 



O The cheerleading team is preparing to host 
a spaghetti dinner as a fund-raising project. 
They have set up and decorated 54 tables 
in the gymnasium. Each table can seat 8 
people. How many people can be seated 
at the spaghetti dinner? 

Solution: 432 people 

O The cheerleaders need to raise $4,260 to 
attend a cheerleader camp. How much 
money must they charge each person if 
they are expecting 400 people at the 
spaghetti dinner? 

Solution: $4 

O To help out the fund-raising project, local 
restaurants have offered $25 gift certificates 
to give as door prizes. One gift certificate will 
be given for each door prize, and there will 
be six door prizes in all. What is the total 
value of all of the gift certificates given by 
the restaurants? 

Solution: $250 



O The total cost of hosting the spaghetti 
dinner will be about $270. If the 
cheerleaders make $3,280 in ticket sales, 
how much money will they have 
after paying for the spaghetti dinner? 

Solution: $3,000 

Eighteen cheerleaders and two coaches 
plan to attend the camp. If each person 
will have an equal share of the $4,260 
expense money, how much money will 
each person have? 

Solution: $562 





W^ f M 



Focus on Problem Solving 117 



J**-' 



4ii«*«**Bi5^; 



Vocabulary 

equivalent fractions 
relatively prime 




In the diagram, | = j| = :^. These are called equivalent fractions 
because they are different expressions for the same nonzero number. 

To create fractions equivalent to a given fraction, multiply or divide 
the numerator and denominator by the same number. 

EXAMPLE [lj Finding Equivalent Fractions 

Find two fractions equivalent to ^. 

Multiply the numerator and denominator by 2. 



U 
16 



14 • 2 _ 2£ 
16-2 32 



M _ 14 H- 2 _ 7 
16 16 -i- 2 8 



Divide the numerator and denominator by 2. 



EXAMPLE 



The fractions ^, ~, and ~ in Example 1 are equivalent, but only ^ is in 
simplest form. A fraction is in simplest form when the numerator and 
denominator are relatively prime. Relatively prime numbers have no 
common factors other than 1. 



53 



Writing Fractions in Simplest Form 

Write the fraction ~| in simplest form. 

Find the GCF of 24 and 36. 

24 = 2 • 2 • 2 • 3 The GCF is 2 • 2 • 3 = 12. 

36 = 2 • 2 • 3 • 3 

Divide the numerator and denominator by 12. 



24 _ 24 H- 12 _ 2 
36 



36^ 12 



118 Chapter 2 integers and Rational Numbers \ 'Mb'j] Lesson Tutorials Online my.hrw.com 



To determine if two fractions are equivalent, find a common 
denominator and compare the numerators. 

EXAMPLE [3] Determining Whether Fractions Are Equivalent 

Determine whether the fractions in each pair are equivalent. 
A I and ^ 

Both fractions can be written with a denominator of 4. 



An improper fraction 
is a fraction whose 
numerator is greater 
than or equal to the 
denominator. 



6 _ 6^2 



9 _ 9 H- 3 



88^24 12 12H-34 

The numerators are equal, so the fractions are equivalent. 

18 and 25 
15 ^"^20 

Both fractions can be written with a denominator of 60. 

18 - 18-4 _ 72 25 _ 25 • 3 _ 75 

15 15-4 60 20 20.3 60 

The numerators are not equal, so the fractions are not equivalent. 



5 is an improper fraction. 
Its numerator is greater than 
its denominator. 



= 1^ 



1 5 is a mixed number. 

It contains both a whole 
number and a fraction. 



EXAMPLE 



e 



Converting Between Improper Fractions and Mixed Numbers 

9 1 

A Write ^ as a mixed number. 

4 

First divide the numerator bv the denominator. 



^ = 21 H- 4 = 5R1 = 5| 

4 4 



B Write 4^ as an improper fraction. 



Use the quotient and remainder 
to write the mixed number. 



First multiply the denominator and whole number, and then add 
the numerator. 



*'4I 



3.4 + 2 



11 
3 



x\y 



Use the result to write the 
improper fraction. 



Think and Discuss 

1. Explain a process for finding common denominators. 

2. Describe how to convert between improper fractions and 
mixed numbers. 



'Mbii\ Lesson Tutorials Online my.hrw.com 2-9 Equivalent Fractions and Mixed Numbers 119 



F > 



2-9 




)jfiU£fcfe 



keyword mtlismWEiM ® 

Exercises 1-44, 45, 57, 59, 61, 
63,65 



GUIDED PRACTICE 



See Example 1 Find two fractions equivalent to the given fraction. 



L 



1. 



21 
42 



33 
55 



12 



15 
40 



See Example 2 Write each fraction in simplest form. 



5. 



13 
26 



6. 



54 
72 



12 

15 



36 
42 



See Example 3 Determine whether the fractions in each pair are equivalent. 



9. I and I 



10. If and f 



11. f andf^ 

b lb 



12. f and if 



See Example 4 Write each as a mixed number. 



13. 



15 



14. 



22 



Write each as an improper fraction. 

17. 6^ 18. iji 



15. 



13 



19. 7| 



16. 



14 



20. 2 



16 



See Example 1 

I 



INDEPENDENT PRACTICE 



Find 
21 



i two fractions equivalent to the given fraction. 
18 22. ^ 23. ^ 



23. ^ 

15 

See Example 2 Write each fraction in simplest form. 

[ 25. g^ 26. 2Y 27. ^ 

See Example 3 Determine whether the fractions i 

i 29 — and — 



48 



in each pair are equivalent. 

30. i^andi 31. Ilandig 



20 ' 



34. pj and || 



L 33. fand^ 

See Example 4 Write each as a mixed number. 
37. f 38. f 



^24 



3 

35 ^ 

-'-'■ on 



Too ^'^'^ 32 



nnri 84 
99 132 



Write each as an improper fraction. 

-- ^^3 «-> , 7 



L 41-25i 



42 4— 

16 



39 ^ 

11 



43. 9| 



24. ^ 

70 



■7Q 100 
-^^^ 250 



32. li^andl^ 

5 8 

36. #and4^ 

15 75 



40. a 



44 4— 

31 



PRACTICE AND PROBLEM SOLVING 



Extra Practice 



See page EP7. 



45. Personal Finance Every month, Adrian pays for his own long-distance 
calls made on the family phone. Last month, 15 of the 60 minutes of long- 
distance charges were Adrian's, and he paid $2.50 of the $12 long-distance 
bill. Did Adrian pay his fair share? 



120 Chapter 2 Integers and Rational Numbers 



Write a fraction equivalent to tlie given number. 




46. 
51. 



55 
10 



47. 6^ 



52. 101 



48. 



9- 

"3 



53. 6 



15 
21 



49. 



54. 



_8_ 
21 

475 

75 



50. 9 
55. 11 



11 

23 
50 



Find the equivalent pair of fractions in each set. 

6 21 3 c-7 7 12 6 CO 2 12 



56 



15' 35' 5 



57. 



12' 20' 10 



58. 



3' 15' 



20 
30' 



15 
24 



59. 



7 9 _32 J72 
4' 5' 20' 40 



There are 12 inches in 1 foot. Write a mixed number to represent each 
measurement in feet. (Example: 14 inches = Ipj feet or l| feet) 



60. 25 inches 



61. 100 inches 



62. 362 inches 



63. 42 inches 



A single bread 
company can make 
as many as 1,217 
loaves of bread each 
minute. 



64. Social Studies A dollar bill is ISyj^^ centimeters long and 6^^ centimeters 
wade. Write each number as an improper fraction. 

■^W Food A baker\' uses 37;^ cups of flour to make 25 loaves of bread each day. 
Write a fraction that shows how many ^ cups of flour are used to make bread 
each day at the bakery. 



i 



:a 



66. Write a Problem Cal made the 
graph at right. Use the graph to write 
a problem involving fractions. 

67. Write About It Draw a diagram 
to show how you can use division to 
write ^ as a mixed number. Explain 
your diagram. 

g^ 68. Challenge Kenichi spent i^ of his 

$100 birthday check on clothes. How 
much did Kenichi's new clothes cost? 



Test Prep and Spiral Review 



How Cal Spends His Day 




Sleep 



i School 



Personal 
time 



69. Multiple Choice Which improper fraction is NOT equivalent to 2^? 



CS) 



CS) 



10 



CT) 



20 



70. Multiple Choice Which fraction is equivalent to |? 



CD 



20 



cg;) 



10 



cb:> t 



CE) 



CD 



25 
10 



24 "^ 18 -^ y --^5 

71. Short Response Maria needs | cups of flour, ^ cups of water, and 

I tablespoons of sugar. Write each of these measures as a mixed number. 



Solve each equation. Check your answer. (Lessons 1-10 and 1-11) 
72. 5b = 25 73. 6 -I- y =18 74. ^- - 57 = 119 

Find the least common multiple (LCM). (Lesson 2-8) 

76. 2,3,4 77. 9, 15 78. 15,20 



75. y = 20 

4 



79. 3, 7, 



2-9 Equivalent Fractions and Mixed Numbers 121 



2-10 



^MMilMiiaA 



Decimals 



Vocabulary 

terminating decimal 
repeating decimal 



In baseball, a player's batting 
average compares the number 
of hits with the number of times 
the player has been at bat. The 
statistics below are for the 2006 
Major League Baseball season 







h 


1 .^^^^^ 


KmBL 


Player 


Hits 


At Bats 


Hits 
At Bats 


Batting Average 
(thousandths) 


Miguel Cabrera 


195 


576 


195 
576 


195 -=- 576 « 0.339 


Ichiro Suzuki 


224 


695 


224 
695 


224 H- 695 = 0.322 



To convert a fraction to a decimal, divide the numerator by the denominator. 



EXAMPLE 







Writing Fractions as Decimals 

Write each fraction as a decimal 
hundredth, if necessary. 





A 


3 
4 




immWi^^m. 






0.75 
4)3.00 
-28 


You can use a 






20 


calculator to check 






-20 


your division: 









3 0400.75 








60501.2 

1 030 0.333... 




3 
4 


= 0.75 



imal. Rounc 


1 to th 


e nearest 


6 
5 


c 


1 

3 


1.2 




0.333 . 


5)6.0 




3)1.000 


-5 




-9 


10 




10 


-10 




-9 







10 
-9 


1=1.2 




1 



= 0.333... 
«0.33 



The decimals 0.75 and 1.2 in Example 1 are terminating decimals 
because the decimals come to an end. The decimal 0.333 ... is a 
repeating decimal because the decimal repeats a pattern forever. You 
can also write a repeating decimal with a bar over the repeating part. 



0.333. 



= 0.3 



0.8333... =0.83 



0.727272. 



= 0.72 



122 Chapter 2 Integers and Rational Numbers \ 'Miu] Lesson Tutorials OnlinE my.hrw.com 



You can use place value to write some fractions as decimals. 
EXAMPLE [2] Using Mental Math to Write Fractions as Decimals 



Write each fract 


A 


2 
5 




5 ^ 2 ~T0 




= 0.4 


B 


7 
25 




7 4 _ 28 
25 4 100 




= 0.28 



Multiply to get a power of ten in the denominator. 



Multiply to get a power of ten in the denominator. 



You can also use place value to write a terminating decimal as a 
fraction. Use the place value of the last digit to the right of the 
decimal point as the denominator of the fraction. 

EXAMPLE rsj Writing Decimals as Fractions 

■ Write each decimal as a fraction in simplest form. 

?7 A 0.036 B 1.28 

. I I 



Reading Math 



;es/ ~i 



You read the decimal 
0.036 as "thirty-six 
thousandths." 



0.036 = 



36 



L 



,000 

36 ^ 4 
1,000-4 P'^ce 



6 is in the 
thousandths 



128 = ^ 
100 



_ 9 
250 



_ 128 H- 4 

~ TOOT4 P'^'^^ 

= 3^ orl^ 

25' 25 



8 is in the 
hundredths 



EXAMPLE [4] Sports Application 



During a football game, Albert completed 23 of the 27 passes 

he attempted. Find his completion rate to the nearest thousandth. 



Fraction 


What the Calculator Shows 


Completion Rate 


23 
27 




0.852 


23 ^9 77imS^ (.8518518519] 



His completion rate is 0.852. 



^^njjjjj^^^^^^n^^mii^^^m^^^^^^i 


Think and Discuss 




1. Tell how to write a fraction as a decimal. 




2. Explain how to use place value to convert 0.2048 to 


a fraction. 



1/jdai; Lesson Tutorials Online mv.hrw.com 



2-70 Equivalent Fractions and Decimals 123 




^-.^H-'i.i ^M^f^ '^Fiy^Ki^-^^f^^ 



;c^j^j33e 



*- 




GUIDED PRACTICE 



See Example 1 Write each fraction as a decimal. Round to the nearest hundredth, if necessary. 



L 



1. 



2. 



21 



See Example 2 Write each fraction as a decimal. 



[ 



25 



10 



11 
6 



20 



See Example 3 Write each decimal as a fraction in simplest form. 

i 9. 0.008 10. -0.6 11. -2.05 



12. 3.75 



See Example 4 13. Sports After sweeping the Baltimore Orioles at home in 2001, the 

Seattle Mariners had a record of 103 wins out of 143 games played. Find 
the Mariners' winning rate. Write your answer as a decimal rounded 
to the nearest thousandth. 



INDEPENDENT PRACTICE 



See Example 1 Write each fraction as a decimal. Round to the nearest hundredth, if necessary. 



I 18. if 



15. 



19. 



32 



500 

500 



See Example 2 Write each fraction as a decimal. 



22. 



23. I 



16. 



20. 



24. 



18 
25 



15 
25 



17. 



21. 



25. 



23 
12 



11 
20 



See Example 3 Write each decimal as a fraction in simplest form. 

i 26. 0.45 27. 0.01 28. -0.25 



30. 1. 



31. 15.25 



32. 5.09 



29. -0.08 
33. 8.375 



See Example 4 34. School On a test, Caleb answered 73 out of 86 questions correctly. 
What portion of his answers was correct? Write your answer as a 
decimal rounded to the nearest thousandth. 



Extra Practice 



See page EP7 



PRACTICE AND PROBLEM SOLVING 



Give two numbers equivalent to each fraction or decimal. 



35. 



39. 15.35 



36. 0.66 



40. 8 



3 



37. 5.05 

41 4 3 
1,000 



Determine whether the numbers in each pair are equivalent. 



43. I and 0.75 
47. 0.275 and I 



40 



44. ^ and 0.45 
48. li^and 1.72 

z5 



45. 



and 0.55 



16 



49. 0.74 and ;^ 



38. 



42. 3^ 



46. 0.8 and I 
50. 0.35 and ^ 



124 Chapter 2 Integers and Rational Numbers 



Economics 



Use the table for Exercises 51 and 52. 



- XYZ Stock Values (October 2006) 


Date 


Open 


High 


Low 


Close 


Oct 16 


17.89 


18.05 


17.5 


17.8 


Oct 17 


18.01 


18.04 


17.15 


17.95 


Oct 18 


17.84 


18.55 


17.81 


18.20 



51 . Write the highest value ot stock XYZ tor each 

, . , , ■ ■ , r Traders watch the 

day as a mixed number m simplest form. ^^^^^^ ,^^5 change 

__ „ 1-11 1-11 ■ r 1 ,,, r^ 1 from the floor of a 

52. On which date did the price ot stock XYZ change ^^^j-i^ exchange 
by ^ of a dollar between the open and close 
of the day? 




53. ^ Write About It UntO recently, prices of stocks 

were expressed as mixed numbers, such as 24|f dollars. The 
denominators of such fractions were multiples of 2, such 
as 2, 4, 6, 8, and so forth. Today, the prices are expressed as 
decimals to the nearest hundredth, such as 32.35 dollars. 

a. Wliat are some advantages of using decimals instead of 
fractions? 

b. The old ticker-tape machine punched stock prices onto a tape. Perhaps 
because fractions could not be shown using the machine, the prices 
were punched as decimals. Write some decimal equivalents of 
fractions that the machine might print. 



Before the days of 
computer technology, 
ticker-tape machines were 
used to punch the stock 
prices onto paper strands. 



54. ^ Challenge Write | and | as decimals. Use the results to 
predict the decimal equivalent of |. 



Test Prep and Spiral Review 



55. Multiple Choice Which is NOT equivalent to 0.35? 



® Too 



^-^ 20 



CD M 

^^ 40 



CE) 



25 
80 



56. Gridded Response Write yl as a decimal rounded to the nearest hundredth. 

Determine whether the given value of the variable is a solution. (Lessors 1 -9) 

57. A- = 2for3x-4= 1 58. .v = 3 for 5x + 4 = 19 59. x = 14 for 9(4 + x) = 162 



Write each as an improper fraction. (Lesson 2-9) 



60. 4^ 



61. 3^ 



62. If 



63. 6:^ 



2-70 Equivalent Fractions and Decimals 125 



Comparing and Ordering 
Rational Numbers 



7.1.6 Identify, write, rename, compare and order rational and common irrational 
numbers and plottliem on a number Ime. 

Wliich is greater, ^ or ^? 

To compare fractions with 
the same denominator, just 
compare the numerators. 



Vocabulary 

rational number 



j^ > ^ because 7 > 2. 



_ 7 
9 



To compare fractions witli 
unhke denominators, first 
write equivalent fractions with 
common denominators. Then 
compare the numerators. 




"I would like an extra-large pizza with 
y pepperoni, -^ sausuage, -- anchovies on 
the pepperoni side, y- pineapple, :j^ doggie 
treats, j- catnip . . . and extra cheese." 



EXAMPLE 1 



Comparing Fractions 

Compare the fractions. Write < or >. 

A t 



10 



The LCM of the denominators 6 and 10 is 30. 



5 _ 5 -5 _ 25 

6 6-5 30 

J7_ _ 7-3 _ 21 
10 10-3 30 



''5 ''I 1 5 7 

3^>-,andso->jQ. 



Write equivalent fractions 
witii 30 as the denominator. 



Compare the numerators. 



B --. 



Both fractions can be written witli a denominator of 45. 



A fraction less than 
can be written as 
-1 zA or ^ 

5' 5 ' -5- 



3 
5 


= 


-3-9 
5-9 


_ -27 
45 


5 
9 


= 


-5- 5 
9- 5 


_ -25 
45 


-27 
45 


< 


^1^, and so - 



3 <- _5 
5 9' 



Write equivalent fractions 
with 45 as the denominator. 
Put the negative signs in 
the numerators. 



126 Chapter 2 Integers and Rational Numbers \ ^Mbu] Lesson Tutorials Online my.hrw.com 



To compare decimals, line up the decimal points and compare digits 
from left to right until you find the place where the digits are different. 



EXAMPLE [Vj Comparing Decimals 

Compare the decimals. Write < or >. 
0.84 



A 0.81 

0.81 



t 



0.84 

Since 0.01 < 0.04, 0.81 < 0.84 

B 0.34 0.342 

0.34 = 0.3434 . . . 



Line up the decimal points. 
The tenths are the same. 
Compare the hundredths: 1 < 4. 



0.342 



0.34 is a repeating decimal. 

Line up the decimal points. 

The tenths and hundredths are the same. 

Compare the thousandths: 3 > 2. 



Since 0.003 > 0.002, 0.34 > 0.342. 



A rational number is a number that can be written as a fraction with 
integers for its numerator and denominator. When rational numbers 
are written in a variety of forms, you can compare the numbers by 
writing them all in the same form. 



EXAMPLE [3] Ordering Fractions and Decimals 



The values on a 
number line increase 
as you move from 
left to right. 



Order |, 0.77, -0.1, and 1 ^ from least to greatest. 



3 _ 



0.60 



0.77 « 0.78 Write as decimals with the 

same number of places. 



1^ = 1.20 



-0.1 = -0.10 

Graph the numbers on a number line. 



-0.1 

I I I I » 



0.77 



I I I I I » I 



I I I ♦ I I I 



-0.5 0.5 

-0.10 < 0.60 < 0.78 < 1.20 



1.0 1.5 

Compare the decimals. 



3 — 1 

From least to greatest, the numbers are -0.1, -p, 0.77, and 1^^. 



Think and Discuss 

1. Tell how to compare two fractions with different denominators. 

2. Explain why -0.31 is greater than -0.325 even though 2 > 1. 



Vld^!^ Lesson Tutorials OnliriE 



2-7 7 Comparing and Ordering Rational Numbers 127 



2-11 



1!S HomeworkHelpOnlinego.hrw.com, 



keyword IKHIiKaiB ® 
Exercises 1-30, 31, 33, 35, 37, 41 



GUIDED PRACTICE 



See Example 1 Compare the fractions. Write < or >. 

i 1. 3 4 2. -^ -^ 

L 5 "^'-' 5 8 8 

See Example 2 Compare the decimals. Write < or >. 



^■-1 



L 



5. 0.622 0.625 



6. 0.405 0.45 



i 4 3^ 3^ 

7 5 3 



7. -3.822 -3.819 



See Example 3 Order the numbers from least to greatest. 



8. 0.55, |, 0.505 



9. 2.5,2.05, - 



13 



10. i, -0.875,0.877 



INDEPENDENT PRACTICE 



See Example 1 Compare the fractions. Write < or >. 



11. 



12. 



16. - 



11 IT 

L ''■ f f - 4 4 

See Example 2 Compare the decimals. Write < or >. 

j 19. 3.8 3.6 20. 0.088 0.109 

^ 22. -1.902 0.920 23. -0.7 -0.07 

See Example 3 Order the numbers from least to greatest. 



13. -^ 

6 



17. - -i 

4 4 



14. 10| 10| 

4 D 

18. -I '4 



21. 4.26 4.266 
24. 3.08 3.808 



25. 0.7,0.755,1 
o 

28. -3.02, -3.02, l\ 



26. 1.82, 1.6, 1^ 

5 

29. 2.88, -2.98, -2 



10 



27. -2.25,2.05 

30. f I. 0.82 
b 5 



2i 
10 



Extra Practice 



See page EP7. 



PRACTICE AND PROBLEM SOLVING 



Choose the greater number. 



31. 4 or 0.7 

4 



32. 0.999 or 1.0 



33 lor — 



35. 0.32 or 0.088 36. 



or -0.05 



37. 



9 7 

To °'' -8 



34. -0.93 or 0.2 
38. 23.44 or 23 



39. Earth Science Density is a measure of mass in a specific unit of space. The 
mean densities (in g/cm') of the planets of our solar system are given in the 
table below. Rearrange the planets from least to most dense. 



Planet 


Density 


Planet 


Density 


Planet 


Density 


Mercury 


5.43 


Mars 


3.93 


Uranus 


1.32 


Venus 


5.20 


Jupiter 


1.32 


Neptune 


1.64 


Earth 


5.52 


Saturn 


0.69 


Pluto* 


2.05 



'designated a dwarf planet in 2006 



128 Chapter 2 Integers and Rational Numbers 



.•Q"-. 40. 




Algae that grows 
in sloths' fur make 
them look slightly 
green. This helps 
them blend Into 
the trees and stay 
out of sight from 
predators. 



43. 



044. 
@45. 



Multi-Step Twenty-four karat gold is considered pure. 

a. Angie's necklace is 22-karat gold. What is its purity as a fraction? 

b. Luke's ring is 0.75 gold. If Angie's necklace and Luke's ring weigh 
the same amount, which contains more gold? 

Life Science Sloths are tree-dwelling animals that live in South and 
Central America. They generally sleep about | of a 24-hour day. Humans 
sleep an average of 8 hours each day. Which sleep the most each day, 
sloths or humans? 



42. Ecology Of Beatrice's total household water 
use, I is for bathing, toilet flushing, and 
laundr\'. How does her water use for these 
purposes compare with that shown in the 
graph? 

What's the Error? A recipe for a large 
cake called for 4^ cups of flour. The chef 
added 10 one-half cupfuls of flour to the 
mixture. Wiat was the chef's error? 

Write About It Lxplain how to compare 
a mixed number with a decimal. 



Average Daily Household 
Use of Water 



5 Bathing, 
toilet flushing, 
laundry 



25 Lawn watering, 
car washing, pool 
maintenance 




25 Drinking, 

cooking, washing dishes, 
running garbage disposal 



Challenge Scientists estimate that Earth is approximately 4.6 billion 
years old. We are currently in what is called the Phanerozoic eon, which 
has made up about ^ of the time that Earth has existed. The first eon, 
called the Hadean, made up approximately 0.175 of the time Earth has 
existed. Which eon represents the most time? 



Test Prep and Spiral Review 



CS) f 



46. Multiple Choice Which number is the greatest? 

CA) 0.71 CD I C£) 0.65 

o 

47. Multiple Choice Which shows the order of the animals from fastest to slowest? 
CE) Spider, tortoise, snail, sloth 
CS) Snail, sloth, tortoise, spider 
CH) Tortoise, spider, snail, sloth 
CD Spider, tortoise, sloth, snail 



Maximum Speed (mi/h) 


Animal 


Snail 


Tortoise 


Spider 


Sloth 


Speed 


0.03 


0.17 


1.17 


0.15 



Compare. Write <, >, or =. (Lesson 2-1) 
48. |-14| -12 49. -7 -8 

Simplify. (Lessons 2-2 and 2-3) 

52. -13 + 51 53. 142 - (-27) 



50. 



54. -118 - (-57) 



51. 3 



55. -27 + 84 



2-7 7 Comparing and Ordering Rational Numbers 129 



CHAPTER 



2 



SECTION 2C 




Ready To Go On? 



.^y Learn It Online 



Quiz for Lessons 2-9 Through 2-11 

^^ 2-9 ] Equivalent Fractions and Mixed Numbers 

Determine whether the fractions in each pair are equivalent. 
1. 4 and I 2. A and 4 3. 



fandf 



j^and^ 



^and|j 



ResourcesOnlinego.hrw.com, 
■5S«MS10RTGO2CIgH 



4. ^and^ 
9 45 



5. There are 2-^, centimeters in an inch. When asked to write this value 



'100 



as an improper fraction, Aimee wrote —-. Was slie correct? Explain 



Qj 2-IOj Equivalent Fractions and Decimals 

Write each fraction as a decimal. Round to the nearest hundredth, if necessary. 



^- To ^- i ^-1 

Write each decimal as a fraction in simplest form. 
10. 0.22 11. -0.135 12. -4.06 



U 
15 



13. 0.07 



14. In one 30-gram serving of snack crackers, there are 24 grams of 
carbohydrates. Wliat fraction of a serving is made up of carbohydrates? 
Write your answer as a fraction and as a decimal. 

1 5. During a softball game, Sara threw 70 pitches. Of those pitches, 29 were 
strikes. Miat portion of tlie pitches that Sara threw were strikes? Write 
your answer as a decimal rounded to the nearest thousandth. 



^; 2-11] Comparing and Ordering Rational Numbers 
Compare the fractions. Write < or >. 



16. 



9 1 9 

- 17. -- — ~ 

4 8 11 

Compare the decimals. Write < or >. 

20. 0.521 0.524 21. 2.05 -2.50 
Order the numbers from least to greatest. 



18. 



24. |, -0.372, -|, 0.5 



26. -5.36,2.36, -5|, -2| 

3 6 



19. -1: 



22. 3.001 3.010 23. -0.26 



2^, |, 2.91, 0.9 



25 

27. 8.75,^.0 



-0.626 



28. Rafael measured the rainfall at his house for 3 days. On Sunday, it rained 
I in. On Monday, it rained | in. On Wednesday, it rained 0.57 in. List the 
days in order from the least to the greatest amount of rainfall. 



130 Chapter 2 Integers and Rational Numbers 



CONNECTIONS 





Amphibians and Reptiles of Arizona The desert cUmate 

of Arizona makes the state an ideal habitat for amphibians and 
reptiles. In fact, the state has more than 140 different species of 
lizards, turtles, snakes, frogs, and toads. Visitors to the state may 
even see one of the 11 species of rattlesnakes found in Arizona. 

1. Most reptiles can survive only in temperatures between -4 °C 
and 36 °C. What is the difference between these temperatures? 

2. In Arizona, there are 28 
species of amphibians 
and 52 species of 
snakes. An employee at 
a museum is arranging 
photos of these species 
on a wall. The photos 
will be placed in rows. 
Each row will have the 
same number of species 
of amphibians and the «s^ 
same number of species 
of snakes. 

a. The employee wants to make as many rows of photos as 
possible. How many rows can the employee make? 

b. How many photos of amphibians will be in each row? 
How many photos of snakes will be in each row? 

For 3-5, use the table. 

3. Write the length of the Gila monster as a 
decimal. 

4. Write the length of the desert iguana as a 
mixed number in simplest form. 

5. List the five species of lizards in order 
from shortest to longest. Explain how 
you put the species in order. 



ARIZONA 




Gila monster 



Lizards of Arizona 


Species 


Length (cm) 


Gila Monster 


^H 


Desert Iguana 


14.6 


Great Plains Skink 


133 
10 


Common Chuckwalla 


22.9 


Zebra-Tailed Lizard 


51 
5 




Real-World Connections 131 




Magic Squares 



A magic square is a grid with numbers, such that the numbers in 
each row, column, and diagonal have the same "magic" sum. 
Test the square at right to see an example of this. 

You can use a magic square to do some amazing calculations. 
Cover a block of four squares (2 x 2) with a piece of paper. 
There is a way you can find the sum of these squares without 
looking at them. Try to find it. {Hiiit:\Nhat number in the 
magic square can you subtract from the magic sum to give 
you the sum of the numbers in the block? Wliere is that 
number located?) 

Here's the answer: To find the sum of any block of four numbers, 
take 65 (the magic sum) and subtract from it the number that is 
diagonally two squares away from a corner of the block. 




18 


10 


22 


14 


1 




18 


10 


22 


14 


1 


12 


4 


16 


8 


25 


12 


4 


16 


S 


25 


6 


23 


\ 


2 


19 


5 23 


15 2 


19 


5 


17 


9 


21 


13 


5 17 


9 21 


13 


24 


11 


3 


20 


7 


24 


11 


3 


20 


7 



The number you 
subtract must fall on an 
extension of a diagonal 
of the block. For each 
block that you choose, 
there will be only one 
direction you can go. 



65 - 21 = 44 65 - 1 = 64 

Try to create a 3 x 3 magic square with the numbers 1-9. 



Modified Tic-Tac-Toe 



The board has a row of nine squares numbered 
1 through 9. Players take turns selecting squares. 
The goal of the game is for a player to select 
squares such that any three of the player's squares 
add up to 15. The game can also be played with 
a board numbered 1 through 1 6 
and a sum goal of 34. /C*^ '■"''" '* 0"''"^ 

*^ GameTimeExtrago.hrw.com 
A complete copy of the rules and a IJ I J I Iiyii ll il lMTiOGan-iesI 
game board are available online. 

132 Chapter 2 Integers and Rational Numbers 





Materials 

• 3 sheets of 
decorative paper 
(8^ in. by 8^ in.) 

• scissors 

• clear tape 

• markers 



^<^ 



PROJECT 



Flipping Over Integers 
and Rational Numbers 



Create your own flip-flop-fold book and use 
it to write definitions, sample problems, and 
practice exercises. 

Directions 

O Stack the sheets of decorative paper. Fold the 
stack into quarters and then unfold it. Use 
scissors to make a slit from the edge of the 
stack to the center of the stack along the left- 
hand crease. Figure A 

Place the stack in front of you with the slit on 
the left side. Fold the top left square over to the 
right side of the stack. Figure B 

Q Now fold down the top two squares from the 
top right corner. Along the slit, tape the bottom 
left square to the top left square. Figure C 

O Continue folding around the stack, always 
in a clockwise direction. When you get to the 
second layer, tape the slit in the same place 
as before. 

Taking Note of the Math 

Unfold your completed booklet. This time, 
as you flip the pages, add definitions, sample 
problems, practice exercises, or any other 
notes you need to help you study the 
material in the chapter. 







■\ 


\ 


il 






i 


[ 




V 




J 


i 







• • 







• • 



133 



^ ARTE 

2 



Study Guide: Revi^^^ 



Vocabulary 

absolute value 73 

additive inverse 72 

composite number 104 

equivalent fractions •] 1 3 

greatest common 
factor (GCF) 



108 



integer 72 

least common 

multiple (LCM) -,12 

multiple •]-|2 

opposite 72 

prime 

factorization -104 



prime number 104 

rational number 127 

relatively prime 113 

repeating decimal 122 

terminating decimal .122 



Complete the sentences below with vocabulary words from the list above. 

1 . A(n) V can be written as the ratio of one ? to another 
and can be represented by a repeating or ? 

2. The ? are the set of whole numbers and their 



_(S). 



EXAMPLES 

Z-f] Integers (pp. 72-75) 

■ Use a number line to order the integers 
from least to greatest. 

3,4,-2,1,-3 

■ I I l»*l |»|»»| ! ■ 



EXERCISES 



-6-4-2 2 4 6 

-3. -2, 1.3,4 



2-2 ) Adding Integers (pp. 80-83) 

■ Find the sum. 

-7+ (-11) 

— 7 + ( — 11) The signs are the same. 
-18 



Compare the integers. Use < or >. 

3. -8 -15 4. -7 7 

Use a number line to order the integers 
from least to greatest. 

5. -6,4,0, -2,5 6. 8, -3,2, -8, 1 

Use a number line to find each absolute 
value. 

7. lol 8. 1-17 I 9. |6| 



Find each sum. 

10. -8 + 5 11. 7 + (-6) 

12. -16 + (-40) 13. -9 + 18 

14. -2 + 16 + (-4) 15. 12 + (-18) + 1 

16. The temperature was -9 °F at 5 a.m. 
and rose 20° by 10 a.m. What was the 
temperature at 10 a.m.? 



134 Chapter 2 Integers and Rational Numbers 



EXAMPLES 



EXERCISES 



2-3] Subtracting Integers (pp. 86-89) 
E ■ Find the difference. 

|i -5 - (-3) 

■ — 5 + 3 = — 2 Add the opposite of -3. 



Find eacli difference. 

17. 8-2 18. 10 - 19 

19. -6- (-5) 20. -5-4 

21. 6 -(-5) -8 22. 10- (-3) - (-1) 



2-4] Multiplying and Dividing Integers (pp 92-95) 

Find each product or quotient. 

23. 5 • (-10) 24. -27 -^ (-9) 

25. -2 • (-8) 26. -40 h- 20 



Find each product or quotient. 

■ 12 • ( — 3) The signs are different, so 

the product is negative. 



■36 



— 16 ^ (—4) The signs are the same, so 
4 the quotient is positive. 



27. -3-4 



28. 45 ^ (-15) 



2-5] Solving Equations Containing Integers (pp 98 101) 



Solve. 




■ A- - 12 


= 4 


+ 12 


+ 12 


X 


= 16 


m -10 


= -2/ 


-10 
-2 


_-2/ 
-2 


5 


= / 



Add 12 to each side. 



Divide each side by -2. 



2-6] Prime Factorization (pp. 104-107) 

Write the prime factorization of 56. 

■ 56 = 8 • 7 = 2 • 2 • 2 • 7, or 2' • 7 



2-7] Greatest Common Factor (pp. 108-111) 



Find the GCF of 32 and 12. 

Factors of 32: 1, 2,08, 16, 32 
Factors of 12: 1,2,3,06, 12 
The GCF is 4. 



Solve. 

29. 7v = 70 
31. /• + 23 = 
33. -26= - 



30. c/ - 8 = 6 

32 ^ = 2 
36 

34. 28 = -7m 



35. A scuba diver is at the -30 foot level. 
How many feet will she have to rise to 
be at the - 12 foot level? 



Write the prime factorization. 

36. 88 37. 27 38. 162 39. 96 

40. Find two composite numbers that each 
have prime factors with a sum of 10. 



Find the greatest common factor. 

41. 120,210 42. 81, 132 

43. 36, 60, 96 44. 220, 440, 880 



yiii-:iu\ Lesson Tutorials OnlinE mv.hrw.com 



Study Guide: Review 135 



EXAMPLES 



EXERCISES 



2-8] Least Common Multiple (pp. 112-115) 

■ FindtheLCMofSandlO. 

Multiples of 8: 8, 16, 24, 32,® 
Multiples of 10: 10,20,30, 
The LCM is 40. 



Find the least common multiple. 

45. 5, 12 46. 4, 32 47. 3, 27 

48. 15, 18 49. 6, 12 50. 5, 7,9 

51 . Two tour buses leave the visitor's 

center at 10:00 a.m. Bus A returns to the 
visitors' center every 60 minutes. Bus B 
returns eveiy 45 minutes. At what time 
will the buses be together again at the 
center? 



2-9 j Equivalent Fractions and Mixed Numbers (pp. 118-121) 



Write 5^ as an improper fraction. 

r-2 _ 3-5 + 2 _ 1? 
3 3 3 

Write ^ as a mixed number. 



11= 17 H-4 = 4R1 =4i 



Divide the 
numerator 
by the 
denominator. 



Write each as an Improper fraction. 



52. 4 



53. 3 



5 "■ "6 

Write each as a mixed number. 



55. 



10 



56. I 



54. lof 

4 



57. 



Find two fractions equivalent to the given 
fraction. 



58. 



16 



59. 



24 



60. 



48 
63 



2-I0] Equivalent Fractions and Decimals (pp. 122-125) 



Write 0.75 as a fraction in simplest form. 

r.-,n-J5__ 75 + 25 _ 3 
100 100 + 25 4 

Write I as a decimal. 

1 = 5^4= 1.25 



Write each decimal as a fraction in 
simplest form. 

61. 0.25 62. -0.004 63. 0.05 



Write each fraction as a decimal. 



64. 



65. 3 

5 



66. I 



2-11] Comparing and Ordering Rational Numbers (pp. 126-129) 



Compare. Write < 


or >. 


3 


2 




4 


3 


Write as fractions 


3 3 

4 3 

-f^< 


2 4 

3 4 

8 
12 


with common 


denominators. 



Compare. Write < or > 



67. 



69. - 



0.81 



■1.5 



68. 0.22 



70. 1 



3_ 

20 



71. Order ^, 0.58, -0.55, and | from least 
to greatest. 



136 Chapter 2 Integers and Rational Numbers 



Chapter Test 



Use a number line to order the integers from least to greatest. 

1. -4,3, -2,0, 1 2. 7, -6,5, -8, -3 



CHAPTER 




Use a number line to find each absolute value. 

3. 111! 4. |-5| 5. |-74| 



6. -1 



Find each sum, difference, product, or quotient. 

7. -7 + (-3) 8.-6-3 9. 17 -(-9) -8 10. 102 + ( -97) + 3 

11.-3-20 12. -36 H- 12 13. -400 ^ (-10) 14. -5 -(-2) -9 



Solve. 

15. w- 4= -6 



16. .v + 5 = 



17. -6(7 = 60 



18. ^=12 



19. Kathr^aVs tennis team has won 52 matches. Her team has won 9 more 
matches than Rebecca's team. How many matches has Rebecca's 
team won this season? 



Write the prime factorization of each number. 
20. 30 21. 66 22. 



23. 



Find the greatest common factor (GCF). 
24. 18,27,45 25. 16,28,44 



26. 14,28,56 



27. 24,36,64 



Find the least common multiple (LCM). 
28. 24,36,64 29. 24,72, 144 



30. 12, 15,36 



31. 9, 16,25 



Determine whether the fractions in each pair are equivalent. 

34 30 ^ 35 

24 28 



32. ^and '^ 



33 iland^O 
6 ^^- 20 24 



Write each fraction as a decimal. Write each decimal as a fraction in 
simplest form. 



36. 



50 



37. 



25 
10 



38. 3.15 



35. #and' 



39. 0.004 



40. The Drama Club has 52 members. Of these members, 18 are in the 
seventh grade. What fraction of the Drama Club is made up of seventh- 
graders? Write your answer as a fraction and a decimal. Round the 
decimal to the nearest thousandth. 



Compare. Write < or >. 



41. 



0.62 



42 1 5 1— 



43. 



44. 



11 



H 



Chapter 2 Test 137 



CHAPTER 



2 



B 



ISTEP+ 

Test Prep 




^f*5' Learn It Online 

'** StateTestPracticego.hrw.com, 



Applied Skills Assessment 

Constructed Response 

1. The sponsors of the marching band 
provided 128 sandwiches for a picnic. 
After the picnic, s sandwiches were left. 

a. Write an expression that shows how 
many sandwiches were handed out. 

b. Evaluate your expression for s - ^S. 
What does your answer represent? 

2. Casey said the solution to the equation 
X + 42 = 65 is 107. Identify the error 
that Casey made. Explain why this 
answer is unreasonable. Show how 

to solve this equation correctly. 
Explain your work. 

Extended Response 

3. Mary's allowance is based on the 
amount of time that she spends 
practicing different activities each 
week. This week Mary spent 12 hours 
practicing and earned $12.00. 

a. Mary spent the following amounts 
of time on each activity: | practicing 
flute, I studying Spanish, ^ playing 
soccer, and :j^ studying math. Write 
an equivalent decimal for the 
amount of time that she spent on 
each activity. Round to the nearest 
hundredth, if necessary. 

b. For each activity, Mary earned the 
same fraction of her allowance as the 
time spent on a particular activity. 
This week, she was paid $2.00 for 
studying Spanish. Was this the correct 
amount? Explain how you know. 

c. Order the amount of time that Mary 
spent practicing each activity from 
least to greatest. 



Multiple-Choice Assessment 

4. During a week in January in Cleveland, 
Ohio, the daily high temperatures were 
-4 °F, -2 °F, -12 °F, 5 °F, 12 °F, 16 °F, and 
20 °F. Which expression can be used 
to find the difference between the 
highest temperature of the week and 
the lowest temperature of the week? 

A. 20 - 2 C 20 - 12 



B. 20 - (-2) 



D. 20 - (-12) 



5. Find the greatest common factor of 
16 and 32. 

A. 2 C. 32 

B. 16 D. 512 



The fraction I is found between which 
pair of fractions on a number line? 



and 



10 



B. land^ 



C^andA 
D-il^ndA 



Maxie earns $210 a week working as a 
lifeguard. After she gets paid, she gives 
each of her three sisters $20, and her 
mom $120 for her car payment. Which 
equation can be used to find p, the 
amount of money Maxie has left after 
she pays her mom and sisters? 

A. p = 210 - (3 X 20) - 120 

B. p = 210 - 20 - 120 

C. p = 120 - (3 X 20) - 120 . 

D. p = 3 X (210 - 20 - 120) 



138 Chapter 2 Integers and Rational Numbers 



8. Which expression can be used to 
represent a pattern in the table? 



-3 4 

-5 2 

-7 

-9 -2 



A. X + 2 

B. -2x 



C. x-(-7) 

D. X - 7 



Which of the following shows a list 
of numbers in order from least to 
greatest? 

A. -1.05, -2.55, -3.05 

B. -2.75, 2|, 2.50 

b 

C. -0.05, -0.01, 3^ 

b 

D. -^l -it 1.05 



13. Simplify the expression (—5)^ -3-4. 

A. -112 C. 13 

B. -37 D. 88 

14. Evaluate a - b for a = -5 and b — 3. 

A. -8 C. 2 

B. -2 D. 8 



Gridded responses cannot be negative 
numbers. If you get a negative value, you 
kely made an error Check your work! 



Gridded Response 

15. Find the missing value in the table. 




t _f + 3.5 


5 


10 


10 


7 



10. Which of the following is an example 
of the Associative Property? 

A. 5 + (4 + 1) = (5 + 4) + 1 

B. 32 + (2 + 11) = 32 + (11 +2) 

C. (2 X 10) + (2 X 4) = 2 X 14 

D. 4(2 X 7) = (4 X 2) + (4 X 7) 

11. There are 100 centimeters in 1 meter. 
Which mixed number represents 
525 centimeters in meters? 



A. 6^ meters 

B. 6| meters 



C. 6| meters 

D. 6| meters 



12. An artist is creating a design with 
6 stripes. The first stripe is 2 meters 
long. The second stripe is 4 meters long, 
the third stripe is 8 meters long, and the 
fourth stripe is 16 meters long. If the 
pattern continues, how long is the 
sixth stripe? 



A. 24 meters 

B. 32 meters 



C. 64 meters 

D. 128 meters 



16. Solve for x and y in each equation. 
Grid the sum of x and y. 



X + 6 = -4 



-3y = -39 



17. Garrett dusts his bedroom every four 
days and sweeps his bedroom every 
three days. If he does both today, how 
many days will pass before he does 
them both on the same day again? 

18. What is the power of 10 if you write 
5,450,000,000 in scientific notation? 

19. What is the value of 8^ ? 



Cumulative Assessment, Chapters 1-2 139 



CHAPTER 



3 




• i I 



niumbers 



3A Decimal Operations 
and Applications 

3-1 Estimating with Decimals 

3-2 Adding and Subtracting 
Decimals 

LAB Model Decimal 
Multiplication 

3-3 Multiplying Decimals 

LAB Model Decimal Division 

3-4 Dividing Decimals 

3-5 Solving Equations 
Containing Decimals 

3B Fraction Operations 
and Applications 

3-6 Estimating with Fractions 

LAB Model Fraction Addition 
and Subtraction 

3-7 Adding and Subtracting 
Fractions 

3-8 Adding and Subtracting 
Mixed Numbers 

LAB Model Fraction 
Multiplication and 
Division 







Multiplying Fractions and 
Mixed Numbers 

Dividing Fractions and 
Mixed Numbers 



3-9 

3-10 

3-11 Solving Equations 



Conta 



ning Fractions 



7.1.7 

7.1.7 
7.2.1 



7.1.7 



7.1.7 



7.1.7 



7.2.1 



'-^A!-- 



Why Learn This? 



By using operations with decimals, you can 
determine statistics for football players and 
teams. 



Learn It Online 

Chapter Project Online go.hrw.com, 



keyword ■MHllM^M ® 




140 Chapter r\ 



Are You Ready? 






Learn It Online 

Resourtes Online go.hrw.com, 

l!fflJ^S10AYR3 ■'go: 



0^ Vocabulary 

Choose the best term from the list to complete each sentence 

1. A(n) 1 is a number that is written using the 

base-ten place value system. 

2. An example of acn) I is ^. 

3. A(n) I is a number that represents a part of 

a whole. 



decimal 
fraction 

improper fraction 
mixed number 
simplest form 



Complete these exercises to review the skills you will need for this chapter. 

Simplify Fractions 

Write each fraction in simplest form. 



4 24 

^- 40 


5 M 

^- 84 


s-i 


7 64 
192 


8 21 
^- 35 


9 ii 
^- 99 


lo-i 


11 20 

30 



Q} Write Mixed Numbers as Fractions 

Write each mixed number as an improper fraction. 



12. 7^. 



16. 3^ 



13. 27 



17. 8:^ 



14. 1 



18. 4i 



15. 3. 



19 5— 



Write Fractions as Mixed Numbers 

Write each Improper fraction as a mixed number. 

20. 

24. 



23 
6 


21. 


17 
3 


22. 


29 

7 


48 
5 


25. 


82 
9 


26. 


69 
4 



23. 



27. 



39 
4 

35 



Q) Add, Subtract, Multiply, or Divide Integers 

Find each sum, difference, product, or quotient. 

28. -11 + (-24) 29. -11-7 30. -4 -(-10) 

31. -22 + (-11) 32. 23 + (-30) 33. -33-74 

34. -62 • (-34) 35. 84 + (-12) 36.-26-18 




Applying Rational Numbers 




Where You've Been 

Previously, you 

• added, subtracted, multiplied, 
and divided whole numbers. 

• used models to solve equations 
with whole numbers. 



In This Chapter 



You will study 

• using models to represent 
multiplication and division 
situations involving fractions 
and decimals. 

• using addition, subtraction, 
multiplication, and division to 
solve problems involving 
fractions and decimals. 

• solving equations with 
rational numbers. 



Key 

Vocabulary /Vocabulario 

compatible numbers numeros compatibles 



reciprocal 



reciproco 



Vocabulary Connections 

To become familiar with some of the 
vocabulary terms in the chapter, consider 
the following. You may refer to the chapter, 
the glossary, or a dictionary if you like. 

1. Wlien two things are compatible, they 
make a good match. You can match a 
fraction with a number that is easier to 
work with, such as 1, ;^, or 0, by rounding 
up or down. How could you use these 
compatible numbers to estimate the 
sums and differences of fractions? 

2. When fractions are reciprocals of each 
other, they have a special relationship. The 
fractions | and | are reciprocals of each 
other. What do you think the relationship 
between reciprocals is? 



Where You're Going 

You can use the skills 
learned in this chapter 

• to estimate total cost when 
purchasing several items at the 
grocery store. 

• to find measurements in fields 
such as carpentry. 



142 Chapter 3 Applying Rational Numbers 




Reading ../ 
and WriMKa 
MathX -^ 




Study Strategy: Use Your lUotes Effectively 

Taking notes helps you understand and remember information from 
your textbook and lessons in class. Listed below are some steps for 
effectively using your notes before and after class. 



Step 1: Before Class 

• Read through your 
notes from the last 
class. 

• Then look ahead to the 
the next lesson. Write 
down any questions 
you have. 



Step 2: During Class 

• Write down mam points 
that your teacher 
stresses. 

• If you miss something, 
leave a blank space and 
keep taking notes. 

• Use abbreviations. Make 
sure you will understand 
any abbreviations later. 

• Draw pictures or 
diagrams. 



y 



7 z, -prime Factorization 

IO/Z/07 Lesson L b 

Ho^ do I kno^ -f.- I hc.3 ^o.n6 th. pr.r.. 
factorization of a number? 

p.,.enamber-wbolenumber>lthat^- 
..octly Z factors: I and itself £- ^ ^^ 

Composite namber-wbole num. that has more 
ihan Z iocfors £^ '^ 6. 9 

The number I has e^ocily om factor. 

fVot prime and not composite 

Prime factorization -o composite num. written 

OS the product of its pr.r.e factors 



Step 3: After Class 

• Fill in any information 
you may have missed. 

• Highlight or circle the 
most important ideas, 
such as vocabulary, 
formulas and rules, or 
steps. 

• Use your notes to quiz 
a friend or yourself. 




Try This 



1. Look at the next lesson in your textbook. Think about how the new 
vocabulary terms relate to previous lessons. Write down any questions 
you have. 

2. With a classmate, compare the notes you took during the last class. Are 
there differences in the main points that you each recorded? Then 
brainstorm two ways you can improve your note-taking skills. 



Applying Rational Numbers 143 



Vocabulary 

compatible numbers 



Jessie earned $26.00 for baby-sitting. She 
wants to use the money to buy a ticicet to 
an aquarium for $14.75 and a souvenir 
T-shirt for $13.20. 

To find out if Jessie lias enough money to 
buy both items, you can use estimation. To 
estimate the total cost of the ticket and the 
T-shirt, round each price to the nearest dollar, 
or integer. Then add the rounded values. 



$14.75 7> 5, so round to $15. 

$13.20 2 < 5, so round to $13. 



$15 

-I- $13 

$28 






■I ■. ^'^^ 




the Georgia Aquarium in Atlanta, GA, is the 
world's largest aquarium, with more than 
8.1 million gallons of water. 



The estimated cost is $28, so Jessie does not have enough money to 
buy both items. 

To estimate decimal sums and differences, round each decimal to the 
nearest integer and then add or subtract. 



EXAMPLE 



CD 



j.:j:iJ.JJ^JJ.^J^J': 

To round to the 
nearest integer, look 
at the digit in the 
tenths place. If it is 
greater than or 
equal to 5, round to 
the next integer. If 
it is less than 5, keep 
the same integer. 
See Skills Bank p. SBl. 



Estimating Sums and Differences of Decimals 

Estimate by rounding to the nearest integer. 

A 86.9 + 58.4 

86.9 > 87 9>S, so round to 87. 





-1- 58.4 


+ 58 


4< 5, so round to 58. 






145 


t Estimate 


B 


10.38 - 6.721 








10.38 ► 


10 


3 < 5, so round to 10. 




- 6.721 


-7 


7 > 5, so round to 7. 






3 


-< Estimate 


C 


-26.3 -f 15.195 








-26.3 


-26 


3 < 5, so round to -26 




+ 15.195 


-1- 15 


1 < 5, so round to 15. 






-11 


< Estimate 



You can use compatible numbers when estimating. Compatible 
numbers are numbers that are close to the given numbers that make 
estimation easier. 



144 Chapter 3 Applying Rational Numbers 



'Ma 



Lessor Tutorials Online mv.hrw.com 



Guidelines for Using Compatible Numbers 



When multiplying . . . 

round numbers to the nearest 
nonzero integer or to numbers 
that are easy to multiply. 



When dividing . . . 

round numbers so that they 
divide without leaving a remainder. 



EXAMPLE [zj Estimating Products and Quotients of Decimals 

Use compatible numbers to estimate. 



ll}JJ3Si 



A prime number has 
exactly two factors, 
1 and itself. So the 
factors of 37 are 
1 and 37. 



32.66 • 


7.69 




32.66 


»■ 


30 


X 7.69 


> 


X 8 
240 


36.5 H- 


(-8.241) 


36.5 


*■ 


36 


-8.241 


*■ 


-9 


36^ 


(-9) = 


-4 -^ 



Round to the nearest multiple of W. 

6> 5, so round to 8. 

Estimate 



37 is a prime number, so round to 36. 
-9 divides into 36 without a remainder. 
Estimate 



When you solve problems, using an estimate can help you decide 
whether your answer is reasonable. 



EXAMPLE [ij School Application 



On a math test, a student worked the problem 6.2)55.9 and got 
the answer 0.9. Use estimation to check whether the answer is 
reasonable. 



6.2 .- 6 

55.9 60 

60 ^ 6 = 10 



2 < 5, so round to 6. 

6 divides into 60 without a remainder. 

Estimate 



The estimate is more than ten times the student's answer, so 0.9 is not 
a reasonable answer. 



Tfimk and Discuss 

1. Explain whether your estimate wall be greater than or less than 

the actual answer when you round botli numbers down in an addition 
or multiplication problem. 

2. Describe a situation in which you would want your estimate 
to be greater than the actual amotmt. 



VJJa;; Lesson Tutorials Online mv.hrw.com 



3-1 Estimating with Decimals 145 



3-1 



illiJj'iJ-EQS 



[•Tiiiiii 

Homework Help Online go.hrw.com, 



keyword ■MBiBKaM ® 
Exercises 1-20, 21, 27, 33, 35, 
37,39,41 



GUIDED PRACTICE 



See Example 1 Estimate by rounding to the nearest integer. 

L 1. 37.2 + 25.83 2. 18.256-5.71 

See Example 2 Use compatible numbers to estimate. 

L 4. 8.09 • 28.32 5. -3.45 • 73.6 



-9.916+ 12.4 



6. 41.9 + 6.391 



See Example 3 7. School A student worked the problem 35.8 • 9.3. The student's answer was 
[ 3,329.4. Use estimation to check whether this answer is reasonable. 



INDEPENDEN 



See Example 1 Estimate by rounding to the nearest integer. 

i 8. 5.982 + 37.1 9. 68.2 + 23.67 



11. 15.23 -6.835 



12. 6.88 + (-8.1) 



See Example 2 Use compatible numbers to estimate. 

14.51.38-4.33 15.46.72 + 9.24 



17. -3.45 -43.91 



18. 2.81 • (-79.2) 



10. -36.8+ 14.217 

13. 80.38 - 24.592 

16. 32.91 - 6.28 

19. 28.22 + 3.156 



See Example 3 20. Ann has a piece of rope that is 12.35 m long. She wants to cut it into smaller 
pieces that are each 3.6 m long. She thinks she will get about 3 smaller pieces 
L of rope. Use estimation to check whether her assumption is reasonable. 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EPS. 



Estimate. 

21. 5.921 - 13.2 

24. 98.6 + 43.921 

27. -48.28 + 11.901 

30. 69.7 - 7.81 

33. -8.9 - (-24.1) 



22. -7.98 - 8.1 
25. 4.69 • (-18.33) 
28. 31.53 + (-4.12) 
31. -6.56- 14.2 
34. 6.92 - (-3.714) 



23. -42.25 + (-17.091) 

26. 62.84 -35.169 

29. 35.9 - 24.71 

32. 4.513 + 72.45 

35. -78.3 + (-6.25) 



36. Jo needs 10 lb of ground beef for a party. She has packages that weigh 
4.23 lb and 5.09 lb. Does she have enough? 

37. Consumer Math Ramon saves $8.35 each week. He wants to buy a video 
game that costs $61.95. For about how many weeks will Ramon have to save 
his money before he can buy the video game? 

38. Multi-Step Tickets at a local movie theater cost $7.50 each. A large bucket of 
popcorn at the theater costs $4.19, and a large soda costs $3.74. Estimate the 
amount that 3 friends spent at the theater when they saw one movie, shared 
one large bucket of popcorn, and had one large soda each. 



146 Chapter 3 Applying Rational Numbers 




Ringette is a 
team sport 
originally 
developed to be 
played by girls. 
Players wear ice 
skates and use 
straight sticks to 
pass, carry, and 
shoot a 
rubber ring to 
score goals. 



39. Transportation Kayla stopped for gasoline at a station that was charging 
$2,719 per gallon. If Kayla had $14.75 in cash, approximately how many 
gallons of gas could she buy? 



Languages Spoken 
in Canada 



French 

23.2% 



40. Social Studies The circle graph 
shows the languages spoken in 
Canada. 

a. Which language do approximately 
60% of Canadians speak? 

b. What is the approximate 
difference between the percent of 
people who speak English and the 
percent who speak French? 

41. Astronomy Jupiter is 5.20 astronomical units (AU) from the Sun. Neptune is 
almost 6 times as far from the Sun as Jupiter is. Estimate Neptune's distance 
from the Sun in astronomical units. 




4 



Sports Scott must earn a total of 27 points to advance to the final round in 
an ice-skating competition. He earns scores of 5.9, 5.8, 6.0, 5.8, and 6.0. Scott 
estimates that his total score will allow him to advance. Is his estimate 
reasonable? Explain. 

43. Write a Problem Write a problem that can be solved by estimating with 
decimals. 

44. Write About It Explain how an estimate helps you decide whether an 
answer is reasonable. 

^ 45. Challenge Estimate. 6.35 - 15.512 + 8.744 - 4.19 - 72.7 + 25.008 



i 



Test Prep and Spiral Review 



46. Multiple Choice Wliich is the best estimate for 24.976 ^ (-4.893)? 
CA) 20 CD -6 CD -5 CS) 2 

47. Multiple Choice Steve is saving $10.50 from his allowance each week to 
buy a printer that costs S150. Which is the best estimate of the number of 
weeks he will have to save his money until he can buy the printer? 



CE> 5 weeks 



CS) 10 weeks 



CE) 12 weeks 



CT) 15 weeks 



48. Short Response Joe's restaurant bill was $16.84. He had $20 in his 

wallet. Explain how to use rounding to estimate whether Joe had enough 
money to leave a $2.75 tip. 



Simplify each expression. (Lessons 2-3 and 2-4) 

49. -5 + 4-2 50. 16 • (-3) -I- 12 



52. -90- (-6) • (-8) 



53. -7-3-1 



51. 28- (-2) • (-3) 
54. -10 • (-5) -I- 2 



3-1 Estimating witli Decimals 147 



L 



3-2 



Adding and Subtracting 
Decimals 



One of the coolest summers 
on record in the Midwest 
was in 1992. The average 
summertime temperature 
that year was 66.8 °F. 
Normally, the average 
temperature is 4 °F higher 
than it was in 1992. 

To find the normal average 
summertime temperature 
in the Midwest, you can add 
66.8 °F and 4 °F. 




Interactivities Online ► 



+ 4 



t 



Use zero as a placeholder so that both numbers have 
the same number of digits after their decimal points. 



Add each column just as you would add integers. 



Line up the decimal points. 
The normal average summertime temperature in the Midwest is 70.8 °F. 



EXAMPLE [T] Adding Decimals 



Add. Estimate to check whether each answer is reasonable. 



A 3.62 + 18.57 

3.62 
+ 18.57 





22.19 




Estimate 




4 + 19 = 23 


B 


9 + 3.245 




9.000 




+ 3.245 



12.245 

Estimate 

9 + 3 = 12 



Line up the decimal points. 
Add. 

22. 19 is a reasonable answer. 



Use zeros as placeholders. 
Line up the decimal points. 
Add. 



12.245 Is a reasonable answer. 



148 Chapter 3 Applying Rational Numbers 



Vldao Lesson Tutorials Online mv.hrw.com 



When adding 
numbers with the 
same sign, find the 
sum of their absolute 
values. Then use the 
sign of the numbers. 



Add. Estimate to check whether each answer is reasonable. 
C -5.78 + (-18.3) 



-5.78 + (-18.3) 

5.78 
+ 18.30 
24.08 
-5.78 + (-18.3) = -24.08 
Estimate 
-6 + (-18) = -24 



Think: 5.78 + 18.3. 

Line up the decimal points. 

Use zero as a placeholder. 

Add. 

Use the sign of the two numbers. 

-24.08 is a reasonable answer. 



EXAMPLE [Tj Subtracting Decimals 



Caution! 



You will need to 
regroup numbers m 
order to subtract in 
Example 2B. 



Subtract. 

A 12.49 - 7.25 

12.49 

- 7.25 

5.24 



14 - 7.32 

13 910 

-7.32 



6.68 



Line up the decimal points. 
Subtract. 



Use zeros as placeholders. 
Line up the decimal points. 
Subtract. 



EXAIV1PLE [31 Transportation Application 



During one month in the United States, 492.23 million commuter 
trips were taken on buses, and 26.331 million commuter trips were 
taken on light rail. How many more trips were taken on buses than 
on light rail? Estimate to check whether your answer is reasonable. 

492.230 Use zero as a placeholder. 

— 26.331 Line up the decimal points. 

465.899 Subtract. 

Estimate 

490 - 30 = 460 465.899 is a reasonable answer. 

465.899 million more trips were taken on buses than on light rail. 



^^^^^^^^^^^^^^^^^^^^^^^^^^^^^B 


Think and Discuss 


12.3 


1. Tell whether the addition is correct. If it is not, 


+ 4.68 


explain why not. 


5.91 


2. Describe how you can check an answer when adding 




and subtracting decimals. 





y'ulBD Lesson Tutorials Online my.hrw.com 



3-2 Adding and Subtracting Decimals 149 



3-2 



iSk^iisiMi^iS&dMSSiMiSii 



.i^^j'ilM^ 




ri(gg. y^.;^S¥r'A'4l^A7T\fr 



GUIDED PRACTICE 




Homework Help Online go.hrw.com, 



keyword ■BHIileaiM ® 

Exercises 1-27, 29, 31, 33, 35, 
37,39,43 



See Example 1 Add. Estimate to check whether each answer is reasonable. 

I 1.5.37+16.45 2.2.46+11.99 3.7 + 5.826 4. -5.62 +(-12.9) 



See Example 2 Subtract. 

L 5. 7.89 - 5.91 



6. 17 - 4.12 



7. 4.9^ 



3.2 8. 9 - 1.03 



See Example 3 9. In 1990, international visitors to the United States spent $58.3 billion. In 
1999, international visitors spent $95.5 billion. By how much did spending 
by international visitors increase from 1990 to 1999? 



INDEPENDENT PRACTICE 



See Example 1 Add. Estimate to check whether each answer is reasonable. 

12. 4.917 + 12 



10. 7.82 + 31.23 11. 5.98 + 12.99 
14. 6 + 9.33 15. 10.022 + 0.11 



16. 8 + 1.071 



13. -9.82 +(-15.7) 
17. -3.29 +(-12.6) 



See Example 2 Subtract. 

18. 5.45 -3.21 

22. 5 - 0.53 



19. 12.87 -3.86 
23. 14 - 8.9 



20. 15.39 - 2.6 
24. 41 - 9.85 



21. 21.04 - 4.99 
25. 33 - 10.23 



See Example 3 26. Angela runs her first lap around the track in 4.35 minutes and her second 
lap in 3.9 minutes. What is her total time for the two laps? 

27. A jeweler has 122.83 grams of silver. He uses 45.7 grams of the silver to 
make a necklace and earrings. How much silver does he have left? 



Extra Practice 



See page EPS. 



PRACTICE AND PROBLEM SOLVING 



Add or subtract. Estimate to check whether each answer is reasonable. 

28. -7.238 + 6.9 29. 4.16-9.043 30. -2.09-15.271 

31. 5.23 - (-9.1) 32.-123-2.55 33.5.29-3.37 



34. 32.6- (-15.86) 
37. 5.9 - 10 + 2.84 



2.55 

35. -32.7 + 62.82 
38. -4.2 + 2.3 - 0.7 



36. -51 + 81.623 

39. -8.3 + 5.38 - 0.537 



40. Multi-Step Students at Hill Middle School plan to run a total of 2,462 mi, 
which is the distance from Los Angeles to New York City. So far, the sixth 
grade has run 273.5 mi, the seventh grade has run 275.8 mi, and the eighth 
grade has run 270.2 mi. How many more miles must the students run to 
reach their goal? 

41. Critical Thinking Why must you line up the decimal points when adding 
and subtracting decimals? 



1 50 Chapter 3 Applying Rational Numbers 






Physical Science 




Egg-drop 
competitions 
challenge students 
to build devices 
that will protect 
eggs when they are 
dropped from as 
high as 100 ft, 



Weather The graph shows the five 
coolest summers recorded in the Midwest. 
The average summertime temperature in 
the Midwest is 70.8 °F. 

42. How mucli warmer was the average 
summertime temperature in 1950 
than in 1915? 

43. In what year was tlie temperature 
4.4 °F cooler than the average 
summertime temperature in 
the Midwest? 



< 



Summers When 


the Midv 
olest 


irest 


Was Co 


69 

or 68 






68.0 68.0 
67 6 




V 67 
= 66 

1 ^^ 
-64 

2i 63 


66.4 


I 1 
1 


66.8 


S, 62 

5; 61 

< 60 ^ 


1 


1903 1915 1927 1950 


1992 


Year 





Source Midwestern Regional Climate Center 



Physical Science To float in water, an 

object must have a density' of less than 

1 gram per milliliter. The densit\' of a 

fresh egg is about 1.2 grams per milliliter. If the density of a spoiled egg is 

about 0.3 grams per milliliter less than that of a fresh egg, what is the density 

of a spoiled egg? How can you use water to tell whether an egg is spoiled? 

^^ 45. Choose a Strategy How much larger 
in area is Agua Fria than Pompeys Pillar? 

CS) 6.6 thousand acres 

CD 20.1 thousand acres 

CSD 70.59 thousand acres 

CS) 71.049 thousand acres 

^1 46. Write About It Explain how to find the sum or difference of two decimals. 

^ 47. Challenge Find the missing number. 5.11 + 6.9 - 1 5.3 + =20 



National 
Monument 


Area 
(thousand acres) 


Agua Fria 


71.1 


Pompeys Pillar 


0.051 




Test Prep and Spiral Review 



48. Multiple Choice In the 1900 Olympic Games, the 200-meter dash was won 
in 22.20 seconds. In 2000, the 200-meter dash was won in 20.09 seconds. 
How many seconds faster was the winning time in the 2000 Olympics? 

CA) 1.10 seconds CEj 2.11 seconds (Xj 2.29 seconds CS) 4.83 seconds 

49. Multiple Choice John left school with $2.38. He found a quarter on his 
way home and then stopped to buy a banana for $0.89. How much money 
did he have when he got home? 



CD $1.24 



CD $1.74 



CE) $3.02 



Solve each equation. Check your answer. (Lesson 2-5) 
50. A- - 8 = -22 51. -3j' = -45 



52. f = -8 



Estimate. (Lesson 3-1) 
54. 15.85 -^ 4.01 



55. 18.95 + 3.21 



56. 44.217 - 19.876 



CD $3.52 



53. 29 = -10 + /; 



57. 21.43 • 1.57 



3-2 Adding and Subtracting Decimals 151 



CKV\6S>-0 




p 



Model Decimal 
Multiplication 



KEY 



Use with Lesson 3-3 



Ok Lai 



Learn It Online 

Lab Resources Online go.hrw.com, 
■ lUii.li.lMMOlahrgGT " 



/y:r','i'y,i'r 


M 


H±-± 


3=1 s 


mm? 


w 



= 1 





"ffl 


::!:: 


±i| 


- + -■ 


iffl 



0.1 



= 0.01 s= 0.001 



REMEMBER 

• When using base-ten 
blocks, always use 
the largest value 
block possible. 



You can use base-ten blocks to model multiplying decimals by 
whole numbers. 



Activity 1 



O Use base-ten blocks to find 3 • 0.1. 

Multiplication is repeated addition, so 3 • 0.1 = 0.1 -1-0.1 -)- 0.1. 



-^^ 


^.m^ 


F^ 


lE-spsr 


R f 


■1 1 -1 1 1 1. 1 J j : 

■~f-\- ' ■ --r* 


--h~l 


_L-,. 'J 




— +4- 


-^ 










it,^ '• 








1 








! 1 ; 1 1 [ 




I 






























: ~^^4 


. ZT 














■'■' 'ii'.tM 










1 





3-0.1 =0.3 



Q Use base-ten blocks to find 5 ■ 0.03. 

5 • 0.03 = 0.03 -1- 0.03 -I- 0.03 + 0.03 -I- 0.03 



m m 






10 • 0.01 = 0.1 



t~ 






h 


n 




M 


m 
















1 


























1 




































1 




























































1 



3 S (3 (31 S 



5 • 0.03 = 0.15 



I 



1 52 Chapter 3 Applying Rational Numbers 



^•g""ia(ES 



Think and Discuss 



1. Why can't you use base- ten blocks to model multiplying a decimal by 
a decimal? 

2. Is the product of a decimal between and 1 and a whole number 
less than or greater than the whole number? Explain. 



Try This 



Use base-ten blocks to find each product. 

1. 4-0.5 2. 2 • 0.04 3. 3 • 0.16 

5. 3 • 0.33 6. 0.25 • 5 7. 0.42 • 3 



4. 6 • 0.2 
8. 1.1 -4 



You can use decimal grids to model multiplying decimals by decimals. 



Activity 2 



O Use a decimal grid to find 0. 4 • 0. 



Shade 0.4 horizontally. 



Shade 0.7 vertically. 



0.4 



0.7 



The area where the shaded 
regions overlap is the answer. 



0.28 



Think and Discuss 



1. Explain the steps you would take to model 0.5 • 0.5 with a decimal grid. 

2. How could you use decimal grids to model multiplying a decimal by a 
whole number? 



Try This 



Use decimal grids to find each product. 

1. 0.6- 0.6 2. 0.5 • 0.4 

4. 0.2 • 0.8 5. 3 • 0.3 

7. 2- 0.5 8. 0.1 -0.9 



3. 0.3 • 0.8 
6. 0.8 • 0.8 
9. 0.1 -0.1 



3-3 Hands-On Lab 1 53 



3-3 







7.1.7 Solve problems that involve multiplication and division with integers, fractions, decimals 
and combmations of the four operations 

You can use decimal grids to model multiplication of decimals. Each 
large square represents 1. Each row and column represents 0.1. Each 
small square represents 0.01. The area where the shading overlaps 
shows the product of the two decimals. 



0.7 



■ c_ 

»^ 



0.8 



0.56 



Interactivities Online ► 



To multiply decimals, multiply 
as you would with integers. To 
place the decimal point in the 
product, count the number of 
decimal places in each factor. 
The product should have the 
same number of decimal places 
as the sum of the decimal places 
in the factors. 




Same digits 



@,'F ' decimal place 
X O.S + ' decimal place 



Ql.S S 2 decimal places 



i X A M P L E [l] Multiplying Integers by Decimals 

Multiply. 



A 6-0.1 

6 

X 0.1 

0.6 



decimal places 

1 decimal place 
0+1=1 decimal place 



B -2 • 0.04 

—2 
X 0.04 
-0.08 

C 1.25 • 23 

1.25 

X 23 

3 75 

+ 25 00 

28.75 



decimal places 
2 decimal places 
+ 2 = 2 decimal places. Use zero as a placeholder. 



2 decimal places 
decimal places 



2 + = 2 decimal places 



154 Chapter 3 Applying Rational Numbers 



faJBi)] Lessor Tutorials Online my.hrw.com 



EXAMPLE [zj Multiplying Decimals by Decimals 

Multiply. Estimate to check whether each answer is reasonable. 



A 1.2 • 1.6 

1.2 
X 1.6 



-2.224 

Estimate 

-3 • 1 = -3 



7 decimal place 
1 decimal place 



72 




120 




1.92 


1 + 1=2 decimal places 


Estimate 




1-2 = 2 


1.92 is a reasonable answer 


B -2.78 • 0.8 




-2.78 


2 decimal places 


X 0.8 


1 decimal place 



2+1=3 decimal places 



-2.224 is a reasonable answer. 



EXAMPLE 



[3 



Nutrition Application 

On average, Americans eat 
0.25 lb of peanut butter per 
month. How many pounds of 
peanut butter are eaten by 
the approximately 302 million 
Americans living in the United 
States per month? 

302 decimal places 

X 0.25 2 decimal places 

1510 

6040 




75.50 
Estimate 

300-0.3 = 90 -^5.50 



+ 2 = 2 decimal places 



reasonable answer 



Approximately 75.50 million (75,500,000) pounds of peanut butter are 
eaten by Americans each month. 



Think and Discuss 

1. Explain whether the multiplication 2.1 • 3.3 = 69.3 is correct. 

2. Compare multiplying integers with multiplying decimals. 



'J'aib'j Lesson Tutorials Online my.hrw.com 



3-3 IVIultiplying Decimals 155 



3-3 



liM^i^iiMiiiaMiMMMSiisii^ 





keyword MteiWKflM ® 
Exercises 1-27, 31, 33, 37, 39, 
41,43,47 



GUIDED PRACTICE 



See Example 1 Multiply. 

L 1. -9 -0.4 



2. 3 • 0.2 



3. 0.06 • 3 



4. -0.5 • 2 



See Example 2 Multiply. Estimate to check whether each answer is reasonable. 

L 5. 1.7-1.2 6. 2.6-0.4 7. 1.5 -(-0.21) 8. -0.4-1.17 

See Example 3 9. If Carla is able to drive her car 24.03 miles on one gallon of gas, how far 
L could she drive on 13.93 gallons of gas? 



INDEPENDENT PRACTICE 



See Example 1 Multiply. 
10. 8 -0.6 

14. 6 • 4.9 



11. 5 -0.07 
15. 1.7- (-12) 



12. -3 -2.7 
16. 43 • 2.11 



13. 0.8 - 4 

17. -7- (-1.3) 



See Example 2 Multiply. Estimate to check whether each answer is reasonable. 

18.2.4-3.2 19.2.8-1.6 20.5.3-4.6 21.4.02-0.7 



22. -5.14 -0.03 



23. 1.04- (-8.9) 



24. 4.31 - (-9.5) 



25. -6.1 • (-1.01) 



See Example 3 26. Nicholas bicycled 15.8 kilometers each day for 18 days last month. How 
many kilometers did he bicycle last month? 

27. Wliile walking, Lara averaged 3.63 miles per hour. How far did she walk in 
1.5 hours? 



Extra Practice 



See page EPS. 



PRACTICE AND PROBLEM SOLVING 



Multiply. Estimate to check whether each answer is reasonable. 
28. -9.6-2.05 29. 0.07-0.03 30. 4-4.15 

31. -1.08 • (-0.4) 

34. -325.9- 1.5 

37. -7.02 - (-0.05) 



32. 1.46- (-0.06) 
35. 14.7-0.13 
38. 1.104- (-0.7) 



33. -3.2 -0.9 

36. -28.5 • (-1.07) 

39. 0.072 - 0.12 



40. Multi-Step Bo earns $8.95 per hour plus commission. Last week, he 
worked 32.5 hours and earned $28.75 in commission. How much money 
did Bo earn last week? 

41. Weather As a hurricane increases in intensity, the air pressure within its 
eye decreases. In a Category 5 hurricane, which is the most intense, the air 
pressure measures approximately 27.16 inches of mercury. In a Category 1 
hurricane, which is the least intense, the air pressure is about 1.066 times 
that of a Category 5 hurricane. What is the air pressure within the eye of a 
Category 1 hurricane? Round your answer to the nearest hundredth. 



1 56 Chapter 3 Applying Rational Numbers 



Boom on U.S. Rivers 



Rafting/ 




■ 7.6 


! 1 








Tubing 


20.1 














J 7.0 


Canoeing 






19.7 


Kayaking 


11.3 


6.6 





10 15 20 

Millions of people 



25 




1994-1995 
999-2000 



1 Todsy 



42. Estimation The graph shows 
the results of a survey about 
river recreation activities. 

a. A report claimed that about 
3 times as many people 
enjoyed canoeing in 
1999-2000 than in 
1994-1995. According to 
the graph, is this claim 
reasonable? 

b. Suppose a future survey 
shows that 6 times as many 
people enjoyed kayaking 
in 2016-2017 than in 
1999-2000. About how 
many people reported that 

they enjoyed kayaking in 2016-201" 



Multiply. Estimate to check whether each answer is reasonable. 

43. 0.3 • 2.8 • (-10.6) 44. 1.3 • (-4.2) • (-3.94) 

45. 0.6 • (-0.9) • 0.05 46. -6.5 • (-1.02) • (-12.6) 

47. -22.08 • (-5.6) • 9.9 48. -63.75 • 13.46 • 7.8 

^p 49. What's the Question? In a collection, each rock sample has a mass of 
4.35 kilograms. There are a dozen rocks in the collection. If the answer is 
52.2 kilograms, what is the question? 

^ 50. Write About It How do the products 4.3 • 0.56 and 0.43 • 5.6 compare? 
Explain. 

@51. Challenge Evaluate (0.2)\ 



Test Prep and Spiral Review 



52. Multiple Choice Which expression is equal to -4.3? 

CS) 0.8 -(-5.375) Cl:> -1.2 -(-3.6) CD -0.75-5.6 



(^ 2.2- (-1.9) 



53. Gridded Response Julia walked 1.8 mi each day from Monday through 
Friday. On Saturday, she walked 2.3 mi. How many miles did she walk in all? 



Write the prime factorization of each number. L'son 2-6) 
54. 20 55. 35 56. 120 



57. 64 



Add or subtract. Estimate to check whether each answer is reasonable. (Lesson 3 2) 

58. -4.875 + 3.62 59. 5.83 - (-2.74) 60. 6.32 + (-3.62) 61. -8.34 - (-4.6) 

62.9.3 + 5.88 63. 32.08-12.37 64. 19-6.92 65. -75.25 + 6.382 



3-3 Multiplying Decimals 157 



Model Decimal 
Division 



Use with Lesson 3-4 



£?, 



Learn It Online 

Lab Resources Online go.hrw.com, 



KEY 



p.|^f.i^::.=m-(^r|!b|:..}r..[ [5 




rn 




Ti r 






-f-\ 


































































































































l-i=! ■ ■ .. ■ ,■ '■ 








B 1 1 






1 

one 




0.1 
1 tenth 


0.01 
1 hundredth 



You can use decimal grids to model dividing decimals by integers 
and by decimals. 



Activity 



O Use a decimal grid to find 0.6 -r 2. 

Shade 6 ':o!- :/:-::-■_. !■) represent 0.6. 



Divide the 6 columns into 2 equal groups. 



: 


v. 


















-V 






















































1 


















^ 


















i 




































y 



































There are 3 columns, or 30 squares, 
in each group. 3 columns = 0.3 
0.6 -^ 2 = 0.3 



Use decimal grids to find 2.25 -h 5. 



Shade 2 grids and 25 squares of a third grid to represent 2.25. 



Divide the grids and squaies into 5 equal groups. Use scissors to cut 
apart the qrids. Think: 225 squares ^ 5 = 45 squares. 



There are 45 squares, or 4.5 columns, in each group. 4.5 columns = 0.45 
2.25 H- 5 = 0.45 



158 Chapter 3 Applying Rational Numbers 



Q Use decimal grids to find 0.8 -^ 0.4. 

Shade 8 columns to represent 0.8. 



11 


a 


^ 




1 


1 


s 




- 
































































































































..■\ :..' 










-■■', 









Divide the 8 columns into groups that each 
contain 0.4 of a decimal grid, or 4 columns. 



There i. l z groups that each contain 0.4 of a grid. 
0.8 -^ 0.4 = 2 



Q Use decimal grids to find 3.9 -^ 1.3. 

Shade 3 grids and 90 squares of a fourth grid to represent 3.9. 



uiviae me gnas ana squares inio groups mat eacii LUiiiain 
1.3 of a decimal arid, or 13 columns. 



There are 3 groups that each contain 1.3 grids. 
3.9 H- 1.3 = 3 



Think and Discuss 



1. Explain why you think division is or is not commutative. 

2. How is dividing a decimal by a whole number different 
from dividing a decimal by another decimal? 



Try This 



Use decimal grids to find each quotient. 

1. 0.8 -^ 4 2. 0.6 H- 4 

5. 4.5 -=-9 6. 1.35 H- 3 



3. 0.9 ^ 0.3 
7. 3.6 -^ 1.2 



4. 0.6 ^ 0.4 
8. 4.2 ^ 2.1 



3-4 Hands-On Lab 159 



rf>^xW'Wrfr"^v'^r^^'^ 



d 




IM Solve problems that involve multiplication and division with integers, 
fractions, decimals and combinations of the four operations 

Sandy and her family traveled from 
Columbus, Ohio, to Chicago, Illinois, 
to visit Millennium Park. They used 
14.95 gallons of gas for their 
358.8-mile drive. 

To find the number of miles per gallon 
the car got, you will need to divide a 
decimal by a decimal. 



.UL-ULiUuuwiiiiii ulIii. JAJ tauWl^B^ 



When you divide two numbers, you can multiply both numbers by 
the same power often without changing the final answer. 

Multiply both 0.6 and 0.3 by 10: 0.6 • 10 = 6 and 0.3 • 10 = 3 

0.6 H- 0.3 = 2 and 6^3 = 2 

By multiplying both numbers by the same power often, you can 
make the divisor an integer. Dividing by an integer is much easier 
than dividing by a decimal. 



EXAMPLE 1 






jji 



Multiply both 
numbers by the least 
power of ten that 
will make the divisor 
an integer. 



Dividing Decimals by Decimals 


Divide. 






A 4.32 - 3.6 






4.32^3.6 = 43.2 


^36 




1.2 






36)43.2 






-36 






72 






-7 2 













B 12.95 H- (-1.25) 






12.95^ (-1.25) = 

10.36 
125)1,295.00 


1295 


- (-125) 


-125 







45 
-37 5 
7 50 
- 7 50 

12.95^ (-1.25) = -10.36 



Multiply both numbers by W 
to make the divisor an integer. 
Divide as with whole numbers. 



Multiply both numbers by 100 
to make the divisor an integer 

Use zeros as placeholders. 
Divide as with whole numbers. 



The signs are different. 



1 60 Chapter 3 Applying Rational Numbers 



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EXAMPLE 



[2j Dividing Integers by Decimals 

Divide. Estimate to check whether each answer is reasonable. 



A 9 ^ 1.25 

9.00^-7- 1.25.= 900 -f 125 

7.2 

125)900.0 

-875 

25 

-25 



Estimate 9 h- 1 = 9 

B -12 -^ (-1.6) 

-12.0 -H (-1.6) = -120 H- (-16) 



7.5 


6)120.0 


-112 


80 


-8 




-12 ^ (-1.6) = 7.5 
Estimate -12 ^ (-2) = 6 



Multiply both numbers by 100 
to make the divisor an integer. 

Use zero as a placeholder. 
Divide as with whole numbers. 



7.2 is a reasonable answer. 



Multiply both numbers by W 
to make the divisor an integer. 

Divide as with whole numbers. 



The signs are the same. 
7.5 is a reasonable answer. 



EXAMPLE 



(3 



Transportation Application 

If Sandy and her family used 14.95 gallons of gas to drive 
358.8 miles, how many miles per gallon did the car get? 

Multiply both numbers by 100 
to make the divisor an integer. 

Divide as with whole numbers. 



"^riClr.iiiTrfinr^^^ 


358.80^ 14.95 = 35,880 -f 1,495 
24 


To calculate miles per 
gallon, divide the 
number of miles 
driven by the 
number of gallons 
of gas used. 


1,495)35,880 

-29 90 

5 980 

-5 980 




I 


The car got 24 miles per gallon. 



^^^^^^^^^^^^^^^^^^^^^^^^^^1 


ThiHk and Discuss 




1. Explain whether 4.27 -=- 0.7 is the same as 427 4 


7. 


2. Explain how to divide an integer by a decimal. 





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3-4 Dividing Decimals 161 



3-4 



A.i3j'i}-:ii}3. 



y 



Homework Help Online go.hrw.com, 



keyword ■WHIiKgM ® 
Exercises 1-27, 31, 33, 35, 37, 
39,41 



GUIDED PRACTICE 



See Example 1 Divide. 

1. 3.78 -^ 4.2 






4. 1.06^ 0.2 



2. 13.3 4-(-0.38) 
5. -9.76 -=- 3.05 



3. 14.49 ^3.15 
6. 263.16 H-(-21. 5) 



See Example 2 Divide. Estimate to check whether each answer is reasonable. 

7. 3 H- 1.2 8. 84 H- 2.4 9. 36 -=- (-2.25) 



10. 24 ^(-1.2) 



11. -18 ^ 3.75 



12. 189 -^ 8.4 



See Example 3 13. Transportation Samuel used 14.35 gallons of gas to drive his car 
L 401.8 miles. How many miles per gallon did he get? 



INDEPENDENT PRACTICE 



See Example 1 Divide. 

i 14. 81.27 4- 0.03 

I 17. -1.12 H- 0.08 



15. -0.408 H- 3.4 
18. 27.82 -^ 2.6 



16. 38.5 -H (-5.5) 
19. 14.7 -^3.5 



See Example 2 Divide. Estimate to check whether each answer is reasonable. 

20. 35 H- (-2.5) 21. 361 ^ 7.6 22. 63 H- (-4.2) 



23. 5 -=- 1.25 



24. 14 -^ 2.5 



25. -78 -^ 1.6 



See Example 3 26. Transportation Lonnie used 26.75 gallons of gas to drive his truck 

; 508.25 miles. How many miles per gallon did he get? 

27. Mitcliell walked 8.5 laps in 20.4 minutes. If he walked each lap at the 

L same pace, how long did it take him to walk one full lap? 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EPS. 



Divide. Estimate to check whether each answer is reasonable. 

28. -24-=- 0.32 29. 153^6.8 30. -2.58 ^ (-4.3) 

31. 4.12 H- (-10.3) 32. -17.85 H- 17 33. 64 -=- 2.56 

Simplify each expression. Justify your steps using the Commutative, Associative, 
and Distributive Properties when neccessary. 

34. 2- • (6.8 -=- 3.4) • 5 35. 11.7 ^ (0.7 -I- 0.6) • 2 



36. 4 • 5(0.6 -I- 0.2) • 0.25 



37. (1.6 -J- 3.2) • (4.2 -I- 8.6) 



38. Critical Thinking A car loan totaling $13,456.44 is to be paid off in 
36 equal monthly payments. Lin Yao can afford no more than $350 
per month. Can she afford the loan? Explain. 



162 Chapter 3 Applying Rational Numbers 



.'Q" 

iW 



n^ 





The Blue Ridge 
Parkway is the 
longest, narrowest 
national park in 
the world. Starting 
in Virginia, it 
covers 469 miles 
and ends at the 
entrance of the 
Great Smoky 
Mountains NP in 
North Carolina. 



39. Earth Science Glaciers form when snow accumulates faster than it melts 
and thus becomes compacted into ice under the weight of more snow. Once 
the ice reaches a thickness of about 18 m, it begins to flow. If ice were to 
accumulate at a rate of 0.0072 m per year, how long would it take to start 
flowing? 



40. 



< 



Critical Thinking Explain why 
using estimation to check the 
answer to 56.21457 -^ 7 is useful. 

Recreation The graph shows the 
approximate number of total visits to 
the three most visited U.S. national 
parks in 2006. Wliat was the average 
number of visits to tliese three parks? 
Round your answer to the nearest 
hundredth. 



Most Popular National Parks 



Blue Ridge g 
Parkway , ^ 

Golden 
Gate Nat, g 
Recreation 



20.9 



='«;a'«L*^Aft^4.^«»'4. «'««^«.. 



13.5 



Area 



^«^&»>4fc&fc4«.feft.&9 



Great Smoky gffScgggigsrggS 

Mountains * 1" t' 1111 1 f ■ 11 21 

Nat Park t-fctfe**.**-**,**.*.*** **.«*.«. 

%= 2 million 



Source National Park Service 



-.^ 



42. 



43. 



Write a Problem Find some supermarket advertisements. Use the ads to 

WTite a problem that can be solved by dividing a decimal by a whole number. 

Write About It Can you use the Commutative Property' when dividing 
decimals? Explain. 



(^ 44. Challenge Use a calculator to simplify the expression 



il 



'.5 + 3.69) H- 48.25 -=- [1.04 - (0.08 • 2)]. 



f 



^ 



Test Prep and Spiral Review 



45. Multiple Choice Which expression is NOT equal to -1.34? 

® -6.7 -=- 5 Ci;> 16.08 -f (-12) 'Cc;) -12.06 H- (-9) Q;) -22.78 H- n 

46. Multiple Choice A deli is selling 5 sandwiches for $5.55, including tax. A school 
spent $83.25 on roast beef sandwiches for its 25 football players. How many 
sandwiches did each player get? 



CD 1 



CS) 2 



CE) 3 



CT) 5 



47. Gridded Response Rujuta spent a total of $49.65 on 5 CDs. What was the 
average cost in dollars for each CD? 



Simplify each expression. (Lessori 14) 
48. 2 + 6-2 

51. 10 - (5 -3)- + 4 + 2 



49. 3^-8-0 
52. 2^+ (7+ 1) 



50. (2 - D' + 3-2- 
53. 6-2-3 + 5 



Multiply. Estimate to check whether each answer is reasonable. ison 3-3) 
54. -2.75-6.34 55. 0.2 - (-4.6) - (-2.3) 56. 1.3 - (-6.7) 



57. -6.87- (-2.65) 



58. 9-4.26 



59. 7.13- (-14) 



3-4 Dividing Decimals 163 




B 



Solving Equations 
Containing Decimals 



7.2.1 Use variables and appropriate operations to write an expression, equation or 
inequality that represents a verbal description. 

Students in a physical education class 
were running 40-yard dashes as part of a 
fitness test. The slowest time in the class 
was 3.84 seconds slower than the fastest 
time of 7.2 seconds. 



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You can write an equation to represent this 
situation. The slowest time 5 minus 3.84 is 
equal to the fastest time of 7.2 seconds. 

5 - 3.84 = 7.2 




[ EXAMPLE 



You can solve an 
equation by 
performing the same 
operation on both 
sides of the equation 
to isolate the 
variable. 



Solving Equations by Adding or Subtracting 

Solve. Justify your steps. 

AS- 3.84 = 7.2 

s - 3.84 = 7.20 Use the Addition Property of Equality. 

+ 3.84 + 3.84 Add 3.84 to both sides, 

s = 1 1.04 



B y+ 20.51 =26 

5 9 10 

j' + 20.51 = 2^.66 
- 20.51 - 20.51 



Use the Subtraction Property of Equality. 
Subtract 20.51 from both sides. 



y 



= 5.49 



EXAMPLE 12 i SoSwing Equations by IVIuitiplying or Dividing 

Solve. Justify your steps. 



3.9 



1.2 





"'=12 
3.9 ^-^ 




^ • 3.9 = 1.2 • 3.9 




((' = 4.68 


B 


4 = 1.6c 




4 = 1.6c 




4 1.6c 




1.6 1.6 




li- 




2.5 = f 



Use the Multiplication Property of Equality. 
Multiply by 3.9 on both sides. 



Use the Division Property of Equality. 
Divide by 1.6 on both sides. 

Think: 4-^ 1.6 = 40-^ 16. 



164 Chapter 3 Applying Rational Numbers 



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EXAMPLE 



(H 



PROBLEM 




PROBLEM SOLVING APPLICATION 

Yancey wants to buy a new snowboard that costs $396.00. If she 
earns S8.25 per hour at work, how many hours must she work to 
earn enough money to buy the snowboard? 

P^ Understand the Problem 

Rewrite the question as a statement. 

• Find the number of hours Yancey must work to earn $396.00. 

List the important information: 

• Yancey earns $8.25 per hour. 

• Yancey needs $396.00 to buy 
a snowboard. 

Make a Plan 

Yancey's pay is equal to lier 
hourly pay times the number 

of hours she works. Since you know how much money she needs to 
earn, you can write an equation with /; being the number of hours. 
8.25/; = 396 



*Q Solve 

8.25/; = 396 

—^ = 1^ Use the Division Property of Equality. 

h = 48 
Yancev must work 48 hours. 




Q Look Back 

You can round 8.25 to 8 and 396 to 400 to estimate how many hours 
Yancey needs to work. 
400 H- 8 = 50 

So 48 hours is a reasonable answer. 



^^^^^^^^^^^^^m^^^^^^^^^^^^^^H 


Think and Discuss 










1. Describe how to solve the equation 


-1.25 + .v= 1. 


25 


Then 


solve. 


2. Explain how you can tell if 1.01 is a 


solution of 105 


- 


-10.1 




without solving the equation. 











m-i 



Lesson Tutorials Onlin€ mv.hrw.com 3-5 Solving Equations Containing Decimals 165 



3-5 



;| 



U£5 



[•Jllllll 

Homework Help Online go.hrw.com, 



keyword ■BHIifcBiM ® 

Exercises 1-23, 29, 33, 35, 39, 
41,43,45 



GUIDED PRACTICE 



See Example 1 Solve. Justify your steps. 

1 1. ;('- 5.8 = 1.2 



See Example 2 



3. k + 3.91 = 28 

5. 

7. 3.lr= 27.9 



A -3-6 



2. .v + 9.15 = 17 
4. /; - 1.35 = 19.9 



6- 0^ = 7.2 
8. 7.5 = 5v 



See Example 3 

L 



9. Consumer Math Jeff bought a sandwich and a salad for lunch. His total 
bill was $7.10. The salad cost $2.85. How much did the sandwich cost? 



INDEPENDENT PRACTICE 



See Example 1 Solve. Justify your steps. 

I 10. i'+ 0.84 = 6 

I 13. 3.52 + ci = 8.6 

See Example 2 16. 3.2f = 8 



11. f- 32.56= 12 
14. w-9.0\ = 12.6 
17. 72 = 4.5z 



12. d - 14.25 = -23.9 
15. ^ + 30.34 = -22.87 
18. 21.8.v= -124.26 



L 



19. -^ = 42 

2 8 



20. 



0.19 



= 12 



21. 



a 
21.23 



= -3.5 



See Example 3 22. At the fair, 25 food tickets cost $31.25. What is the cost of each ticket? 

23. To climb the rock wall at the fair, you must have 5 ride tickets. If each ticket 
costs $1.50, how much does it cost to climb the rock wall? 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EP9. 



25. ^=-0.6 



26. w- 4.1 = -5 



Solve. Justify your steps. 

24. 1.2j'= -1.44, 

27. /■+ 0.48 = 1.2 

30. a + 0.81 = -6.3 

33 '- =-5 2 

-0.18 

36. -A: = 287.658 
39. 15.217 -;■ = 4.11 



42. The Drama Club at Smith Valley Middle School is selling cookie dough in 
order to raise money for costumes. If each tub of cookie dough costs $4.75, 
how many tubs must members sell to make $570.00? 

43. Consumer Math Gregory bought a computer desk at a thrift store for $38. 
The regular price of a similar desk at a furniture store is 4.5 times as much. 
What is the regular price of the desk at the furniture store? 



28. 


.V - 5.2 = -7.3 


29. 


1.05 = -7 m 


31. 


60k = 54 


32. 


37T ^ °-^^ 


34. 


7.9 = d+ 12.7 


35. 


-1.8 + v= -3.8 


37. 


-11 = -12.254 


38. 


0.64/= 12.8 


40. 


-2.1 =p+ (-9.3) 


41. 


^ = 54.6 



166 Chapter 3 Applying Rational Numbers 




.'Q**?. 44. Physical Science Pennies minted, or created, before 1982 are made 

1 1 n iL'*^ mostly of copper and have a density of 8.85 g/cm '. Because of an increase 

npnnVHP in the cost of copper, the density' of pennies made after 1982 is 1.71 g/cm^ 

'M^xs^H jggg What is the densiry of pennies minted today? 

Social Studies The table shows 
the most common European 
ancestral origins of Americans (in 
millions), according to a Census 
2000 supplementary' survey. In 
addition, 19.6 million people 
stated that their ancestry was 
"American." 

a. How many people claimed 
ancestr\' from the countries 
listed, according to the survey? 

b. If the data were placed in order 
from greatest to least, between which two nationalities would 
"American" ancestry be placed? 

@ 46. What's the Error? A student's solution to the equation m + 0.63 = 5 was 
1)1 = 5.63. What is the error? What is the correct solution? 

47. Write About It Compare the process of solving equations containing 
integers with the process of solving equations containing decimals. 

^ 48. Challenge Solve the equation -2.8 + ib- 1.7) = -0.6 • 9.4. 



From 1892 to 

1924, more than 
22 million 
immigrants came 
to Ellis Island, 
New York. 



Ancestral Origins of Americans 


European Ancestry 


Number (millions) 


English 


28.3 


French 


9.8 


German 


46.5 


Irish 


33.1 


Italian 


15.9 


Polish 


9.1 


Scottish 


5.4 







Test Prep and Spiral Review 



49. Multiple Choice What is the solution to the equation -4.55 + .v = 6.32? 

CD X = -1.39 CS) X = 1.77 Cc:' .V = 10.87 CE) x = 28.76 

50. Multiple Choice The pep squad is selling tickets for a raffle. The 
tickets are $0.25 each or 5 for $1.00. lulie bought a pack of 5 tickets. 
Which equation can be used to find how much Julie paid per ticket? 



CD 5.V = 0.25 



CG) 0.25x= 1.00 



(H) 5a- = 1.00 



CD 1.00x = 0.25 



51. Extended Response Write a word problem that the equation 6.25x - 125 
can be used to solve. Solve the problem and explain what the solution means. 



Write each number in scientific notation. Lesson 1-3) 
52. 340,000 53. 6,000,000 

Simplify each expression. (Lesson 3-4) 

55. 6.3 H- 2.1 - 1.5 56. 4 • 5.1 -^ 2 + 3.6 



58. (-5.4 + 3.6) -=- 0.9 



59. -4.5 H- 0.6- (-1.2) 



54. 32.4 • 10- 

57. (1.6 + 3.8) -H 1.8 
60. 5.8 -t- 3.2 + (-6.4) 



3-5 Solving Equations Containing Decimals 167 




To Go On? 



,1^ Learn It Online 

^^'* ResoLircesOnlinego.hrw.com, 



Quiz for Lessons 3-1 Through 3-5 

Q) 3-1 j Estimating with Decimals 



Estimate. 

1. 163.2 • 5.4 



2. 37.19 + 100.94 



3. 376.82 - 139.28 4. 33.19 H- 8.18 



5. Brad worked the homework problem 1 19.67 m -h 10.43 m. His answer was 
1 1.47 m. Use estimation to clieck whether this answer is reasonable. 



Qy 3-2 ] Adding and Subtracting Decimals 

Add or subtract. 

6. 4.73 + 29.68 7. -6.89 - (-29.4) 



8. 23.58 - 8.36 



9. -15 + (-9.44) 



^ 3-3 ] Multiplying Decimals 

Multiply. 



10. 3.4 -9.6 



11. -2.66-0.9 



12. 



(-0.06) 13. 6.94 • (-24) 



14. Cami can run 7.02 miles per hour. How many miles can she run in 
1.75 hours? Round your answer to the nearest hundredth. 

^) 3-4 ] Dividing Decimals 

Divide. 

15. 55 H- 12.5 16. -126.45 H- (-4.5) 17. -3.3 H- 0.11 18. -36 + (-0.9) 



19. 10.4 + (-0.8) 20. 18 H- 2.4 



21. -45.6+12 22. -99.36 + (-4) 



23. Cynthia ran 17.5 laps in 38.5 minutes. If she ran each lap at the same pace, 
how long did it take her to run one full lap? 

24. A jewelry store sold a 7.4-gram gold necklace for $162.18. How much was the 
necklace worth per gram? Round your answer to the nearest tenth. 



Q<) 3-5 ] Solving Equations Containing Decimals 



Solve. 

25. 3.4 + 11 = 



26. A- 1.75 = -19 27. -3.5=-5.v 28.10.1 = 



29. Pablo earns $5.50 per hour. His friend Raymond earns 1.2 times as much. 
How much does Raymond earn per hour? 



168 Chapter 3 Applying Rational Numbers 



Focus on Problem Solving 




t 



Look Back 




Does your solution answer the question in 
the problem? 

Sometimes, before you solve a problem, you first need to use the 
given data to find additional information. Any time you find a 
solution for a problem, you should ask yourself if your solution 
answers the question being asked, or if it just gives you the 
information you need to find the final answer. 



t 



Read each problem, and determine whether the given solution 
answers the question in the problem. Explain your answer 



O At one store, a new CD costs $15.99. At a 
second store, the same CD costs 0.75 as 
much. About how much does the second 
store charge? 

Solution: The second store charges 
about $12.00. 

Bobbie is 1.4 feet shorter than 
her older sister. If Bobbie's 
sister is 5.5 feet tall, how 
tall is Bobbie? 

Solution: 

Bobbie is 4.1 feet tall. 



-u^^ 




O Juanita ran the 100-yard dash 

1.12 seconds faster than Kellie. Kellie's 
time was 0.8 seconds faster than Rachel's. 
If Rachel's time was 15.3 seconds, what 
was luanita's time? 

Solution: Kellie's time was 14.5 seconds. 

O The playscape at a local park is located in 
a triangular sandpit. Side A of the sandpit 
is 2 meters longer than side B. Side B is 
twice as long as side C. If side C is 
6 meters long, how long is side A? 

Solution: Side B is 12 meters long. 

Both Tyrone and Albert walk to and from 
school every day. Albert has to walk 
1.25 miles farther than Tyrone does each 
way. If Tyrone's house is 0.6 mi from 
school, how far do the two boys walk 
altogether? 

Solution: Albert lives 1.85 mi from 
school. 



Focus on Problem Solving 169 



a 



B 



Estimating with 
Fractions 



7.1.7 Solve problems that involve multiplication and division with integers, 
fractions, decimals and combinations of the four operations. 

One of the largest cheese wheels ever 
produced was made in Alkmaar, 
Netherlands, and weighed about 
1,2504; lb. About how much heavier 

50 

was this than the average cheese 
wheel, which may weigh about 6 lb? 

Sometimes, when solving problems, 

you may not need an exact answer. 

To estimate sums and differences of 



Interactivities Online ► 




fractions and mixed numbers, round each fraction to 0, \, or 1. You can 
use a number line to help. 

*H — I — I \ — ^— I \ 1 — I \ — f-* 



2^ ± 
5 2 



I Is closer to y than to 0. 



You can also round fractions by comparing numerators with denominators. 



Benchmarks for Rounding Fractions 



Round to if the 

numerator is much 

smaller than the 

denominator. 



Examples: |, ^, ^ 



Round to i if the 

numerator is about half 

the denominator. 



Examples: |, ^, ^ 



Round to 1 if the 

numerator is nearly 

equal to the 

denominator. 



Examples:! If, ^ 



EXAMPLE 







Measurement Application 

One of the largest wheels of cheese ever made weighed about 
1,250^ lb. Estimate how much more this wheel of cheese weighed 
than an average 6 lb wheel. 



l,250pir - 6 

50 
1 



1 250-^ 



■1,250 
1,250 - 6 = 1,244 



Round the mixed number. 
Subtract. 



The cheese wheel weighed about 1,244 lb more than an average 
cheese wheel. 



1 70 Chapter 3 Applying Rational Numbers 



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EXAMPLE [^ Estimating Sums and Differences 



Estimate each sum or difference. 





A 


4 13 
7 16 












1-^2 


ii-1 




Round each fraction. 




B 


5- ' = 4 

33 ,3! 


4- 


4 


Subtract. 


:fi;j3JWIlimK u-^ 




Round lto~ since it 


Round each mixed nurvber 


is closer to | than 0. 


C 


3i + 3^ = 7 
^8 ^ 5J 






Add. 






4—6 




-5 — 


^_l 




Round each number. 






6-14)- 


4 




Add. 



EXAMPLE 



You can estimate products and quotients of mixed numbers by 
rounding to the nearest whole number. If the fraction in a mixed 
number is greater than or equal to i, round the mixed number up to 
the next whole number. If the fraction is less than -,, round down to a 
whole number by dropping the fraction. 



|3j Estimating Products and Quotients 

Estimate each product or quotient. 



A 4 



10 

► 4 



4 • 7 = 2E 



10 



Round each mixed number to 
the nearest whole number. 
Multiply. 



B llf^2i 



11 = 



12 



12 -;- 2 = 6 



Round each mixed number to the 
nearest whole number. 
Divide. 



Think and Discuss 

1. Demonstrate how to round 



11 and 5I. 



2. Explain how you know that 25^ • 5t^ > 125. 



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3-6 Estimating with Fractions 171 




■? Homework Help Online go.hrw.com, 



keyword ■MiaiiKgiM ® 

Exercises 1-26, 27, 29, 31, 35, 
37,39,43 



GUIDED PRACTICE 



See Example 1 1. The length of a large SUV is 18||j feet, and the length of a small SUV is 

1 15^ feet. Estimate how much longer the large SUV is than the small SUV. 



See Example 2 Estimate each sum or difference. 



2. ^ + ^ 
6 12 



3 ^ 

^' 16 



See Example 3 Estimate each product or quotient. 

7. 2li-7i 



6 1^-9^ 
25 7 



4 2i + 3~ 



8.31^^4 



5 5^-2^ 



9. 12^ -31 



INDEPENDENT PRACTICE 



See Example 1 10. Measurement Sarah's bedroom is 14| feet long and 12| feet wide. 

L Estimate the difference between the length and width of Sarah's bedroom. 

See Example 2 Estimate each sum or difference. 





11 i + ^ 
"• 9^5 


12 2- + 1- 

9 8 


13. 8^ - ei 

4 5 


14. 


4 + l-r) 

3 I 6j 




1M-S 


16. 15i-10| 


17.83^+21 


18. 


f + 4 


See Example 3 


t Estimate each 


product or quotient. 










19. 23f^3| 


20. 10§ ^ 4| 


21. 2l.l4| 


22. 


4- 1^1 




23- 5f^2| 


24. 12^ -31 

6 7 


25. 8l ^ iZ 


26. 


15i^ • 1^ 
^^15 V 




r PRACTICE AND PlPBiiM SOLVING 






or quotient. 






rExtra Practice" 


Estimate each 


sum, difference, product. 




See page EP9. 






^■l-l 


28. l + f 


29. 2f . 8A 


30. 


1 6^ - 3- 
20 9 




31. -l|.4J| 


32- 5i - 4i 


33- 3| + f;3 


34. 


"!-(-=§] 




35. 1 + 3| + 6 

/ 


1 36. 8| + 6jL 


+ 3f 37. 


»i- 


>|-"M 



38. Kevin has 3^ pounds of pecans and 6| pounds of walnuts. About how 
many more pounds of walnuts than pecans does Kevin have? 

39. Business October 19, 1987, is known as Black Monday because the stock 
market fell 508 points. Xerox stock began the day at $70| and finished at 
$56^. Approximately how far did Xerox's stock price fall during the day? 

40. Recreation Monica and Paul hiked 5| miles on Saturday and 4--^ miles 
on Sunday. Estimate the number of miles Monica and Paul hiked. 

41. Critical Thinking If you round a divisor down, is the quotient going to 
be less than or greater than the actual quotient? Explain. 



172 Chapter 3 Applying Rational Numbers 



Life Science The diagram shows the wingspans of different species of birds. 
Use the diagram for Exercises 42 and 43. 



Blue jay 




42. 

43. 

@44. 



945. 
^46. 



Approximately how much longer is the wingspan of an albatross than the 
wingspan of a gull? 

Approximately how much longer is the wingspan of a golden eagle than 
the wingspan of a blue jay? 

Write a Problem Using mixed numbers, write a problem in whicli an 
estimate is enough to solve the problem. 

Write About It How is estimating fractions or mixed numbers similar to 
rounding whole numbers? 

Challenge Suppose you had bought 10 shares of Xerox stock on October 16, 
1987, for $73 per share and sold ihem at the end of the day on October 19, 1987, 
for $56t per share. Approximately how much money would you have lost? 



£ 



Test Prep and Spiral Review 



47. Multiple Choice For which of the following would 2 be the best estimate? 

CS) 8| • 4| CD 4i ^ 2| CT) 8| • 2i CE) 8^ ^ 4| 

48. Multiple Choice The table shows the distance Maria hiked each day last week. 



Day 


Mon 


Tue 


Wed 


Thu 


Fri 


Sat 


Sun 


Distance (mi) 


'4 


gll 


^l 


^l 





< 


H 



Which is the best estimate for the total distance Maria hiked last week? 

CD 40 mi CD 44 mi CD 48 mi CD 52 mi 



Solve each equation. Check your answer. (Lessons 1-10 and 1-11) 
49. A- + 16 = 43 50. V - 32 = 14 51. 5??? = 65 



Solve. (Lesson 3-5) 
53. -7.1.v= -46.15 



54. 



.7 = 1'+ (-4.6) 



55. 



(] _ 



-5.4 



3.6 



52. f = U 



56. r- 4 = -31.2 



3-6 Estimating with Fractions 173 



-ih 



Model Fraction Addition 
and Subtraction 



Use with Lesson 3-7 



J^ Learn It Online 



Fraction bars can be used to model addition and subtraction of fractions. 



Lab Resources Online go.hfw.com, 
liWTMsTo Lab3 ■Go] 



Activity 



You can use fraction bars to find | + 1. 

o o 



Use fraction bars to represent both fractions. Place the fraction bars 
side by side. 



-% ^^B 



3 + 2-5 
8 8 8 



O Use fraction bars to find each sum. 
a. 



i + i 

3 3 



b. 4 + 1 



12 12 



d. i + ^ 
5 5 



You can use fraction bars to find ^ + 4. 

Use fraction bars to represent both fractions. Place the fraction 
bars side by side. Which kind of fraction bar placed side by side will 
fit below I and i? (Hint: What is the LCM of 3 and 4?) 



1 + 1 = ^ 

3 4 12 




O Use fraction bars to find each sum. 



a. 



+ 



b. k + 



i + i 

3 6 



d. i + i 
4 6 



You can use fraction bars to find 1 + 1. 

3 b 

Use fraction bars to represent both fractions. Place the fraction bars 
side by side. Which kind of fraction bar placed side by side will fit 
below \ and ^? [Hint: What is the LCM of 3 and 6?) 



H 



1 , 5 _ 7 
3 6 6 



174 Chapter 3 Applying Rational Numbers 



When the sum is an improper fraction, you can use the 1 bar along with 
fraction bars to find the niLxed-number equivalent. 




l=ll 



Use fraction bars to find each sum. 
3 _L 3 |j 2 _L 1 



a- f + 4 



3 "*" 2 



5 + 1 

6 4 



d-i + f 



You can use fraction bars to find 1-4. 

Place a ^ bar beneath bars that show |, and find which fraction fills in 
the remaining space. 






1 _ 1 

2 6 



O Use fraction bars to find each difference. 



^- 3 3 



"•4 6 



'■'2 3 



"4 3 



Think and Discuss 



1. Model and solve | - |- Explain your steps. 

2. Two students solved ^ + i in different ways. One got y^ for the answer, 
and the other got i. Use models to show which student is correct. 

3. Find three different ways to model ~ + \. 

4. If you add two proper fractions, do you always get a sum that is greater 
than one? Explain. 



Try Til is 



Use fraction bars to find each sum or difference. 



1 i + i 

2 2 

5 ^-i 

12 3 



2 ^ + i 
3 6 

6-2-4 



3 -i + i 

^•4 6 

7. ^-i 
4 6 



4 -!- + — 
3 12 



9. You ate | of a pizza for lunch and | of the pizza for dinner. How much of 
the pizza did you eat in all? 

10. It is I mile from your home to the library. After walking | mile, you stop 
to visit a friend. How much farther must you walk to reach the library? 



3-7 Hands-On Lab 175 



3-7 



Fractions 




From Januaiy 1 to 

March 14 of any given 

year, Earth completes 

approximately ^ of its 

circular orbit around 

the Sun, while Venus 

completes approximately 

^ of its orbit. To find out 

how much more of its 

orbit Venus completes 

than Earth, you need to subtract fractions 




January 1 



Venus on 
arch 14 



Earth on 
March 14 



EXAMPLE 




Adding and Subtracting Fractions with Like Denominators 

Add or subtract. Write each answer in simplest form. 
A 

iU lU 

Add the numerators and 

keep the common denominator. 



3_ + ± 
10 10 




10 10 


3 + 1 
10 



_ J_ _ 2 
10 5 



1-4 
9 9 

7 _ 4 _ 7_ 
9 9 



_ 3 _ 1 
9 3 



Simplify. 



Subtract the numerators and 
/ceep the common denominator. 



Simplify. 



To add or subtract fractions with different denominators, you must 
rewrite the fractions with a common denominator. 



HelpfuiihilB 



The LCM of two 
denominators is the 
lowest common 
denominator (LCD) 
of the fractions. 



Two Ways to Find a Common Denominator 



Method 1: Find the LCM (least 
common multiple) of the 
denominators. 



1 



^+ I The LCM of the 

denominators is 4. 



2 + 1 = 3 

4 4 4 



Method 2: Multiply the 
denominators. 



1 + 1 = K1 + iri Multiply the 
denominators. 



4 ^ 2 _ 6 



1 76 Chapter 3 Applying Rational Numbers 



\T\!i-i!j\ Lesson Tutorials Online mv.hrw.com 



EXAMPLE 



Adding and Subtracting Fractions with Unlike Denominators 

Add or subtract. Write each answer in simplest form. 



A 

1 


3,5 
8 12 




3, 5 _3-3, 5-2 
8 12 8 • 3 12-2 




_ 9 , 10 _ 19 

24 24 24 




Estimate i + i = i 


B 


1 5 
10 8 




1 5 _ 1 • 4 5-5 
10 8 10-4 8-5 




_ 4 25 _ 21 
40 40 40 




Estimate o - ^ = -^^ 


fc 


3^8 




2,7_ 2-8,7-3 
3 8 3-8 8-3 




_ 16 , 21 _ 5 

24 24 24 



The LCM of the denominators is 24. 
Write equivalent fractions. Add. 



19 
24 



is a reasonable answer. 



Estimate 



1 + 1=0 



The LCM of the denominators is 40. 
Write equivalent fractions. Subtract. 
~~^ is a reasonable answer. 



Multiply the denominators. 
Write equivalent fractions. Add. 
~ is a reasonable answer. 



EXAMPLE [3j Astronomy Application 



From January 1 to March 14, Earth completes about 5 of its orbit, 
while Venus completes about ] of its orbit. How much more of its 
orbit does Venus complete than Earth? 



3 5 3-5 5-3 

= A _ J_ 

15 15 



The LCM ot the denominators is 15. 
Write equivalent fractions. 
Subtract. 



Venus completes -j^ more of its orbit than Earth does. 



Think and Discuss 

1. Describe the process for subtracting fractions with different 
denominators. 

2. Explain whether | + | = | is correct. 



'fi'Jb'j Lesson Tutorials Online mv.hrw.com 



3-7 Adding and Subtracting Fractions 177 




Zi3l^JS3^ 



diiictrniiii 

(P^ HomeworkHelpOnlinego.hrw.com, 



keyword ■BHMcaa ® 
Exercises 1-27, 29, 31, 37, 47, 
49,51,55 



GUIDED PRACTICE 



See Example 1 Add or subtract. Write each answer in simplest form. 



L 
See Example 2 

See Example 3 



1. 


2 1 

3 3 


2. 


12 12 


5. 


i + i 
6 ^ 3 


6. 


9 3 

10 4 



16 
21 



7-! + 



4. ^ + 11 

17 17 

8. t-4 



9. Parker spends ^ of his earnings on rent and | on entertainment. How much 
more of liis earnings does Parker spend on rent than on entertainment? 



INDEPENDENT PRACTICE 



See Example 1 Add or subtract. Write each answer in simplest form. 

12. t + l 



10. - + - 

3 3 



14. 



See Example 2 18. f + ^ 



22. 



j_ 

5 

21 
24 



11. -^ + -L- 

20 20 

15. I- i 



19 - + — 
6 12 

23. ^ - ii 
4 12 



IK 8 5 

^^- 9"9 

20-1 + ! 

24. 1 - - 

2 7 



13 _6_ + A 

15 15 



17. 
21. 
25. 



25 25 



2 "*" 8 

X _ i 

10 6 



See Example 3 26. Seana picked | quart of blackberries. She ate pj quart. How much was left? 

1 27. Armando lives ^ mi from his school. If he has walked \ mi already this 
L morning, how much farther must he walk to get to his school? 



PRACTICE AND PROBLEM SOLVING 



Extra Practice 



See page EP9 



Find each sum or difference. Write your answer in simplest form. 



28. i + f 
32 



36. 
40. 
44. 



5+i 
7 3 

7 T S 
- + - + - 

8 3 6 



2 8 7 



9_ 

35 



_5_ 
14 



29. 
33. 
37. 
41. 
45. 



1 

9 

J_ 
12 



3 4- J_ _ 3 
5 10 4 

i + 3_l 
3 7 9 



21 



30. 
34. 
38. 
42. 
46. 



1 

2 


3 

4 






3 
4 


+ 1 






3 
10 


-1 


+ 


1 
5 


2 
9 


7 
18 


+ 


1 
6 


- 


9 ~ 


1 
12 


- 



31. 


2+A 
3 15 


35. 


9 1 
14 7 


39. 


2_1 + A 
5 6 10 


43. 


A + i + i 
15 9 3 


47. 


T d R 
— - + - + - 

3 5 8 



48. Cooking One fruit salad recipe calls for | cup of sugar. Another recipe calls 
for 2 tablespoons of sugar. Since 1 tablespoon is j^ cup, how much more 
sugar does the first recipe require? 



49. 



50. 



It took Earl ~ hour to do his science homework and | hour to do his math 
homework. How long did Earl work on homework? 

Music In music written in 4/4 time, a half note lasts for l^ measure and an 
eighth note lasts for ~ measure. In terms of a musical measure, what is the 
difference in the duration of the two notes? 



178 Chapter 3 Applying Rational Numbers 



Person 


Distance (mi) 


Rosalyn 


1 
8 


Cai 


3 
4 


Lauren 


2 
3 


Janna 


7 
10 








Fitness Four friends had a competition to see how far 
they could walk while spinning a hoop around their 
waists. The table shows how far each friend walked. 
Use the table for Exercises 51-53. 

51 . How much farther did Lauren walk than Rosalyn? 

52. What is the combined distance that Cai and 
Rosal\ai walked? 

53. \Vlio walked farther, Janna or Cai? 

54. Measurement A shrew weighs -j| lb. A hamster weighs | lb. 

a. ITow many more pounds does a hamster weigh 
than a shrew? 

b. There are 16 oz in 1 lb. How many more ounces ^ 
does the hamster weigh than the shrew? 

55. Multi-Step To make | lb of mixed nuts, 
how many pounds of cashews would you 
add to ~ lb of almonds and ^ lb of peanuts? 

56. Mal<e a Conjecture Suppose the pattern 
1. ^. T. Ii ^ ••■ is continued forever. Make a 

o 4 o Z 

conjecture about the rest of the numbers in the pattern. 

57. Write a Problem Use facts you find in a 
newspaper or magazine to write a problem that 

can be solved using addition or subtraction of fractions. 

58. Write About It Explain the steps you use to add or subtract fractions 
that have different denominators. 




59. Challenge The sum of two fractions is 1. If one fraction is ^ greater than 
the other, what are the two fractions? 



c 



Test Prep and Spiral Review 



60. Multiple Choice What is the value of the expression 4 + |? 

® i ® i ®i ®i 

61. Gridded Response Grace has I, pound of apples. Julie has ^ pound of 
apples. They want to combine their apples to use in a recipe that calls for 
1 pound of apples. How many more pounds of apples do they need? 



Find the greatest common factor (GCF). (Lesson 2 7) 

62. 5,9 63. 6,54 64. 18,24 



Estimate each sum or difference. (Lesson 3-6) 
66. 



7 9 



"• "s - 4 



68.7i-(-3|) 



65. 12,36.50 



69. 6^ + 2| 

o / 



3-7 Adding and Subtracting Fractions 179 



3-8 



Mixed Numbers 



Beetles can be found all over the world in a 
fabulous variety of shapes, sizes, and colors. 
The giraffe beetle from Madagascar can 
grow about 6| centimeters longer than the 
giant green fruit beetle can. The giant green 
fruit beetle can grow up to Ip centimeters 
long. To find the maximum length of the 
giraffe beetle, you can add 6^ and l^. 



EXAMPLE Wi Measurement Application 




The giraffe beetle can grow about 6| centimeters longer than the 
giant green fruit beetle can. The giant green fruit beetle can grow 
up to l| centimeters long. What is the maximum length of the 
giraffe beetle? 



6^+11 = 7 + ^ 
a 5 5 



Add the fractions, and then add the integers. 
Add. 
The maximum length of the giraffe beetle is 7^ centimeters. 



-n 



EXAMPLE 



fjJj^MJJllil' 




Adding Mixed Numbers 

Add. Write each answer in simplest form. 



Add the fractions 


A 


3| + 4| 


first in case an 




5 5 


improper fraction 
needs to be 




3| + 4f = 7 + | 


rewritten. 




= 7+11 






= »i 




B 


= 8 + 1 

= ^30 = ^^ 



Add the fractions, and then add the integers. 

Rewrite the improper fraction 

as a mixed number. 

Add. 



Find a common denominator. 

Add the fractions, and then add the integers. 

Add. Then simplify. 



180 Chapter 3 Applying Rational Numbers 



VhJiLi Lesson Tutorials Onlln€ mv.hrw.com 



Sometimes, when you subtract mixed numbers, the fraction portion 
of the first number is less than tlie fraction portion of the second 
number. In these cases, you must regroup before subtracting. 



Remeni 



M 



Any fraction in which 
the numerator and 
denominator are the 
same is equal to 1. 



' REGROUPING MIXED NUMBER 


^- . ■ - ... 


Words 


Numbers 


Regroup. 

Rewrite 1 as a fraction with 
a common denominator. 

Add. 

V 


7^ = 6+1+^ 



EXAMPLE [sj Subtracting Mixed Numbers 

Subtract. Write each answer in simplest form. 



'' l«i-4| 



10^ 



4- = 6^ 
9 9 



B I 2^ - S^ 



12^ 



5I2 = 1221 _ 5I7 
^24 ^24 24 

' 1 1 



C 72| - 63| 
5 5 

72|-63^ = 7l|-63| 

5 5 5 5 



Subtract the fractions, and then 
subtract the integers. 



Find a common denominator. 

Subtract the fractions, and then 
subtract the integers. 



Simplify. 



Regroup. 72^ = 71 + ^ + ^ 

Subtract the fractions, and then 
subtract the integers. 



Think and Discuss 

1. Explain whether it is possible for the sum of two mixed numbers 
to be a whole number. 

2. Explain whether 2| + l| = 3| is correct. Is there another way to 
write the answer? 

3. Demonstrate how to regroup to simplify 6| - 4|. 



Ii'Jb'j Lesson Tutorials OnliriE 



3-8 Adding and Subtracting Mixed Numbers 181 



3-8 



GUIDED PRACTICE 



Homework Help Online go.hrw.com, 



keyword ■««!««;< ® 
Exercises 1-26, 27, 31, 33, 37, 
41,43,45 



See Example 1 1. Measurement Chrystelle's mother is l| ft taller than Chrystelle. If 
L Chrystelle is 3^ ft tall, how tall is her mother? 



See Example 2 Add. Write each answer in simplest form 

1 3 2- + 3- 

3. ^g -I- j^ 



! 2 3- + 4 

L 5 5 8 4 9 

See Example 3 Subtract. Write each answer in simplest form 



4. l| + 4^ 



6- ef - 5| 



5-!- - 2^ 
6 6 



8. 3§ - 2| 



5 5! + oi 



9- 1 - 3i 



INDEPENDENT PRACTICE 



See Example 1 10. Sports The track at Daytona hiternational Speedway is =^ mi longer 
than the track at Atlanta Motor Speedway. If the track at Atlanta is l|^ mi 
long, how long is the track at Daytona? 



L 



See Example 2 


Add. Write each 


answer in simplest form 


• 












11. 64 + 8f 

4 4 


12. 3| + ?! 

5 


13. 


3i + if 




14. 


2- + 4- 




1 "i ''— + A- 


16-6^ + 4 


17. 


^h^f-o 




18. 


^5 ^ ^4 


See Example • 


t Subtract. Write each answer in simplest 


brm 












19 2— - 1~ 

14 14 


20 4— - 1 — 


21. 


8-2f 




22. 


U ^3 




23. 8f - 6i 

_ 4 5 


24 3- - 2- 


25. 
1— 


^5 -^2 




26. 


11 -6| 




r PRACTICE AND PROBLEM SOLVING 






mplest form. 

29. 9I + 4f 




30. 






• Add or subtract. Write each answer in sii 

27. 7^ + 8^ 28. 14| - 8^ 




[Extra Practice 




See page EP9. 


91_8 C5I 

'' 12 2 




3 1. Jg -|-Z^2 


32. 25i + 3| 


33. 


lZ_iZ 

9 18 




34. 


4 + 4 




35 1— + 2— 
^^- ^15 ^^10 


36- 14 - i 


37. 


4^ + 1- + 3- 
^3 I- ig -1- ^2 


38. 


5l + 8|-9i 




Compare. Write <, >, or =. 














39. 12i - lOf 


2 10 


40. 


4^ + 3^ 
^2 ^-^5 


4f 


.31 






41. 13f-2| 


■^6 ^ *9 


42. 


4l_2i 
^3 4 


3^ 

-^4 


-'i 





43. The liquid ingredients in a recipe are water and olive oil. The recipe 
calls for 3^ cups of water and l| cups of olive oil. How many cups of 
liquid ingredients are included in the recipe? 



182 Chapter 3 Applying Rational Numbers 



.-QH 




Travel The table shows the distances in miles 
X between four cities. To find the distance between 
two cities, locate the square where the row for 
one citv' and the column for the other 
city intersect. 



,.9 



New Zealand 
is made of two 
main islands in 
the soutliwestern 
Pacific Ocean. 
The native Maori 
people refer to 
New Zealand 
as Aoetearoa, 
or "The Land of 
the Long White 
Cloud." 



44. 



45. 



< 



How much farther is it from 
Charleston to Dixon than from 
Atherton to Baily? 

If you drove from Charleston to 
Atherton and then from Atherton 
to Dixon, how far would you drive? 

Agriculture In 2003, the United 

States imported j^ of its tulip bulbs 

from the Netherlands and ^ of its tulip bulbs from New Zealand. Wliat 



Atherton 


X 


40 1 


loof 


16l 


Baily 


40 1 


X 


210| 


30 1 


Charleston 


lOOf 


210| 


X 


98 1 


Dixon 


16l 


30 1 


98 1 


X 



fraction more of tulip imports came from the Netherlands? 

47. Recreation Kathy wants to hike to Candle Lake. The waterfall trail is 1 

long, and the meadow trail is 1| miles long. Wiiicii route is shorter and by how 
much? 



I miles 



48. Choose a Strategy Spiro needs to draw a 6-inch-long line. He does not 
have a ruler, but he has sheets of notebook paper that are 8^^; in. wide and 

1 1 in. long. Describe how Spiro can use the notebook paper to measure 6 in. 

49. Write About It Explain why it is sometimes necessary to regroup a mixed 
number when subtracting. 

50. Challenge Todd had d pounds of nails. He sold 3^ pounds on Monday 
and 5^ pounds on Tuesday. Write an expression to show how many pounds 
he had left and then simplify' it. 



m 



Test Prep and Spiral Review 



lUUlUUIUU 



51. Multiple Choice Which expression is NOT equal to 2^ ? 



® 4 + if 



® 5Jf-3^ 



CS:> 6-3^ 



CE) li+li 



52. Short Response Wliere Maddie lives, there is a S^-cent state sales tax, a 
l|-cent county sales tax, and a |-cent city sales tax. The total sales tax is 
the sum of the state, county, and cit\' sales taxes. What is the total sales tax 
where Maddie lives? Show your work. 



Find each sum. 


(Lesson 2-2) 


















53. 


-3 + 9 


54. 


6+ (- 


■15) 


55. 


-4 + ( 


-8) 




56. - 


11+5 


Find each sum or difference. Write 


your 


answer in simplest 


form. 


(Lesson 3-7) 




57. 


1 + ^ 


58. 


3 1 
7 3 




59. 


3 + _L 

4 18 






^«l 


4 
5 



3-8 Adding and Subtracting Mixed Numbers 183 



^m<^<>-S 



Model Fraction 
Multiplication and Division 



Use with Lessons 3-9 and 3-10 



You can use grids to model fraction multiplication and division. 



JT?. 



Learn It Online 

Lab Resources Online go.hrw.com, 
■lMM510Lab3tGo-M 



Activity 1 



Use a grid to model 



4 2" 



Think of ^ • ^ as ~ of ^. 

Model \ by shading half of a grid. 



The denominator tells you to divide the grid into 2 parts. 
The numerator tells you how many parts to shade. 



Divide the grid into 4 equal horizontal sections. 



Use a different color to shade | of the same grid. 







What fraction of the whole 
is shaded? 

3.1 = 3 

4 2 8 



The denominator tells you to divide the grid into 4 parts. 
The numerator tells you how many parts to shade. 



To find the numerator, think: How many parts overlap? 

To find the denominator, think: How many total parts are there? 



Think and Discuss 



1. Are I • i and ^ • | modeled the same way? Explain. 

2. When you multiply a positive fraction by a positive fraction, the product 
is less than either factor. Wliy? 



184 Chapter 3 Applying Rational Numbers 



Try This 



Use a grid to find each product. 



1. 



1 I 

2 ' 2 



4 3 



5 i 
8 3 



2 5 
5 ' 6 



Activity 2 



Use grids to model 4 



^ 2 
3 ■ 3" 



Divide 5 grids into thirds. Shade 4 grids and :^ of a fifth grid to represent 4^^. 







-i 



Think: How many groups of^ are in 4|? 



Divide the shaded grids into equal groups of: 






2 
3 






There are 6 groups of ^, wath ^ left over. This piece is ^ of a group of |. 
Thus there are 6 + ^ groups of | in 4|. 

3 3 2 



Think and Discuss 



1. Are ^ -^ i and g -^ j modeled the same way? Explain. 

2. When you divide fractions, is the quotient greater than or less than 
the dividend and the divisor? Explain. 



Try This 



Use grids to find each quotient. 



1. 



12 ■ 6 



2. ^-^ 
5 10 



2^4 
3 ■ 9 



4 3^^^ 
^••^5 5 



3-9 Hands-On Lab 185 



'Atk.. 




Mixed Numbers 



7.1.7 Solve problems that involve multiplication and division with integers, 
fractions, decimals and combinations of the four operations. 

The original Sunshine 
Sicyway Bridge connecting 
St. Petersburg and Palmetto, 
Florida, opened in 1954 and 
had a toll of $1.75. The current 
Sunshine Skyway Bridge 
opened in 1987, replacing the 
original. In 2007, the toll for 
a car crossing the bridge was 
I of the toll in 1954. To find 
the toll in 2007, you will need 
to multiply the toll in 1954 by 
a fraction. 




To multiply fractions, multiply the numerators to find the product's 
numerator. Then multiply the denominators to find the product's 
denominator. 



EXAMPLE 



9 



Multiplying Fractions 

Multiply. Write each answer in simplest form. 
A -15 



-i: 



J ■ o = ~ 



_K3 1 

1 * 3 



_ _ l.q- 2 
1-3, 

= -V^ 
= -10 



Write -15 as a fraction. 
Simplify. 

IVIultiply numerators. Multiply denominators. 



iI!i)i:uJJJjjjj 



The product of two 
positive proper 
fractions is less than 
either fraction. 



i . i 

4 * 5 

1 . 4 _ 1 -,4^ 
4*5 ,4-5 

_ 1 

5 



(4) 



3 • 1 

4-2 



Simplify. 

Multiply numerators. Multiply denominators. 



The signs are different, so the answer will 
be negative. 

Multiply numerators. Multiply denominators. 



186 Chapter 3 Applying Rational Numbers 



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EXAMPLE 



12] Multiplying Mixed Numbers 

Multiply. Write each answer in simplest form. 



A 8-2^ 



o3 _ 8 11 

4 1 4 

_' X- 11 



Write mixed numbers as improper fractions. 
Simplify. 



= =Y= = 22 Multiply numerators. Multiply denominators. 



B i-4i 

3 2 



i . 4i = i . 9 
3 2 3 2 



Write the mixed number as an improper fraction. 
Simplify. 



3 1 

= 77 or 1;^ Multiply numerators. Multiply denominators. 



^ -^5 '12 

33 . 1^ = 18 . 13 
■^5 ^12 5 12 



Write mixed numbers as improper fractions. 



2A Simplify. 



5- ^2, 



on q 

= yjT or 3jx Multiply numerators. Multiply denominators. 



EXAMPLE [3] Transportation Application 



In 1954, the Sunshine Skyway Bridge toll for a car was Si. 75. In 
2007, the toil was ^ of the toll in 1954. What was the toll in 2007? 



7 

1 75 . i = 1^:^ = i3 . 4 

= 7.4 
4 7 



I I 



= 1 



Write the decimal as a fraction. 

Write the mixed number as an improper 
fraction. 

Simplify. 

Multiply numerators. Multiply denominators. 



The Sunshine Sk^avay Bridge toll for a car was $1.00 in 2007. 



Think and Discuss 

1. Describe how to multiply a mixed number and a fraction. 

2. Explain why ^ ' \' \ = 4x is or is not correct. 

3. Explain why you may want to simplify before multiplying 
What answer will you get if you don't simplify first? 



2 3 



m-d 



Lessor Tutorials Online 



3-9 IVIultiplying Fractions and IVIixed Nunnbers 187 



3-9 



»rtTiiih'iii>7^iV{ri'Viitiiitiiiiiiiii«i^^^^^ 




Homework Help Online go.hrw.com, 



keyword MHIiltgjM ® 
Exercises 1-27, 33, 39, 43, 45, 
49,53,55 



(SUlDEiC) PRACTICE 



See Example 1 Multiply. Write each answer in simplest form. 



See Example 2 
See Example 3 



1. -8 



2. 



3. 



5. 4 
9 



7. U-1 



4.f.(-15) 
8. 2^- (-7) 



On average, people spend ^ of the time they sleep in a dream state. If Maxwell 
slept 10 hours last night, how much time did he spend dreaming? Write your 
answer in simplest form. 



INDEPENDENT PRACTICE 



See Example 1 Multiply. Write each answer in simplest form. 





10. 


= 4 




14. 


2 5 

5 7 


See Example 2 


! 18. 


7^-2^ 




22. 


2 ^91 

3 '^ 4 


See Example ■ 


t 26. 


Sherr\' 



11. 


4 


1 
8 


15. 


3 
8 


2 
3 


19. 


6 


1 



12. 3 



20. 2|.i 



23. U-1 



24. 7-5 



13. 6 
17. - 
21. 2 



2 
3 

5 . 2 

6 ' 3 



8 3 

25. 3f • 2\ 
4 5 



26. Sherr\' spent 4 hours exercising last week. If ^ of the time was spent jogging, 
how much time did she spend jogging? Write your answer in simplest form. 

27. Measurement A cookie recipe calls for | tsp of salt for 1 batch. Doreen is 
making cookies for a school bake sale and wants to bake 5 batches. How 
much salt does she need? Write your answer in simplest form. 



Extra Practice 



See page EP10. 



PRACTICE AND PROBLEM SOLVING 



Multiply. Write each answer in simplest form. 



^^- 8 5 



32. 



36. 3i • 5 

40. -2 . li . 2 

3 2 3 



29 4- • - 

" 7 6 

33 -^-^ 

■^^ 4 9 



37. 



41. 



2 3 5 

8 . A . 33 

9 11 40 



30. 



2 a 



34 4- • 2- 



38. 



42. |-6-8| 

b 3 



31. 
35. 



2-i 

'^ 6 



9 I 16/ 

1.3.7 
! 5 9 



39 li.3.Z 
2 5 9 



Complete each multiplication sentence. 



44. 



48. p 



__ _ A 

8 16 

3 _ 1 

4 



45. 



49. 



_ _ 1 

4 2 

i = A 

5 15 



46. 



50. 



5 _ A 
8 12 

9^ _ A 
11 



47. 



51. 



3 
5 

Is 



_ _ 3 

7 7 

3 _ J_ 
5 25 



52. Measurement A standard paper clip is 1^ in. long. If you laid 75 paper 
clips end to end, how long would the line of paper clips be? 



188 Chapter 3 Applying Rational Numbers 



53. Physical Science The weight of an object on the moon is | its weight on 
Earth. If a bowling ball weighs 12t^ pounds on Earth, how much would it 
weigh on the moon? 



Radio 



54. In a survey, 200 students were asked 
what most influenced them to 
download songs. The results are 
shown in the circle graph. 

a. How many students said radio 
most influenced them? 

b. How many more students were 
influenced by radio than by a 
music video channel? 

c. How many said a friend or 
relative influenced them or they 
heard the song in a store? 

55. The Mississippi River flows at a rate 
of 2 miles per hour. If Eduardo floats 
down the river in a boat for 5| hours, 
how far will he travel? 

© 56. Choose a Strategy What is the product oil 
<S)i CD 5 CT) 1 



Influences for Downloading Songs 



Friend/ 
relative 



Other 




Live 
performance 



Heard/saw 
n store 



Music video 
channel 



4? 
5" 



CD) 



d 



5 "^ " "^20 

57. Write About It Two positive proper fractions are multiplied. Is the product 
less than or greater than one? Explain. 

^ 58. Challenge Write three multiplication problems to show that the product of 
two fractions can be less than, equal to, or greater than 1. 



Test Prep and Spiral Review 



59. Multiple Choice W^iich expression is greater than 5|? 



® 8-4 



CE) 



(-«?) 



cr:' 3 



1 . 5 



CE) - 



3 14 



16 ^-^ 9 \ 7 1 "—"2 7 ^-^ 7 27 

60. Multiple Choice The weight of an object on Mars is about j| its weight on 
Earth. If Sam weighs 85 pounds on Earth, how much would he weigh on Mars? 

CE) 11 pounds CS) 3l| pounds CH) 120| pounds CD 226^ pounds 

Use a number line to order the integers from least to greatest. (Lesson 2-1 ) 

61. -7, 5, -3, 0, 4 62. -5, -10, -15, -20, 63. 9, -9, -4, 1, -1 



Add or subtract. Write each answer in simplest form. (Lesson 3-8) 

4 65. 2f - li 66. 5^ + 3^ 

3 5 4 3 7 14 



64. 4^ + 2| 65. 2^ - l4 66. 5^ + 3A 67. 4| + 2-? 



3-9 Multiplying Fractions and Mixed Numbers 189 



3-10 



Mixed Numbers 







7.1.7 Solve problems that involve multiplication and division with integers, 
fractions, decimals and combinations otthe four operations. 

Reciprocals can help you divide by fractions. Two numbers are reciprocals 
or multiplicative inverses if their product is 1. Tlie reciprocal of ^ is 



3 because 



3 = 



3 _ 3 _ 



1. 



Vocabulary 

reciprocal 
multiplicative inverse 



3 1 3 

Dividing by a number is the same as multiplying by its reciprocal. 

— Reciprocals • 



r 

3 



2 



6.1 

-Same answer- 



2 

J 



Interactivities Online ^ You can use this rule to divide by fractions. 



EXAMPLE 



1 I Dividing Fractions 

Divide. Write each answer in simplest form. 



A 


2 
3 


. 1 
■ 5 








2 
3 


. 1 _ 2 5 

■5 3 1 

2-5 

3- 1 


3^ 
-^3 


Multiply by ttie reciprocal of ^ 


B 


3 
5 


4-6 








3 

5 


-6 = ^-1 
5 6 

'3-1 

5-62 




Multiply by the reciprocal of 6 
Simplify. 






_ 1 
10 







I EXAMPLE 



(3 



Dividing Mixed Numbers 

Divide. Write each answer in simplest form. 



A 4i^2i 

41^2^ = -^ 
3 2 3 



= 13 2 
3 5 

= ^ or lii 

15 15 



Write mixed numbers as improper fractions. 
Multiply by the reciprocal of |. 



190 Chapter 3 Applying Rational Numbers 



VliJau Lesson Tutorials Online my.hrw.com 



Divide. Write each answer in simplest form. 



EXAMPLE 



^ 



5 . 

6 ■ 


4 




5 . 

6 ■ 


7 


5 . 50 

6 ■ 7 




= 


5 7 
6*50 




= 


V-7 
6-50,0 




— 


7 
60 


*i 


. 6 
■ 7 




4 


^ 6 _ 

7 


24 . 6 

5 ■ 7 




= 


24 7 
5 6 



l/l/r/te 7y as an improper fraction. 
Multiply by the reciprocal of^. 
Simplify. 



_ 24-7 
5 -gi 

= ^ or 5^ 
5 "' ^5 



l/l/r/te 4| as an improper fraction. 
Multiply by the reciprocal of | . 
Simplify. 



Social Studies Application 

Use the bar graph to 

determine how many times 

longer a S100 bill is expected 

to stay in circulation than a 

$1 bill. 

The life span of a $1 bill is 

U years. The life span of a 

$100 bill is 9 years. 

Think: How many l^'s are there 

in 9? 



Life Spans of Bills 




$50 $100 



9^1^ = ^ 
2 1 



_ 9 2 
1 3 

= f ore 



Write both numbers as improper fractions. 
Multiply by the reciprocal of ^. 
Simplify. 



A $100 biU is expected to stay in circulation 6 times longer than a $1 bill. 



Think and Discuss 

1. Explain whether ^ -^ I is the same as 2 



3' 



2. Compare the steps used in multiplying mixed numbers with 
those used in dividing mixed numbers. 



'■Mau Lesson Tutorials Online 



3-10 Dividing Fractions and Mixed Nunnbers 191 



3-10 



GUIDED PRACtltE 



Homework Help Online go.hrw.com, 



keyword laailfcaill ® 

Exercises 1-27, 29, 31, 33, 35, 
37,43,47 



See Example 1 Divide. Write each answer in simplest form. 



See Example 2 



1. 6^ 



5 ^-3i 
^'6 3 



7 3^3 

5 ■ 4 

6. 5f - 4 



3. 



K-^i 



4 _5^2 
9 ■ 5 

8- 2l^^ f 



See Example 3 9. Kareem has 12^ yards of material. A cape for a play takes 3| yards. How many 
L capes can Kareem make with the material? 



INDEPENDENT PRACTICE 



See Example 1 Divide. Write each answer in simplest form. 



10.2^^ 
14. l^-i 



11. 10 ^1 
15. ^^12 



12. 4h-| 



16. 



^6 



13 ^^i 
17. -16 -H 



See Example 2 18. pr -^ 4 



22. 35^9 



iq A — 9_L 

4 ■ "10 

23. 14| H- li 

3 o 



20. 224^4^ 

24 7— -=- 2- 

'10 5 



21. -lO^H- 



25. 



5 8 



See Example 3 26. A juicer holds 43| pints of juice. How many 2|-pint bottles can be filled 
with that much juice? 

27. Measurement How many 24| in. pieces of ribbon can be cut from a roll of 
ribbon that is 147 in. long? 



Extra Practice 



See page EP10. 



PRACTICE AND PROBLEM SOLVING 



30 - -=- - 

3 9 



Evaluate. Write each answer in simplest form. 

28. 6f^| 29. -1^^(^ 

32.1. 4f 33.(2| + 3|). 11 34. (1 . |) . fl 

^^- 2 (5 " U^) + i " 3 

39 2 ^ /5 X] _ 2 . 1 

3 l6 12; 2 



31 _i3^2l 
J I. i^ . z.^ 



35. 



37, 



3 ^ 15 ^ f_4 \ 
7 ■ 28 ■ I 5 J 



38-J^2i 



40. 3 + A 4. 2 

4 20 5 



- 1 



''■{ff 



+ 



_9_ 
10 



42. Three friends will be driving to an amusement park that is 226| mi from their 
town. If each friend drives the same distance, how far will each drive? Explain 
how you decided which operation to use to solve this problem. 

43. Multi-Step How many 1 lb hamburger patties can be made from a lOl lb 
package and an 111 15 package of ground meat? 

44. Write About It Explain what it means to divide | by 1 Use a modefin your 
explanation. 



1 92 Chapter 3 Applying Rational Numbers 



• Q* * . . 




Industrial Arts 



45. 



46. 



47. 



48. 



49. 



Multi-Step The students in Mr. 
Park's woodworking class are making 
birdhouses. The plans call for the side 
pieces of the birdhouses to be 7^ 
inches long. If Mr. Park has 6 boards 
that are 50| inches long, how many side 
pieces can be cut? 

Critical Thinking Brandy is stamping circles 
from a strip of aluminum. If each circle is 
l| inches tall, how many circles can she get 
from an 8|-inch by l^-inch strip of aluminum? 

For his drafting class, Manuel is drawing plans for a bookcase. Because he 
wants his drawing to be | the actual size of the bookcase, Manuel must 
divide each measurement of the bookcase by 4. If the bookcase will be 
3| feet wide, how wide will Manuel's drawing be? 

The table shows the total number of hours that the students in each of 
Mrs. Anwar's 5 industrial arts classes took to complete their final 
projects. If the third-period class has 17 students, how many hours did 
each student in that class work on average? 




^S^ Challenge Alexandra is cutting wood stencils to spell her 
first name with capital letters. Her first step is to cut a square of 
wood that is 3:^ in. long on a side for each letter in her name. Will 
Alexandra be able to make all of the letters of her name from a single piece 
wood that is ll; in. wide and 18 in. long? Explain your answer. 



Period 


Hours 


1st 


200| 


2nd 


179| 


3rd 


199| 


5th 


190| 


6th 


180l 



of 



I 



Test Prep and Spiral Review 



50. Multiple Choice Which expression is NOT equivalent to 2^ -h It^? 



3 13 



Cl> 22^13 



CD 8^13 
3 ■ 8 



51. Multiple Choice What is the value of the expression 



9. . 1 - £? 
5 6 " 5' 



CD 



CDl 



CE) 



15 



^rn 



CD 25 



52. Gridded Response Each cat at the animal shelter gets | c of food every day. If 
Alysse has 16^ c of cat food, how many cats can she feed? 

Find the least common multiple (LCM). (Lesson 2 8) 

53.2,15 54.6,8 55.4,6,18 56.3,4,8 



Multiply. Write each answer in simplest form. (Lesson 3-9) 



57. - 



iL. 5 

15 8 



58. l|,.6 



59. l|-24 



60. 



6-2^ 



3-W Dividing Fractions and Mixed Numbers 193 



Solving Equations 
Containing Fractions 



7.2.1 Use variables and appropriate operations to write an expression, equation or 
inequality that represents a verbal tJescription. 



Gold classified as 24 karat is pure gold, while 
gold classified as 18 karat is only | pure. 
The remaining ^ of 18-karat gold is made 
up of one or more different metals, 
such as silver, copper, or zinc. 

Equations can help you determine the 
amounts of metals in different kinds of 
gold. The goal when solving equations 
that contain fractions is the same as when 
working with other kinds of numbers — to 
isolate the variable on one side of the equation. 

EXAMPLE MIJ Solving Equations by Adding or Subtracting 

Solve. Write each answer in simplest form. 






A 


•^5 5 

^ 5 5 


/ 


You can also isolate 


B 


^ + 11 = -^ 

18 27 

^ + 11=-^ 
18 ^ " 27 




the variable y by 
adding the opposite 




18 18 27 
" 54 


-re 


sides. 




54 





Use the Addition Property of Equality. 
Add. 



Use ttie Subtraction Property of Equality. 
Find a common denominator 
Subtract. 



Recall that the product of a nonzero number and its reciprocal is 1. 
This is called the Multiplicative hiverse Property. 



Multiplicative Inverse Property 


Words 


Numbers 


Algebra 


The product of a nonzero number 
and its reciprocal, or multiplicative 
inverse, is one. 


1.5= 1 
5 4 '■ 


b a 



You can use the Multiplicative Inverse Property to solve multiplication 
equations that contain fractions and whole numbers. 



194 Chapter 3 Applying Rational Numbers 



^Ms'j Lesson Tutorials Online my.hrw.com 



EXAMPLE 



Caution! 



To undo multiplying 
by ^, you must divide 

by ^ or multiply by 

its reciprocal, |. 



Solving Equations by Multiplying 

Solve. Write each answer in simplest form. 



3 5 

3 5 

2^ . 3 J^ . 3 
3 2 5 Zi 

A- = f or U 

3 5 



3j' 



3y = 



3y 



. I=¥. i 

3 7 ^1 

y = l 



L/se t/ie Multiplicative Inverse Property. 
Multiply by the reciprocal of |. Then simplify. 



Use the Multiplicative Inverse Property. 
Multiply by the reciprocal of 3. Then simplify. 



EXAMPLE 



(B 



Physical Science Application 

Pink gold is made of pure gold, 
silver, and copper. There is ^ more 
pure gold than copper in pink gold. 
If pink gold is | pure gold, what 
portion of pink gold is copper? 



Let c represent the amount of copper 
in pinlc gold. 




' ^ 20 4 




Write an equation. 


c + ii_n^3_ 

20 20 4 


11 

20 


Subtract to isolate c. 


-M- 


11 

20 


Find a common denominator. 


-25, 




Subtract. 


-i 




Simplify. 


Pink gold is ^ copper. 







^^^^^^^^^^^^^^^^^^^^^^^^^^^^^B 


Thmk and Discuss 










1. 


Show the first step you 


would use 


to solve m + 3| = 


12^ 




2. 


Describe how to decide 


' whether | 


is a solution of ^y 


_ 3 
5' 




3. 


Explain why solving |c = | by muhiplying both sides 
same as solving it by dividing both sides by |. 


byf 


is the 



VjJ^:;] Lesson Tutorials Online mv.hrw.com 3-11 Solving Equations Containing Fractions 195 



3-11 



;iIJjr'3Jd33 



GUIDED PRACTICE 



Homework Help Online go.hrw.com. 



keyword MiBifcaiM ^ 
Exercises 1-20, 27, 31, 33, 35, 
39,41,43 



See Example 1 Solve. Write each answer in simplest form. 



2. ;h + i = I 

b b 



See Example 2 4. ^.v = 8 5. |r = | 



3- ?^-3 = 6 



6. 3w^ 



2 = 5 
3 



7 



See Example 3 7. Kara has | cup less oatmeal than she needs for a cookie recipe. If she has 
I cup of oatmeal, how much oatmeal does she need? 



INDEPENDENT PRACTICE 



See Example 1 Solve. Write each answer in simplest form. 



11. x + l = 2l 



See Example 2 14. ^.v = 4 

17. 6r = Jr 



9. r-i = j 



12- " + 10=11) 



0. 


,-^ = 1 

24 3 


3. 


y + 5 = 19 
■'6 20 


6. 


^y-fo 


9. 


h = a 



See Example 3 20. Earth Science Carbon- 14 has a half-life of 5,730 years. After 17,190 

years, ^ of the carbon- 14 in a sample will be left. If 5 grams of carbon- 14 
are left after 17,190 years, how much was in the original sample? 





B fKi 


F\l.ll^t: ANU 


KKUKLC 


IVI d< 


JLVINU ^ 




23. 






^ Solve. Write each 

21. ir = l 

5 5 


answer in simplest form. 

22. ,.-i = | 




r Extra Practice 




See page EPIO. 


^' = ! 




24. 


9 18 




25. 


|v=l 




26. 


j-l = \^. 




27. 


^11 = 3^ 

3" •^s 




28. 


- + i = 3^ 
6 15 




29. 


4^ 8 




30. 


- JL + /;/ = - 

26 


7 
13 


31. 


-4 + r=- 


1 

11 


32. 


-' 4 20 




33. 


/, -3 = _il 
8 24 




34. 


~36'^= ~T6 




35. 


13 13 




36. 


4f + p=5l 




37. 


^-4 = 9^ 




38. 


6^k = 13i 



39. Food Each person in Finland drinks an average of 24]^ lb of coffee per year. 
This is IB-j^ lb more than the average person in Italy consumes. On average, 
how much coffee does an Italian drink each year? 

40. Weather Yuma, Arizona, receives 102^^ fewer inches of rain each year 
than Quillayute, Washington, which receives lOSjj inches per year. {Source: 
National Weather Service). How much rain does Yuma get in one year? 



1 96 Chapter 3 Applying Rational Numbers 



41. 






Life Science Scientists have discovered l\ million species of animals. 
This is estimated to be ^ the total number of species thought to exist. 
About how many species do scientists think exist? 



Birthplaces of U.S. Presidents 
in Office, 1789-1845 



Massachusetts 




Virginia 



The Chase Tower 
is the tallest 
skyscraper in 
Indiana. The two 
spires bring the 
building's height 
to 830 feet. One 
of the spires 
functions as a 
communications 
antenna, while 
the other is 
simply decorative. 



History The circle graph shows 
the birthplaces of the United States' 
presidents who were in office 
between 1789 and 1845. 

a. If six of the presidents 
represented in the graph were 
born in Virginia, how many 
presidents are represented in 
the graph? 

b. Based on your answer to a, how 
many of the presidents were 
born in Massachusetts? 

Architecture In Indianapolis, 

the Market Tower has = as many 

stories as the Chase Tower. If the Market Tower has 32 stories, how many 

stories does the Chase Tower have? 

44. Multi-Step Each week, Jennifer saves ^ of her allowance and spends 
some of the rest on lunches. This week, she had ^ of her allowance left 
after buying her lunch each day. What fraction of her allowance did she 
spend on lunches? 

(^ 45. What's the Error? A student solved |.v = | and got x = |. Find the error. 



<^ 




South 
Carolina 



New York 



/*345. 
46. 

47. 



Write About It Solve 3|z = l~. Explain why you need to write mixed 
numbers as improper fractions when multiplying and dividing. 

Challenge Solve ^w — 0.9. Write your answer as a fraction and as a decimal. 



C 



Test Prep and Spiral Review 



48. Multiple Choice Which value of y is the solution to the equation 

V - ^ = ^? 

J 8 5- 



-* 40 



CD y = 



_ 10 



13 



®."='i 



CE) y = 2 



49. Multiple Choice Which equation has the solution .v = -|? 



CDfx=-l 



® -\- = I) 



CE) -f + A- = I CD X - 3| = 3| 



Order the numbers from least to greatest. (Lesson 2-11) 

50. -0.61, -|, -|, -1.25 
5 3 



51. 3.25,3^,3,3.02 



Estimate. (Lesson 3-1) 
53. 5.87 - 7.01 



54. 4.0387 + (-2.13) 



52. i -0.2, -^, 0.04 



55. 6.785 • 3.01 



J- 7 7 Solving Equations Containing Fractions 197 




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Quiz for Lessons 3-6 Through 3-11 

(v) 3-6 j Estimating with Fractions 



Estimate each sum, difference, product, or quotient. 

1.4-^ 2.-1 + 5^ 



3. 4fk • 3i 

15 4 



er 



3-7 j Adding and Subtracting Fractions 

Add or subtract. Write eacli answer in simplest form. 



t + ^ 



6. 



7. 



.1 + 6 
3 9 



4 qL ^ A^ 
^- ^9 • ^5 



a 5 2 
^- 8"3 



(vj 3-8] Adding and Subtracting Mixed Numbers 



Add or subtract. Write each answer in simplest form. 

-5 



1 ? 

9 gl + 2- 



10. i| + ?! 

b 3 



11. 57 



12. 8 



12 



er 



13. A mother giraffe is ISy^j ft tall. She is 5^ ft taller than her young giraffe. 
How tall is the young giraffe? 

3-9] Multiplying Fractions and Mixed Numbers 

Multiply. Write each answer in simplest form. 



14. -12 



15. 



_5 7_ 

14 * 10 



16. 8^ 

5 



10 



17. 10 



18. A recipe calls for 1^ cups flour. Tom is making 2^ times the recipe for 
his family reunion. How much flour does he need? Write your answer 
in simplest form. 

Q) 3-10] Dividing Fractions and Mixed Numbers 
Divide. Write each answer in simplest form. 



19 ^ ^ ^ 

6 6 



20. |.-4 



21 5- ^ ^ 



22. 4f ^ 1^ 



23. Nina has 94 yards of material. She needs l4 yards to make a pillow case. 
How many pillow cases can Nina make with the material? 

^) 3-11] Solving Equations Containing Fractions 

Solve. Write each answer in simplest form. 



24 v-^ = ^ 

3 15 



25. t=-2q 



26- ^'" = I 



27. 



_ 1 



+ P=- 



28. A recipe for Uncle Frank's homemade hush puppies calls for ^ teaspoon 
of cayenne pepper. The recipe calls for 6 times as much salt as it does 
cayenne pepper. How much salt does Uncle Frank's recipe require? 



198 Chapter 3 Applying Rational Numbers 



CONNECTIONS 





Civil Rights In Educatlow Heritage Trail The roots of tree Virginia 

public education in the United States can be traced to southern 
Virginia. A self-guided driving tour of the area takes visitors to 
more than 40 schools, libraries, and other sites that played a 
key role in the story of civil rights in education. 

The Wilson family is driving the Civil Rights in Education Heritage 
Trail. Use the map to solve these problems about their trip. 

1 . The Wilsons drive from Appomattox 
to Petersburg on the first day of their 
trip. How many miles do they drive? 

2. On the second day of the trip, they 
drive from Petersburg to South 
Hill. How much farther do they 
drive on the first day than on the 
second day? 

3. The distance from South Boston 
to Halifax is I of the distance from 
Farmville to Nottoway. What is the 
distance from South Boston to 
Halifax? 

4. The entire trip from Appomattox to Halifax is 202. 1 miles. The 
Wilsons' car gets 21.5 miles to the gallon. How many gallons of 
gas will they use for the trip? 

5. Gas costs $3.65 per gallon. How much will gas cost for the 
entire trip? 




Appomattox 

.29,-omi^ 



Farmville 



Carver-Price 






^ 


.4 5 mi 


School 




R.R. Moton 
High School 


^Vj460j— 






Nottowa 














Mary M. B 
High Sc 


et^ 
ho 


Tune 
ol 




49fomi 



Petersburg 



u 



Virginia State '> 
University 



Halifax 
South Bosti: 



South Hilly 



ss^.- 



T-^^^y^ 



^■ 



V I 



R GXN I A 



Southside Virginia 
Community College 
Christanna Campus 




25 kilometers 



Real-World Connections 199 




Number Patterns i.5>ji#. 

The numbers one through ten form the pattern below. Each arrow _ ^ ^ 

indicates some kind of relationsliip between the two numbers. ^'^'^^^t. 
Four relates to itself. Can you figure out what the pattern is? 



one- 



two 

1 

•three 

t 
six 



seven 



-ten 



eight 



five- 
/ 



four- 



■ nme 



The Spanish numbers "^^° I 

iiiro through diez form 




a similar pattern. In this 

case, ciiico relates to itself, dos (2) 



ocho (8) siete (7) 

I i ^ 

tres (3) ►quatro (4)-« ►seis (6) cinco (5) ) 



diez (10) 



nueve (9) 



Other interesting number patterns involve cyclic numbers. Cyclic 
numbers sometimes occur when a fraction converts to a repeating 
nonterminating decimal. One of the most interesting cyclic 
numbers is produced by converting the fraction 4 to a decimal. 

1 = 0.142857142857142... 

Multiplying 142857 by the numbers 1-6 produces the same digits 
in a different order. 



1 • 142857 = 142857 

2 • 142857 = 285714 



3 • 142857 = 428571 

4 • 142857 = 571428 



5 • 142857 = 714285 

6 • 142857 = 857142 



Fraction Action 



Roll four number cubes and use the numbers 
to form two fractions. Add the fractions and 
try to get a sum as close to 1 as possible. To 
determine your score on each turn, find the 
difference between the sum of your fractions 
and 1 . Keep a running total of your score as 
you play. The winner is the player with the 
lowest score at the end of the game. 

A complete copy of the rules are available online. 




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\ 200 Chapter 3 Applying Rational Numbers 



\\ 



^I# 



Materials 

• file folder 

• ruler 

• pencil 

• scissors 

• markers 



' => ?«^\ 



PROJECT 



Slide notes through the frame to review key concepts 
about operations with rational numbers. 

Directions 

O Keep the file folder closed throughout the project. 
Cut off a 3^-inch strip from the bottom of the 
folder. Trim the remaining folder so that is has no 
tabs and measures 8 inches by 8 inches. Figure A 

Cut out a thin notch about 4 inches long along the 
middle of the folded edge. Figure B 

Cut a 3|-inch slit about 2 inches to the right of the 
notch. Make another slit, also 3| inches long, 
about 3 inches to the right of the first slit. Figure C 

O Weave the 3^-inch strip of the folder into the 
notch, through the first slit, and into the second 
slit. Figure D 

Taking Note of the Math 

As you pull the strip through the frame, divide the strip 
into several sections. Use each section to record 
vocabulaiy and practice problems 
from the chapter. 




Operation 
Slide Through 



D 


N 


_Jt 




^' \ . 




v 


J 



i I 




CHA?r£^ 3 





OPeKaT,0n5 with 
RATIONAL NuMe,Eg5 




Ws in the Bag! 



201 



study Guide: Review 



reciprocal 190 



Vocabulary 

compatible numbers 144 

multiplicative inverse 1 90 

Complete the sentences below with vocabulary words from the list above. 



1. When estimating products or quotients, you can use L 



that are close to the original numbers and easy to use. 

? because they multiply to give 1. 



o o 

2. The fractions ^ and ^ are 

O J 



EXAMPLES 



EXERCISES 



3-1j Estimating with Decimals (pp. 144-147) 



Estimate. 

63.28 - 
+ 16.52 - 

43.55 - 
X 8.65 - 



63 

+ 17 

80 

40 
X 9 
360 



Round each decimal to 
the nearest integer. 

Use compatible 
numbers. 



Estimate. 

3. 54.4 + 55.99 4. 11.48- 5.6 

5. 24.77 • 3.45 6. 37.8 H- 9.3 

7. Helen saves $7.85 each week. She wants 
to buy a TV that costs $163.15. For about 
how many weeks will Helen have to save 
her money before she can buy the TV? 



3-2 ) Adding and Subtracting Decimals (pp. 148-151) 

■ Add. Add or subtract. 

5.67 + 22.44 8. 4.99 + 22.89 

Line up the decimal points. ^q jg 09 — 11 87 

12. 23 - 8.905 



5.67 

+ 22.44 

28.11 



Add. 



9. -6.7 + (-44.5) 
11. 47 + 5.902 
13. 4.68 + 31.2 



3-3] Multiplying Decimals (pp. 154-157) 
■ Multiply. 



1.44- 0.6 

1.44 
X 0.6 
0.864 



2 decimal places 
1 decimal place 
2+1=3 decimal places 



Multiply. 

14. 7 • 0.5 

16. 4.55 • 8.9 

18. 63.4- 1.22 

20. Fred buys 4 shirts at $9.52 per shirt. 
How much did Fred spend? 



15. 


-4.3-9 


17. 


7.88 • 7.65 


19. 


-9.9 • 1.9 



202 Chapter 3 Applying Rational Numbers 



EXAMPLES 



EXERCISES 



3-4] Dividing Decimals (pp. 160-163) 
■ Divide. 



7H- 2.8 

2.5 
28)70i) 

56 
140 

140 


Divide. 

0.96^ 1.6 
0.6 



16)9.6 
-9 6 



Multiply both numbers by 
10 to make the divisior an 
integer. 



Divide. 

21. 16^ 3.2 
23. 48 ^ 0.06 

^ (-12.5) 



25. 
27. 
29. 
31. 
33. 



.65 -H 1. 



22. 50 H- (-1.25) 

24. 31 ^ (-6.2) 

26. 816 ^ 2.4 

28. 9.483 ^ (-8.7) 



126.28 ^ (-8.2) 30. 2.5 H- (-0.005) 



9 -=- 4.5 



32. 13 -^3.25 



Multiply both numbers by 
10 to make the divisor an 
integer. 







In qualifying for an auto race, 
one driver had lap speeds 
of 195.3 mi/h, 190.456 mi/h, 
193.557 mi/h, and 192.757 mi/h. 
What was the driver's average 
speed for these four laps? 



3-5 ) Solving Equations Containing Decimals (pp 164-167) 



Solve. 

/; - 4.77 = 8.60 

+ 4.77 + 4.77 

n = 13.37 



Add to isolate n. 



Solve. 

34. .V + 40.44 = 30 

36. 0.8/; = 0.0056 
38. 3.65 + c^=: -1.4 



35. -j^ = 100 
37. k - 8 = 0.64 



39. 



-0.: 



= 15.4 



40. Sam wants to buy a new wakeboard 
that costs $434. If he makes $7.75 per 
hour, how many hours must he work to 
earn enough money for the wakeboard? 



3-6j Estimating with Fractions (pp. 170-173) 



Estimate. 

73_4i 
4 3 



Estimate each sum, difference, product, 
or quotient. 



41. lU + 12^ 



- 4^ = 3i 
2 2 



43. 9^ + 



(-^^1 



42. 5f - 13if 
44. llf-lli 



11^-3^ 
12 -^5 

11 " 



45 C5I3 . 4! 



46. 



■'•i*(->i 



12 

12 -H3 = 4 



12 



47. Sara ran 2^ laps on Monday and 7| laps 
on Friday. About how many more laps 
did Sara run on Friday? 



\y'i<l:i<j\ Lesson Tutorials OnlinE mv.hrw.com 



Study Guide: Review 203 



EXAMPLES 



EXERCISES 



3-7 ! Adding and Subtracting Fractions (pp. 176-179) 



Add. 

1 , 2 _ ^ , _6_ 
3 5 15 15 



Write equivalent 
fractions using a 
common denominator. 




Add or subtract. Write each answer in 


simplest form. 




^M4 


49 - + - 
"■ 4 5 


=»-A + if 


51 4_1 
^'- 9 3 



3-8] Adding and Subtracting Mixed Numbers (pp. 180-183) 

■ Add. Add or subtract. Write each answer in 



li + 2i = 1| + 2| Add the integers, simplest form. 

and ttien add ttie 52. 3^ + 2^ 
= 3 + - 8 3 

6 fractions. 

^ 54 8- - ''- 



53. 



4 12 



55. 11^- lOi 
4 3 



3-9] Multiplying Fractions and Mixed Numbers (pp. 186-189) 



Multiply. Write the answer in 

simplest form. 

,1 . c3 _ 9 . 23 
4 



Multiply. Write each answer in 
simplest form. 



53 = 9 
4 2 



= ^or25| 



56. l|-4i 
58. 4 -31 



57. 



5 10 



59. 34 • l4 



3-IOJ Dividing Fractions and Mixed Numbers (pp. 190-193) 



Divide. 



2 = 3.5 
5 4 2 



!\/luitiply by tlie 
reciprocal of\. 



Divide. Write each answer in simplest form. 

61. 



60. i ^ 6i 



2 -^4 



62. 



63. 2^ ^ 1 



11 ^ li 
13 ■ 13 

64. A 21-inch long loaf of bread is cut into 
3 



-inch slices. How many slices will 
there be? 



3-11] Solving Equations Containing Fractions (pp. 194-197) 

■ Solve. Write the answer in simplest form. Solve. Write each answer in simplest form. 



lx = l 

4"^ 6 




. Ir- 1 . ■* 
4'^ 6 1 

X = ^ = ^ 
6 3 


Multiply by the 
reciprocal of ^. 





65. 



5 3 



67.1x = f 



66-i + y = t 

68. f + x = f 



69. Ty had 2^ cups of oil and used | cup for 
a recipe. How many cups of oil are left? 



204 Chapter 3 Applying Rational Numbers 



Chapter Test 



CHAPTER 






Estimate. 
1. 19.95 + 21.36 2. 49.17 - 



3. 3.21 • 16.78 



4. 49.1 H- 5.6 



Add or subtract. 

5. 3.086 + 6.152 6. 5.91 + 12. 



7. 3.1 - 2.076 



8. 14.75 - 6.926 



Multiply or divide. 

9. 3.25 • 24 



10. -3.79 • 0.9 



11. 32 -=- 1.6 



12. 3.57 + (-0.7) 



Solve. 

13. w- 5.3 = 7.6 



14. 4.9 = c + 3.7 



15. /)+ 1.8 = 2.1 



16. 4.3/? = 81.7 



Estimate each sum, difference, product, or quotient 
18. 5l-3\ 



17 ^ + ^ 

4 8 



19. 6l-2| 
/ 9 



20. 8l + 3A 



Add or subtract. Write each answer in simplest form. 



21. 



10 "^ 5 



22. 



23. 7^ + 5|i 



24. 9-3^ 



Multiply or divide. Write each answer in simplest form. 



25. 5-4:1 



26. 2^ ■ 2| 



27. 



28. 2^ H- 1| 

5 6 



29. A recipe calls for 4| tbsp of butter. Nasim is making 3^ times the recipe for 
his soccer team. How much butter does he need? Write your answer in 
simplest form. 

30. Brianna has 1 1| cups of milk. She needs 1^ cups of milk to make a pot of 
hot cocoa. How many pots of hot cocoa can Brianna make? 



Solve. Write each answer in simplest form. 



3M-I 



32. 



\c = 980 



33. -^ + w = 



34 z - -^ = - 



35. Alan finished his homework in 1^ hours. It took Jimmy | of an hour longer 
than Alan to finish his homework. How long did it take Jimmy to finish 
his homework? 

36. Mya played in two softball games one afternoon. The first game lasted 
42 min. The second game lasted 1| times longer than the first game. 
How long did Mya's second game last? 



Chapter 3 Test 205 



CHAPTER 





Test Tackier 

STANDARDIZED TEST STRATEGIES 



Gridded Response: Write Gridded Responses 

When responding to a test item that requires you to place your answer 
in a grid, you must fill in the grid on your answer sheet correctly, or the 
item will be marked as incorrect. 




EXAMPLE 



I 


. 


I 


<7 






















• 











® 


(0) 


® 


® 


® 


• 


(T) 


• 


® 


® 


® 


® 


@ 


@ 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


(8) 


® 


® 


® 


® 


® 


® 


• 


® 



Gridded Response: Solve the equation 0.23 + r = 1.42. 

0.23 + /•= 1.42 

- 0.23 - 0.23 

r= 1.19 

• Using a pencO, write your answer in the answer boxes at the top of 
the grid. Put the first digit of your answer in the leftmost box, or put 
the last digit of your answer in the rightmost box. On some grids, 
the fraction bar and the decimal point have a designated box. 

• Put only one digit or symbol in each box. Do not leave a blank 
box in the middle of an answer. 

• Shade the bubble for each digit or symbol in the same column 
as in the answer box. 








5 


/ 


3 











• 




















® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


@ 


@ 


@ 


® 


® 


® 


® 


® 


• 


® 


® 


® 


® 


® 


® 


® 


• 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 


® 



Gridded Response: Divide. 3 -^ l| 



3^a = ^^| 



_ 3 5 
1 9 



= 15 = 5^ ^2 ^^g 
9 3 3 

The answer simplifies to ^, 1= or 1.6. 

• Mixed numbers and repeating decimals cannot be gridded, so 
you must grid the answer as ^. 

• Write your answer in the answer boxes at the top of the grid. 

• Put only one digit or symbol in each box. Do not leave a blank 
box in the middle of an answer. 

• Shade the bubble for each digit or symbol in the same column 
as in the answer box. 



206 Chapter 3 Applying Rational Numbers 




If you get a negative answer to a gridded 
response item, rework the problem 
carefully. Response grids do not include 
negative signs, so if you get a negative 
answer, you probably made a math error. 



Read each statement, and then answer the 
questions that follow. 



Sample A 

A student correctly solved 
an equation for .v and got 
42 as a result. Then the 
student filled in the grid 
as shown. 



Sample C 

A student subtracted 
— 12 from 5 and got an 
answer of - 17. Then the 
student filled in the grid 
as shown. 





















4 


2 





© 
® 
® 
@ 
(D 
® 
(D 
® 
® 
@ 
d; 



© 
@ 

;i; 

(D 
(D 
® 
® 
® 
® 

8 
9 


o 
© 
® 
® 
@ 
@ 
® 
® 
® 
® 

'8 
I 


© 
® 
® 
@ 
@ 
® 
® 
® 
® 

8 

9 



© 
® 
® 
@ 
® 
® 
® 
® 
® 
® 
® 



® 



5. What error did the student make 
when finding the answer? 

6. Explain why you cannot fill in a 
negative number on a grid. 

7. Explain how to fill in the answer 
to 5 - (-12) correctly. 



1. What error did the student make 
when filling in the grid? 

2. Explain a second method of filling in 
the answer correctly. 



Sample B 

A student correctly 
multiplied 0.16 and 0.07. 
Then the student filled in 
the grid as shown. 



O 



© 



3. What error did the student make 
when filling in the grid? 

4. Explain how to fill in the answer 
correctly. 



Sample D 
A student correctly 
simplified | + pj and got 
Ipj as a result. Then the 
student filled in the grid 
as shown. 



1 


q 


/ 


1 


2 








• 








© 


© 


© 


© 


© 


® 


(0) 


® 


® 


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(1) 





• 


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(2) 


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re- 


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'6; 


® 


.7 


7 


7 


7) 


® 


8 


8 


8 


8) 


® 


• 


a 


? 


'9J 



8. What answer is shown in the grid? 

9. Explain why you cannot show a 
mixed number in a grid. 



10. Write two equivalent forms of the 
answer 1^ that co 
the grid correctly. 



answer 1^ that could be filled in 



Test Tackier 207 



CHAPTER 



3 







ISTEP+ 
Test Prep 



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StateTestPracticego.hrw.com, ■ 
■Hf,l|,||.|Msi.iiPstPreplGoa 



Applied Skills Assessment 

Constructed Response 

1. Louise is staying on the 22nd floor of 
a hotel. Her mother is staying on the 
43rd floor. Louise wants to visit her 
mother, but the elevator is temporarily 
out of service. Write and solve an 
equation to find the number of floors 
that Louise must climb if she takes 
the stairs. 

2. Mari bought 3 packages of colored 
paper. She used | of a package to make 
greeting cards and used l| packages for 
an art project. She gave | of a package 
to her brother. How much colored 
paper does Mari have left? Show the 
steps you used to find the answer. 

3. A building proposal calls for 6 acres 
of land to be divided into |-acre lots. 
How many lots can be made? Explain 
your answer. 

Extended Response 

4. A high school is hosting a triple-jump 
competition. In this event, athletes 
make three leaps in a row to try to 
cover the greatest distance. 

a. Tony's first two jumps were 1 1| ft 
and 1 1^ ft. His total distance was 
44 ft. Write and solve an equation 
to find the length of his final jump. 

b. Candice's three jumps were all the 
same length. Her total distance was 
38 ft. What was the length of each 
of her jumps? 



c. The lengths of Davis's jumps were 
1 1 .6 ft, 1 1^ ft, and 1 1| ft. Plot these 
lengths on a number line. What 
was the farthest distance he 
jumped? How much farther was 
this distance than the shortest 
distance Davis jumped? 

Multiple-Choice Assessment 

5. A cell phone company charges $0.05 
per text message. Which expression 
represents the cost of t text messages? 

A. 0.05t C. 0.05 - t 

B. 0.05 + t D. 0.05 -H t 

6. Ahmed had $7.50 in his bank account 
on Sunday. The table shows his account 
activity for each day last week. What 
was the balance in Ahmed's account 
on Friday? 



Day Deposit Withdrawal 


Monday 


$25.25 


none 


Tuesday 


none 


-$108.13 


Wednesday 


$65.25 


none 


Thursday 


$32.17 


none 


Friday 


none 


-$101.50 



A. -$86.96 

B. -$79.46 



C. $0 

D. $96.46 



Natasha is designing a doghouse. She 
wants the front of the doghouse to be 
3^ feet wide, and she wants the side of 
the doghouse to be 2| feet wider than 
the front. Which equation can be used 
to find X, the length of the side of the 
doghouse? 



A. 


3- + 2- = X 
^2 ^ "^4 -^ 


C.3l 


•2i = . 


B. 


3^ - 2^ = X 
^2 ^4 ^ 


D.31 


.2| = x 



208 Chapter 3 Applying Rational Numbers 



What is the value of 5| h- |? 



A. 17 



B. 



C. 10 



D. 5^ 



9. Mrs. Herold has 5^ yards of material to 
make two dresses. The larger dress 
requires 3| yards of material. Which 
equation can be used to find t, the 
number of yards of material remaining 
to make the smaller dress? 



B. 3|.t=5l 



C. 35 



^-H 



D. 3| + t = 5l 



10. Carl is building a picket fence. The 
first picket in the fence is 1 m long, 
the second picket is 1^ m long, and 
the third picket is 1^ m long. If the 
pattern continues, how long is the 
seventh picket? 



l|m 



A 

B. 2 m 



C. 
D. 



2> 
2^ m 



11. Daisy the bulldog weighs 45|| pounds. 
Henry the beagle weighs 2l| pounds. 
How many more pounds does Daisy 
weigh than Henry? 



A. 23|| pounds 

B. 24| pounds 



C. 24j^ pounds 

D. ^7jE pounds 



12. What is the prime factorization of 110? 

A. 55 • 2 C. 11 • 5 • 2 

B. 22 • 5 • 2 D. 110 • 1 



13. 



Joel threw a ball 24| yards. Jamil threw 
the ball 33J^ yards. Estimate how much 
farther Jamil threw the ball than Joel 
did. 

A. 8 yards C. 12 yards 

B. 10 yards D. 15 yards 




When possible, use logic to eliminate 
at least two answer choices. 



14. Which model best represents the 
expression § x ' ? 

O Z 

A. 




k,., ..... 




1 




^^^H 

















15. The table shows the different types 
of pets owned by the 15 students in 
Mrs. Sizer's Spanish class. What fraction 
of the students listed own a dog? 



Type of Pet Number of Students 


Cat 


5 


Dog 


9 


Hamster 


1 



A. 



15 



B. 



D. 



Gridded Response 

16. Frieda earns $5.85 per hour. To find the 
amount of money Frieda earns working 
X hours, use the equation y = 5.85x. 
How many dollars does Frieda earn if 
she works 2.4 hours? 

17. Solve the equation j^x = ^ for x. 

18. What is the value of the expression 
2(3.1) + 1.02(-4) -8 + 3'? 



Cumulative Assessment, Chapters 1-3 209 



'k 



4A 

4-1 
4-2 
4-3 

4-4 

4B 
4-5 
4-6 
4-7 

4C 

LAB 
4-8 

4-9 
4-10 



CHAPTER 



4 



Ratios, Rates, and 
Proportions 

Ratios 

Rates 

Identifying and Writing 
Proportions 

Solving Proportions 

Measurements 

Customary Measurements 
IVletric IVIeasurements 
Dimensional Analysis 

Proportions in 
Geometry 

Make Similar Figures 

Similar Figures and 
Proportions 

Using Similar Figures 

Scale Drawings and 
Scale Models 



D )! 



7.3.5 

7.3.5 
7.3.5 



LAB Make Scale Drawings 



Why Learn This? 



Proportions can be used to find the heights 
of objects that are too tall to measure 
directly, such as a lighthouse. 



£?. 



Learn It Online 

Chapter Project Online go.hrw.com, 




apter 



• Use proportionality to 
solve problems, including 
problems involving 
similar objects, units of 
measurement, and rates. 



apter 4 



^ 



ESSit- 





piiii ~im 




L' ■"'■ii-c ' *saffl»«S?W«!c! V 



m^f, 



;'M!£*Mii 









Are You Ready? 



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*^ ResourcesOnlinego.hrw.com, 

■B«lMS10AYR4-W^ 



0^ Vocabulary 

Choose the best term from the list to complete each sentence. 

1. A(n) ? states that two expressions are equivalent. 

2. To ? an expression is to substitute a number for the 
variable and simplify. 

3. A value of the variable in an equation that makes the 
statement true is acn) ? of the equation. 

4. A(n) ? is a number that can be written as a 
ratio of two integers. 

Complete these exercises to review skills you will need for this chapter. 

Evaluate Expressions 

Evaluate each expression. 

5. .v + 5for.v= -18 



equation 
evaluate 

irrational number 
rational number 
solution 



7. ^ for :: = 96 

— D 

9. -3z + 1 for c = 4 



11. 5 



fory = —3 



6. -9vfor v= 13 
8. w- 9for !('= -13 
10. 3»' + 9for((' = 7 
12. X-+ 1 for.v= -2 



(z) Solve Equations 

Solve each equation. 

13. y + 14 = -3 14. -4y = -72 15. y - 6 = 39 

17. 56 = 8y 18. 26 = y + 2 19. 25 - y = 7 

21. -72 = 3_v 22.25 = ^ 23. 15 + y = 4 



16. ^=-9 



20. 
24. 



121 

V 



= 11 

20 = -2y 



(v) Number Patterns 

Find the next three numbers in the pattern. 

25. 95, 112, 129, 146 26. 85, 65, 60, 40, 35 

27. 20, 20, 100, 100, 500 28. 12, 14, 17, 21, 26 

29. 1,3,5,7,... 30. -19,-12,-5,2,... 

31. 5, -10, 20, -40, 80 32. 0, -10, -5, -15, -10, 



Proportional Relationships 211 



Where You've Been 

Previously, you 

• used ratios to describe 
proportional situations. 

• used ratios to make predictions 
in proportional situations. 

• used tables to describe 
proportional relationships 
involving conversions. 






You will study 

• using division to find unit rates 
and ratios in proportional 
relationships. 

• estimating and finding 
solutions to application 
problems involving 
proportional relationships. 

• generating formulas involving 
unit conversions. 

• using critical attributes to 
define similarity'. 

• using ratios and proportions in 
scale drawings and scale models. 



Where You're Going 



Key 

Vocabulary /Vocabulario 



You can use the skills 
learned in this chapter 

• to read and interpret maps. 

• to find heights of objects that 
are too tall to measure. 



corresponding 
angles 


angulos 
correspondientes 


corresponding 
sides 


lados 
correspondientes 


equivalent ratios 


razones equivalentes 


proportion 


proporcion 


rate 


tasa 


ratio 


razon 


scale 


escala 


scale drawing 


dibujo a escala 


scale model 


modelo a escala 



similar 



semejante 



Vocabulary Connections 

To become familiar with some of the 
vocabulary terms in the chapter, consider the 
following. You may refer to the chapter, the 
glossary, or a dictionary if you like. 

1. "Miles per hour," "students per class," and 
"Calories per serving" are all examples 

of rates. Wliat other rates can you think 
of? How would you describe a rate to 
someone if you couldn't use examples in 
your explanation? 

2. You can select a gear ratio on a bicycle for 
maximum speed. Think of other examples 
where the word ratio is used. What do 
these examples have in common? 

3. Similar means "having characteristics in 
common." If two triangles are similar , 
what might they have in common? 



212 Chapter 4 





Reading / 

and WrLtuva 
Math X -^ 




Writing Strategy: Use Your Own Words 

Using your own words to explain a concept can help you understand 
the concept. For example, learning how to solve equations might seem 
difficult if the textbook does not explain solving equations in the same 
way that you would. 

As you work through each lesson: 

• Identify the important ideas from the explanation in the book. 

• Use your own words to explain these ideas. 




What Sara Reads 



An equation is a 
mathematical statement 
that two expressions are 
equal in value. 

Wlien an equation contahis 
a variable, a value of the 
variable that makes the 
statement true is called a 
solution of the equation. 

If a variable is multiplied 
by a number, you can often 
use division to isolate the 
variable. Di\ide both sides of 
the equation by the number. 



What Sara Writes 



/In eciuafion has an equal sicji-i 
to show that f\A/o expressions 
are ecfual to each other. 

The solution of an equation 
that has a variable in it is the 
number that the variable is 
equal to. 



When the variable is multiplied 
by a number, you can undo the 
multiplication and qet 
the variable alone by 
dividincj both sides 
of the equation by 
the number. 




TVy This 



Rewrite each sentence in your own words. 

1. When solving addition equations involving integers, isolate the 
variable by adding opposites. 

2. When you solve equations that have one operation, you use an 
inverse operation to isolate the variable. 



Proportional Relationships 213 



& 




7.1.9 Solve problems involving ratios and proportions. Express one quantity 
as a fraction of another, given their ratio, and vice versa. Find how. . 

In basketball practice, Kathlene 
made 17 baskets in 25 attempts. 
She compared the number of 
baskets she made to the total 
number of attempts she made 
by using the ratio ~. A ratio is 

Vocabulary a comparison of two quantities 

ratio by division. 



B 



... many times one quantity 
is as large as another, 
given their ratio, and 
vice versa. Express one 
quantity as a traction of 
another given the two 
quantities. Find the whole, 
or one part, when a whole 
IS divided into parts in 
a given ration. Solve 
problems involving two 
pairs of equivalent ratios. 



Kathlene can write her ratio of 
baskets made to attempts in three 
different ways. 




17 



2s 17tol5 17:25 



EXAMPLE 



|1j Writ! 



Writing Ratios 

A basket of fruit contains 6 apples, 4 bananas, and 3 oranges. 
Write each ratio in all three forms. 



A bananas to apples 

number of bananas _ 4 



There are 4 bananas and 6 apples. 



number of apples 6 
The ratio of bananas to apples can be written as j, 4 to 6, or 4:6. 



B bananas and apples to oranges 
number of bananas and apples _ 4 + 6 
number of oranges 3 



3 



The ratio of bananas and apples to oranges can be written 
asf, 10to3, or 10:3. 



C oranges to total pieces of fruit 

number of oranges _ 3 



number of total pieces of fruit 6 + 4 + 3 13 

The ratio of oranges to total pieces of fruit can be written as ~, 
3 to 13, or 3:13. 



214 Chapter 4 Proportional Relationships 



yida 



Lesson Tutorials OnlinE my.hrw.com 



Sometimes a ratio can be simplified. To simplify a ratio, first write it 
in fraction form and then simplify the fraction. 



EXAMPLE 



ii^fJ 



bi/.: 



A fraction is in 
simplest form 
when the GCF of 
the numerator and 
denominator is 1 . 



P^9 



Writing Ratios in Simplest Form 

At Franklin Middle School, there are 252 students in the seventh 
grade and 9 seventh-grade teachers. Write the ratio of students to 
teachers in simplest form. 

students 



teachers 



252 
9 

= 252 ^9 
9^9 



Write the ratio as a fraction. 

Simplify. 

For every 28 students, ttiere is 1 teacher. 



The ratio of students to teachers is 28 to 1. 



To compare ratios, vwite them as fractions with common denominators. 
Then compare the numerators. 



EXAMPLE 



O 



Comparing Ratios 

Tell whether the wallet size photo or the portrait size photo has 
the greater ratio of width to length. 





Width (in.) 


Length (in.) 


Wallet 


3.5 


5 


Personal 


4 


6 


Desk 


5 


7 


Portrait 


8 


10 



Wallet: 



Portrait: 



width (in.) _ 3.5 
length (in.) 5 

width (in.) _ 8 _ 4 
length (in.) ~ TO ~ 5 



Write the ratios as fractions 
with common denominators. 



Because 4 > 3.5 and the denominators are the same, the portrait size 
photo has the greater ratio of width to length. 



Tfiink and Discuss 

1. Explain why the ratio ^ in Example IB is not written as a mixed 
number. 

2. Tell how to simplify a ratio. 

3. Explain how to compare two ratios. 



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4-1 Ratios 215 



4-1 



;i3:?aB33 



S HomeworkHelpOnlinego.hrw.com, 



keyword MiTiHIlEBW ® 

Exercises 1-10, 11, 15, 17, 19 



See Example 1 

[. 
See Example 2 

L 

See Example 3 



GUIDED PRACTICE 



Sun-Li has 10 blue marbles, 3 red marbles, and 17 white marbles. 
Write each ratio in all three forms. 

1. blue marbles to red marbles 2. red marbles to total marbles 

3. In a 40-gallon aquarium, there are 21 neon tetras and 7 zebra danio fish. 
Write the ratio of neon tetras to zebra danio fish in simplest form. 

4. Tell whose DVD collection has the 
greater ratio of comedy movies to 
adventure movies. 





Joseph 


Yolanda 


Comedy 


5 


7 


Adventure 


3 


5 



INDEPENDENT PRACTICE 



See Example 1 Asoccer league has 25 sixth-graders, 30 seventh-graders, and 15 eighth-graders. 
Write each ratio in all three forms. 



5. 6th-graders to 7th-graders 
7. 7th-graders to 8th-graders 



6. 6th-graders to total students 

8. 7th- and 8th-graders to 6th-graders 



See Example 2 



9. Thirty-six people auditioned for a play, and 9 people got roles. Write the ratio 
I in simplest form of the number of people who auditioned to the number of 

! people who got roles. 

See Example 3 10. Tell whose bag of nut mix has the 

greater ratio of peanuts to total nuts. 





Dina 


Don 


Almonds 


6 


11 


Cashews 


8 


7 


Peanuts 


10 


18 



Extra Practice 



See page EPll. 



PRACTICE AND PROBLEM SOLVING 



Use the table for Exercises 11-13 

1 1 . Tell whether group 
1 or group 2 has the 
greater ratio of the 
number of people 
for an open-campus 
lunch to the number of 
people with no opinion. 



Opinions on Open-Campus Lunch 




Group 1 


Group 2 


Group 3 


For 


9 


10 


12 


Against 


14 


16 


16 


No Opinion 


5 


6 


8 



12. Wliich group has the least ratio of the number of people against an 
open-campus lunch to the total number of survey responses? 

13. Estimation For each group, is the ratio of the number of people for an open- 
campus lunch to the number of people against it less than or greater than ^? 



216 Chapter 4 Proportional Relationships 



Physical Science 



The pressure of water at different depths can be 
measured in atniospljeres. or atm. The water pressure 
on a scuba diver increases as the diver descends 
below the surface. Use the table for Exercises 14-20. 

Write each ratio in all three forms. 

14. pressure at -33 ft to pressure at surface 

15. pressure at -66 ft to pressure at surface 

16. pressure at —99 ft to pressure at surface 

17. pressure at -66 ft to pressure at -33 ft 

18. pressure at —99 ft to pressure at —66 ft 

19. Tell whether the ratio of pressure at -66 ft to 
pressure at -33 ft is greater than or less than the 
ratio of pressure at —99 ft to pressure at —66 ft. 

20. 



''^ Challenge Compare the ratio of the beginning 
pressure and the new pressure when a scuba diver goes 
from -33 ft to -66 ft and when the diver goes from the 
surface to -33 ft. Are these ratios of pressures less than 
or greater than the ratio of pressure when the diver goes 
from -66 ft to -99 ft? Use ratios to explain. 




Test Prep and Spiral Review 



21. Multiple Choice Johnson Middle School has 125 sixth-graders, 150 
seventh-graders, and 100 eighth-graders. Which statement is NOT true? 

CS) The ratio of sixth-graders to seventh-graders is 5 to 6. 

CE) The ratio of eighth-graders to seventh-graders is 3:2. 

C£) The ratio of sbcth-graders to students in all three grades is 1:3. 

CE) The ratio of eighth-graders to students in all three grades is 4 to 15. 

22. Short Response A pancake recipe calls for 4 cups of pancake mix for 
every 3 cups of milk. A biscuit recipe calls for 2 cups of biscuit mix for every 
1 cup of milk. Which recipe has a greater ratio of mix to milk? Explain. 



Solve. (Lesson 3-5) 
23. 1.23 + .v= -5.47 



24. 3.8y = 27.36 



25. v- 3.8 = 4.7 



26. On Monday Jessika ran 3^ miles. On Wednesday she ran 4| miles. How much farther 
did Jessika run on Wednesday? (Lesson 3-7) 



4-1 Ratios 217 



^'^■^''"■''■*'**siua;OT 







Vocabulary 

rate 
unit rate 








7.1,9 Solve problems involving ratios and proportions. Express one quantity 
as a fraction of another, given their ratio, and vice versa. Find how... 

The Lawsons are going 
camping at Rainbow Falls, 
which is 288 miles from their 
home. They would like to 
reach the campground in 
6 hours. What should their 
average speed be in miles 
per hour? 



... many times one quantity 
is as large as another, 
given their ratio, and 
vice versa. Express one 
quantity as a fraction of 
another given the two 
quantities. Find the whole, 
or one part, when a whole 
is divided into parts in 
a given ration. Solve 
proljlems involving two 
pairs of equivalent ratios. 



In order to answer the question 
above, you need to find the 
family's rate of travel. A rate is a 
ratio that compares two quantities 
measured in different units. 






The Lawson familv's rate is 



288 miles 



6 hours 

A unit rate is a rate whose denominator is 1 when it is written as 
a fraction. To change a rate to a unit rate, first write the rate as a 
fraction and then divide both the numerator and denominator by the 
denominator. 



EXAMPLE lli Finding Unit Rates 




Interactivities Online ► 



A During exercise, Sonia's heart beats 675 times in 5 minutes. 
How many times does it beat per minute? 

675 beats 



5 minutes 

675 beats ^ 5 
5 minutes ^ 5 

135 beats 



Write a rate that compares heart beats and time. 
Divide the numerator and denominator by 5. 
Simplify. 



1 minute 

Sonia's heart beats 135 times per minute. 

B To make 4 large pizza pockets, Paul needs 14 cups of broccoli 
How much broccoli does he need for 1 large pizza pocket? 

14 cups broccoli 



4 pizza pockets 
14 cups broccoli -^ 4 
4 pizza pockets -^ 4 
3.5 cups broccoli 



Write a rate that compares cups to pocl<ets. 
Divide the numerator and denominator by 4. 
Simplify. 



1 pizza pocket 
Paul needs 3.5 cups of broccoli to make 1 large pizza pocket. 



218 Chapter 4 Proportional Relationships 



y'liiBU] Lesson Tutorials Online mv.hrw.com 



An average rate of speed is the ratio of distance traveled to time. The ratio 
is a rate because the units being compared are different. 

EXAMPLE [2] Finding Average Speed 

!; The Lawsons want to drive 288 miles to Rainbow Falls in 

6 hours. What should their average speed be in miles per hour? 

288 miles 



6 hours 
288 miles ^ 6 _ 48 miles 



Write the rate as a fraction. 

Divide tlie numerator and denominator 
by the denominator. 



6 hours ^ 6 1 hour 

ti 

I Their average speed sliould be 48 miles per hour. 



A unit price is the price of one unit of an item. The unit used depends 
on how the item is sold. The table shows some examples. 



Type of Item 


Examples of Units 


Liquid 


Fluid ounces, quarts, gallons, liters 


Solid 


Ounces, pounds, grams, kilograms 


Any item 


Bottle, container, carton 



EXAMPLE r 3J Consumer Math Application 



The Lawsons stop at a roadside farmers' 
market. The market offers lemonade in 
three sizes. Which size lemonade has the 
lowest price per fluid ounce? 

Divide the price by the number of fluid 

ounces (fl oz) to find the unit price of each 

size. 

$0.89 ^ $0.07 $1.69 _ $0.09 

ISfloz floz 



Size 


Price 


12 fl oz 


$0.89 


18 fl oz 


$1.69 


24 fl oz 


$2.09 



$2.09 ^ $0.09 
24 fl oz fl oz 



12 floz floz 

Since $0.07 < $0.09, the 12 tl oz lemonade has the lowest price 
per fluid ounce. 



Think and Discuss 

1. Explain how you can tell whether a rate represents a unit rate. 

2. Suppose a store offers cereal with a unit price of $0.15 per ounce. 
Another store offers cereal with a unit price of $0.18 per ounce. 
Before determining which is the better buy, what variables must 
you consider? 



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4-2 Rates 219 




^ HomeworkHelpOnlinego.hrw.com, 



keyword ■BHIiEBiM (^ 

Exercises 1-8, 9, 11, 13, 23 



See Example 1 



See Example 2 

L 

See Example 3 



GUIDED PRACTICE 



1. A faucet leaks 668 milliliters of water in 8 minutes. How many milliliters of 
water does the faucet leak per minute? 

2. A recipe for 6 muffins calls for 360 grams of oat flakes. How many grams 
of oat flakes are needed for each muffin? 

3. An airliner makes a 2,748-mile flight in 6 hours. Wliat is the airliner's 
average rate of speed in miles per hour? 

4. Consumer Math During a car trip, the Webers buy gasoline at three 
different stations. At the first station, they pay $18.63 for 9 gallons of gas. 
At the second, they pay $29.54 for 14 gallons. At the third, they pay $33.44 
for 16 gallons. Which station offers the lowest price per gallon? 



See Example 1 



INDEPENDENT PRACTICE 



See Example 2 



See Example 3 



5. An after-school job pays $116.25 for 15 hours of work. How much money 
does the job pay per hour? 

6. It took Samantha 324 minutes to cook an 18 lb turkey. How many 
minutes per pound did it take to cook the turkey? 

7. Sports The first Indianapolis 500 auto race took place in 1911. The 
winning car covered the 500 miles in 6.7 hours. What was the winning 
car's average rate of speed in miles per hour? 

8. Consumer Math A supermarket sells orange juice in three sizes. The 
32 fl oz container costs $1.99, the 64 fl oz container costs $3.69, and the 

96 fl oz container costs $5.85. Which size orange juice has the lowest price 
per fluid ounce? 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EP11. 



Find each unit rate. Round to the nearest hundredth, if necessary. 

9. 9 runs in 3 games 10. $207,000 for 1,800 ft- 11. $2,010 in 6 mo 

12. 52 songs on 4 CDs 13. 226 mi on 12 gal 14. 324 words in 6 min 

15. 12 hr for $69 16. 6 lb for $12.96 17. 488 mi in 4 trips 

18. 220 min 20 s 19. 1.5 mi in 39 min 20. 24,000 km in 1.5 hr 

21. In Grant Middle School, each class has an equal number of students. There 
are 38 classes and a total of 1,026 students. Write a rate that describes the 
distribution of students in the classes at Grant. What is the unit rate? 

22. Estimation Use estimation to determine which is the better buy: 
450 minutes of phone time for $49.99 or 800 minutes for $62.99. 



220 Chapter 4 Proportional Relationships 



Find each unit price. Then decide which is the better buy. 



23. 



$2.52 Q^$3.64 



42 oz 



52 oz 



24. 



$28.40 $55.50 



;yd 



15 yd 



25. 



$8.28 $13.00 
0.3 m 0.4 m 



26. Sports At the track meet, Justin won the 100-meter race in 12.61 seconds. 
Shawn won the 200-meter race in 26.38 seconds. Which runner ran at a faster 
average rate? 



27. Social Studies The population density 
of a country is the average number 
of people per unit of area. Write the 
population densities of the countries in 
the map at right as unit rates. Round your 
answers to the nearest person per square 
mile. Then rank the countries from least 
population density to greatest population 
density. 



fi 



^^cipulatioh Density " 
(people per square mil^ 



France: 

Population 60,876,13 

Area 210.668 




Poland: 

Population 38,S36,869 

Area 117,571 



Germany: 

Population 82,422,299 

Area 135,236 



28. Write a Problem A store sells paper 
towels in packs of 6 and packs of 8. Use 
this information to write a problem about 
comparing unit rates. 

29. Write About It Michael Jordan has the highest scoring average in NBA 
history. During his career, he played in 1,072 games and scored a total 
of 32,292 points. I^xplain how to find a unit rate to describe his scoring 
average. What is the unit rate? 

30. Challenge Mike fills his car's gas tank with 20 gallons of regular gas at 
$2.01 per gallon. His car averages 25 miles per gallon. Serena fills her car's 
tank with 15 gallons of premium gas at S2.29 per gallon. Her car averages 
30 miles per gallon. Compare the drivers' unit costs of driving one mile. 




Test Prep and Spiral Review 



' ' "" 



31. Multiple Choice What is the unit price of a 16-ounce box of cereal that 
sells for $2.48? 



CE) $0.14 



CD $0.15 



CT) $0.0155 



CS? $0,155 



32. Short Response A carpenter needs 3 minutes to make 5 cuts in a board. Each 
cut takes the same length of time. At what rate is the carpenter cutting? 

Multiply. Estimate to check whether each answer is reasonable. (Lesson 3 3) 

33. -4.87 • (-2.4) 34. -6.2 • 130 35. 0.65 • (-2.07) 

36. Julita's walking stick is 3| feet long, and Toni's walking stick is 3| feet long. 
Whose walking stick is longer and by how much? lesson 3-8) 



4-2 Rates 221 



4-3 



Vocabulary 

equivalent ratios 
proportion 



' j^/ Si 



ReaiiiaMgii 



Read the proportion 
f = li by saying 
"six is to four as 
twenty-one is to 
fourteen." 



Identifying and Writ 
Proportions ^ 



Students in Mr. Howell's 
math class are measuring 
the width w and the length 
( of their faces. The ratio of 
€ to w is 6 inches to 4 inches 
for Jean and 21 centimeters 
to 14 centimeters for Pat. 



These ratios can be written as t and 
Since both ratios simplify to ^, they are 
equivalent. Equivalent ratios are ratios 
that name the same comparison. 

An equation stating that two ratios are 
equivalent is called a proportion . The 
equation, or proportion, below states that 
the ratios | and y^ are equivalent. 







Round face, f = * 



6 = 21 
4 14 

If two ratios are equivalent, they are said to be proportional, 
or /;; proportion. 



EXAMPLE |lj Comparing Ratios in Simplest Form 

Determine whether the ratios are proportional. 



2 ^ 

7' 21 

2 

7 

21 



21 H- 3 



^ is already in simplest form. 
Simplify jj. 



Since 5 = ^, the ratios are proportional. 



A _6_ 

24' 20 

24 



20 



8 -f 8 
24 H- 8 

6-^2 
20 H- 2 



1 
3 

3_ 
10 



Simplify 
Simplify 



24' 



20- 



1 q 

Since ^ ^ -tt^. the ratios are tiot proportional. 



222 Chapter 4 Proportional Relationships 



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EXAMPLE [2] Comparing Ratios Using a Common Denominator 



Use the data in the table to 
determine whether the ratios of 
oats to water are proportional for 
both servings of oatmeal. 

Write the ratios of oats to water 
for 8 servings and for 12 servings. 

Ratio of oats to water, 8 ser\'ings: | 
Ratio of oats to water, 12 semngs: | 



Servings of 
Oatmeal 


Cups of 
Oats 


Cups of 
Water 


8 


2 


4 


12 


3 


6 



Write the ratio as a fraction. 
Write the ratio as a fraction. 



2 




9 


6 




12 


4 




4 


6 




24 


3 




3 


4 




12 


6 




6 


4 




24 



Write t/ie fractions witli a common 
denominator, sucii as 24. 



Since both ratios are equal to }^, they are proportional 



You can find an equivalent ratio by multiplying or dividing both 
terms of a ratio by the same number. 

EXAMPLE [3] Finding Equivalent Ratios and Writing Proportions 



.♦Qlli. 




The ratios of the sizes 
of the segments of a 
nautilus shell are 
approximately equal 
to the golden ratio, 
1.618. ...This ratio 
can be found in many 
places in nature. 



Find a ratio equivalent to each ratio. Then use the ratios to write 
a proportion. 

» 8 



14 



8 
14 


= 


8- 

14 • 


20 
20 


_ 160 
280 


8 
14 


= 


160 
280 






4 










IB 










4 
18 


= 


4 4- 
18- 


2 
r 2 


2 
~ 9 


4 
18 


= 


2 
9 







IVIultiply both terms by any number, 

such as 20. 

Write a proportion. 



Divide both terms by a common factor, 

such as 2. 

Write a proportion. 



Think and Discuss 

1. Explain why the ratios in Example IB are not proportional. 

2. Describe what it means for ratios to be proportional. 

3. Give an example of a proportion. Then tell how you know it is 
a proportion. 



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4-3 Identifying and Writing Proportions 223 



4-3 



iicioajsaa 



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keyword ■MMBEgl ® 
Exercises 1-28, 29, 39, 41, 43 



GUIDED PRACTICE 



See Example 1 Determine whether the ratios are proportional. 



1. 



See Example 2 



2 4 
3' 6 

10 15 
12' 18 



5 A _2_ 
10' 18 



3. 



6. 



6 8 



_9_ 15 
12' 20 

3 5 
4' 6 



4 3 A 
4' 12 



8. 



4 6 
6' 9 



See Example 3 Find a ratio equivalent to each ratio. Then use the ratios to write a 
proportion. 



10. 



11. 



12. 



10 



INDEPENDENT PRACTICE 



See Example 1 Determine whether the ratios are proportional. 



13 ^ ^ 

! '^- 8' 14 

See Example 2 17. |, ^ 



14. 



18. 



_8_ U) 
24' 30 

18 15 
12' 10 



15. 



18 _8L 
20' 180 



19 I 14 
8' 24 



16. 



20. 



15 27 
20' 35 

18 K) 
54' 30 



See Example 3 Find a ratio equivalent to each ratio. Then use the ratios to write a proportion. 



21. 



25. 



13 



22. 



26. 



60 



22 



23. 



27. 



104 



24. 
28. 



121 
99 

27 
72 



Extra Practice 



See page EP11. 



PRACTICE AND PROBLEM SOLVING 



Complete each table of equivalent ratios. 



29. angelfish 



tiger fish 



6 



U 



20 



30. squares 



circles 



Find two ratios equivalent to each given ratio. 
31. 3 to 7 32. 6:2 33. 



35. 6 to 9 



36. 



U) 
50 



37. 10:4 



16 



34. 8:4 
38. 1 to 10 



39. Ecology If you recycle one aluminum can, you save enough energy to 
run a TV for four hours. 

a. Write the ratio of cans to hours. 

b. Marti's class recycled enough aluminum cans to run a TV for 

2,080 hours. Did the class recycle 545 cans? Justify your answer using 
equivalent ratios. 

40. Critical Thinking The ratio of girls to boys riding a bus is 15:12. If the 
driver drops off the same number of girls as boys at the next stop, does 
the ratio of girls to boys remain 15:12? Explain. 



224 Chapter 4 Proportional Relationships 



41. Critical Thinking Write all possible proportions using only the numbers 
1, 2, and 4. 

42. School Last year in Kerry's school, the ratio of students to teachers 
was 22:1. Write an equivalent ratio to show how many students and 
teachers there could have been at Keriy's school. 

43. Life Science Students in a 
biolog>' class visited four 
different ponds to determine 
whether salamanders and frogs 
were inhabiting the area. 

a. Wliat was the ratio of 
salamanders to frogs in 
Cypress Pond? 

b. In which two ponds was the ratio of 
salamanders to frogs the same? 



Pond 


Number of 
Salamanders 


Number 
of Frogs 


Cypress Pond 


8 


5 


Mill Pond 


15 


10 


Clear Pond 


3 


'V 


Gill Pond 


2^^ 


.^A, 



/ 



44. Marcus earned $230 for 40 hours of work. Phillip earned $192 for 32 hours 
of work. Are these pay rates proportional? Explain. 

^ 45. What's the Error? A student wrote the proportion ^ = ||j- What did the 
student do wrong? 



46. Write About It Explain two different ways to determine if two ratios are 
proportional. 



fff 47. Challenge A skydiver jumps out of an airplane. After 0.8 second, she has 
fallen 100 feet. After 3.1 seconds, she has fallen 500 feet. Is the rate (in feet 
per second) at which she falls the first 100 feet proportional to the rate at 
which she falls the next 400 feet? Explain. 



i 



Test Prep and Spiral Review 



'*"" 



48. Multiple Choice Which ratio is NOT equivalent to ||? 



CS) 



CE) 



CD 



64 



3 ^-^12 ^^96 

49. Multiple Choice Which ratio can form a proportion with ^? 



6" 



CD 



13 



(S) 



25 
36 



®s 



CD 



CD 



144 



95 
102 



Divide. Estimate to check whether each answer is reasonable. (Lesson 3-4) 

50. 14.35 H- 0.7 51. -9^-2.4 52. 12.505^3.05 53. 427 H- (-5.6) 



Compare. Write <, >, or =. (Lesson 4-1) 
54. 3:5 12:15 55. 33:66 1:3 



56. 9:24 3:8 



57. 15:7 8:3 



4-3 Identifying and Writing Proportions 225 



4-4 



Vocabulary 

cross product 
Interactivities Online ^ 



oivmg Kroporiions 



Density is a ratio that compares a 
substance's mass to its volume. If you 
are given the density of ice, you can 
find the mass of 3 mL of ice by solving 
a proportion. 

For two ratios, the product of the 
first term in one ratio and the 
second term in the other is a 
cross product . If the cross products 
are equal, then the ratios 
form a proportion. 

5 • 6 = 30 



2 • 15 = 30 




Ice floats in water because the density of ice is 
less tfian the density of water. 



CROSS PRODUCTS^ 



In the proportion j — -,< where /; ?t o and d ^ 0, 
the cross products, a • d and b • r, are equal. 



You can use cross products to solve proportions with variables. 
EXAMPLE {ij Solving Proportions Using Cross Products 



Use cross products to solve the proportion ^ = ^. 



10 • 6 = /;• 3 

60 = 3p 

60 _^ 
3 3 

' 20 = p 



The cross products are equal. 
Multiply. 

Divide each side by 3. 



It is important to set up proportions correctly. Each ratio must 
compare corresponding quantities in the same order. Suppose a boat 
travels 16 miles in 4 hours and 8 miles in .v hours at the same speed. 
Either of these proportions could represent this situation. 



Trip 1 



1 1 6 mi ] _ [ 
V 4 h J U h . 



.Trip2 ^^^""'-^h)- 
^ C 8 mi xh )- 



Trip 1 
Trip 2 



226 Chapter 4 Proportional Relationships 



^M-d'j Lesson Tutorials OnliriE iny.hrw.com 



EXAMPLE 



PROBLEM 



5" 

SOLVING 



C3 



PROBLEM SOLVING APPLICATION 

Density is the ratio of a substance's mass to its volume. The 
density of ice is 0.92 g/mL. What is the mass of 3 mL of ice? 



pl> Understand the Problem 

Rewrite the question as a statement. 

• Find the mass, in grams, of 3 mL of ice. 

List the important information: 

1 . mass (B) 

• densit)' = — j ~- 

■^ volume (mL) 



density of ice = 



0-92 g 
1 mL 



Make a Plan 

Set up a proportion using the given information. Let ni represent the 
mass of 3 mL of ice. 

0-92 g ^ _m_ *_ mass 
1 mL 3 mL < — volume 

•HI Solve 

Solve the proportion. 

^^^pX^ Write the proportion. 

/// • 1 = 0.92 • 3 The cross products are equal, 

in = 2.76 Multiply. 

The mass of 3 mL of ice is 2.76 g. 



Q Look Back 



Since the density of ice is 0.92 g/mL, each milliliter of ice has a mass 
of a little less than 1 g. So 3 mL of ice should have a mass of a little 
less than 3 g. Since 2.76 is a little less than 3, the answer is reasonable. 



Think and Discuss 

1. Explain how the term cross product can help you remember how 
to solve a proportion. 

2. Describe the error in these steps: | = j^; 2x = 36; .v = 18. 

3. Show how to use cross products to decide whether the ratios 
6:45 and 2:15 are proportional. 



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4-4 Solving Proportions 227 



^x^i^sMmsMimsiusiis^iiass^iixmaMMiiimi:. . 



11 ai^jQJSQi 



£; 



^ 




^ HomeworkHelpOnlinego.hrw.com, 



keyword MBtelllBBW ® 

Exercises 1-15, 29, 31, 33, 3S, 
37,39 



GUIDED PRACTICE 



See Example 1 



Use cross products to solve each proportion. 



_6_ 
10 



2. i = ^ 
7 /^ 



12.3 _ 75 


4 f _ 1.5 
^- 42 3 


'" 100 



See Example 2 5. A stack of 2,450 one-dollar bills weighs 5 pounds. How much does a stack 
L of 1,470 one-dollar bills weigh? 



See Example 1 



See Example 2 



Extra Practice 



See page EP11. 



INDEPENDENT PRACTICE 



Use cross products to solve each proportion. 

6. A = ^ 7. ^ = i^ 8. 



10 



4 
36 


= 


X 

180 


45 

X 


- 


15 
3 



84 



24 



, _r_ 
52 



11 i. = 96 



12-1 = 



9 J^ = ll 

140 '' 

13. H = 5 



6 16 5 12 " 8 

14. Euro coins come in eight denominations. One denomination is the one- 
euro coin, which is worth 100 cents. A stack of 10 one-euro coins is 
21.25 millimeters tall. How tall would a stack of 45 one-euro coins be? 
Round your answer to the nearest hundredth of a millimeter. 

15. There are 18.5 ounces of soup in a can. This is equivalent to 524 grams, 
lenna has 8 ounces of soup. How many grams does she have? Round your 
answer to the nearest whole gram. 



PRACTICE AND PROBLEM SOLVING 



Solve each proportion. Then find another equivalent ratio. 



16. 
20. 
24. 
28. 



4 _ 


12 


/; 


24 


1 _ 
3 


y 
25.5 


r . 


_ 32.5 


84 


182 



17. 
21. 
25. 



■V 

15 



90 



18 _1 
-V 5 

76 _ a 
304 



A- 



18. 
22. 
26. 



39 _ 
4 

m _ 
4 

9 
500 



I rj 
20 



2,500 



19. 1^ = 



23. 
27. 



16.5 



8.7 
2 


_ (1 

4 


5 _ 


6 
19.8 



A certain shade of paint is made by mixing 5 parts blue paint with 2 parts 
white paint. To get the correct shade, how many quarts of white paint 
should be mixed with 8.5 quarts of blue paint? 

29. Measurement If you put an object that has a mass of 40 grams on one 
side of a balance scale, you would have to put about 18 U.S. dimes on the 
other side to balance the weight. About how many dimes would balance 
the weight of a 50-gram object? 

30. Sandra drove 126.2 miles in 2 hours at a constant speed. Use a proportion 
to find how long it would take her to drive 189.3 miles at the same speed. 

31 . Multi-Step In lune, a camp has 325 campers and 26 counselors. In July, 
265 campers leave and 215 new campers arrive. How many counselors does 
the camp need in July to keep an equivalent ratio of campers to counselors? 



228 Chapter 4 Proportional Relationships 



• U*' 



Life Science 



\ P. P iT\, Arrange each set of numbers to form a proportion. 

32. 10,6,30, 18 33. 4,6, 10, 15 




75,4,3, 100 



36. 30, 42, 5, 7 



34. 12,21,7,4 
37. 5,90, 108,6 



This catfish was 
7 feet, 7 inches 
long and weighed 
212 pounds! She 
was caught and 
re-released in the 
River Ebro, near 
Barcelona, Spain. 



Life Science On Monday a marine biologist took a random sample of 50 
fish from a pond and tagged them. On Tuesday she took a new sample 
of 100 fish. Among them were 4 fish that had been tagged on Monday. 

a. What comparison does the ratio y~ represent? 

b. What ratio represents the number offish tagged on Monday to n, 
the total number offish in the pond? 

c. Use a proportion to estimate the number offish in the pond. 

39. Chemistry The table shows the 
type and number of atoms in one 
molecule of citric acid. Use a 
proportion to find the number of 
oxygen atoms in 15 molecules of 
citric acid. 



Composition of Citric Acid 


Type of Atom 


Number of Atoms 


Carbon 


6 


Hydrogen 


8 


Oxygen 


7 



40. Earth Science You can find your 

distance from a thunderstorm by counting the number of seconds between 
a lightning flash and the thunder. For example, if the time difference is 21 s, 
then the storm is about 7 km away. About how far away is a storm if the 
time difference is 9 s? 

^41. What's the Question? There are 20 grams of protein in 3 ounces of 
sauteed fish. If the answer is 9 ounces, what is the question? 

1, . 42. Write About It Give an example from your own life that can be 
described using a ratio. Then tell how a proportion can give you 
additional information. 

43. Challenge Use the Multiplication Property of Equality and the proportion 
^ = ^ to show that the cross product rule works for all proportions. 



i 



Test Prep and Spiral Review 



44. Multiple Choice Which proportion is correct? 



^^ 8 10 



^^7 15 



^^ 14 30 



^^ 25 18 



45. Gridded Response Find a ratio to complete the proportion | = | so that 
the cross products are equal to 12. Grid your answer in the form of a fraction. 



Estimate. (Lesson 3-1) 
46. 16.21 - 14.87 



47. 3.82 • (-4.97; 



48. -8.7- (-20.1) 



Find each unit rate. (Lesson 4-2) 

49. 128 miles in 2 hours 50. 9 books in 6 weeks 



51. 



;114 in 12 hours 



4-4 Solving Proportions 229 



CHAPTER 



4 



SECTION 4A 



Ready To Go On? 



^^*P Learn It Online 

*■** ResourcesOnlinego.hrw.com 



Quiz for Lessons 4-1 Through 4-4 



er 



4-1 ] Ratios 

1 . The 2007 record for the University of North Carolina Softball team was 46 wins to 
21 losses. Write the ratio of wins to losses in all three forms. 

2. A concession stand sold 14 strawberry, 18 banana, 8 grape, and 6 orange 
fruit drinks during a game. Tell whether the ratio of strawberry to orange 
drinks or the ratio of banana to grape drinks is greater. 



er 



4-2 ] Rates 
Find each unit rate. Round to the nearest hundredth, if necessary. 

3. $140 for 18 ft^ 4. 346 mi on 22 gal 5. 14 lb for $2.99 

6. Shaunti drove 62 1 miles in 11.5 hours. Wliat was her average speed in miles 
per hour? 

7. Agrocei7 store sells a 7 oz bag of raisins for $1.10 and a 9 oz bag of raisins 
for $1.46. Which size bag has the lower price per ounce? 



er 



4-3 ] Identifying and Writing Proportions 

Find a ratio equivalent to each ratio. Then use the ratios to write a proportion. 



8. 



16 



9 ^ 



10. 



12 



11. 



40 
48 



12. Ryan earned $272 for 40 hours of work. Jonathan earned $224 for 32 hours of 
work. Are these pay rates proportional? Explain. 

13. On a given day, the ratio of dollars to euros was approximately 1:0.735. Is the ratio 
20 to 14.70 an equivalent ratio? Explain. 



(^ 4-4 j Solving Proportions 

Use cross products to solve each proportion. 



14 II = -Ul 
8 4 



15. 



20 _ 2^ 
t 6 



16. 



6__ 0J2 

11 z 



17. 



15 _ .V 



24 



10 



18. One human year is said to be about 7 dog years. Cliff's dog is 5.5 years old in 
human years. Estimate his dog's age in dog years. x 



230 Chapter 4 Proportional Relationships 




Focus on Problem Solving 



JS^ 




HI 



• Choose a problem-solving strategy 

The follov\qng are strategies that you might choose to help you 
solve a problem: 

• Make a table • Draw a diagram 

• Find a pattern • Guess and test 

• Make an organized list • Use logical reasoning 



Work backward 
Use a Venn diagram 



Solve a simpler problem 
Make a model 



Tell which strategy from the list above you would use to solve each 
problem. Explain your choice. 



1 A recipe for bluebern,' muffins calls for 

1 cup of milk and 1.5 cups of blueberries. 
Ashley wants to make more muffins than 
the recipe yields. In Ashley's muffin 
batter, there are 4.5 cups of blueberries. 
If she is using the recipe as a guide, how 
many cups of milk will she need? 

2 There are 32 students in Samantha's math 
class. Of those students 18 are boys. Write the 
ratio in simplest form of the number of girls in 
Samantha's class to the number of boys. 



3 Jeremy is the oldest of four brothers. 
Each of the four boys gets an allowance 
for doing chores at home each week. The 
amount of money each boy receives 
depends on his age. Jeremy is 13 years 
old, and he gets $12.75. His 11 -year-old 
brother gets $1 1 .25, and his 9-year-old 
brother gets $9.75. How much money 
does his 7-year-old brother get? 

4 According to an article in a medical journal, 
a healthful diet should include a ratio of 2.5 
servings of meat to 4 servings of vegetables. 
If you eat 7 serangs of meat per week, how 
many servings of vegetables should you eat? 




Focus on Problem Solving 231 




^MMiinfc 



For more on 
measurements, 
see the table of 
measures on the 
inside back cover. 



Measurements 



The king cobra is one of the world's 
most poisonous snakes. Just 2 fluid 
ounces of the snake's venom is enough 
to kill a 2-ton elephant. 

You can use the following benchmarks 
to help you imderstand fluid ounces, 
tons, and other customary 
units of measure. 








Customary Unit 


Benchmark 




Length 


Inch (in.) 


Length of a small paper clip 






Foot (ft) 


Length of a standard sheet of paper 




■■ii-- 


Mile (mi) 


Length of 4 laps around a track 




Weight 


Ounce (oz) 


Weight of a slice of bread 




Pound (lb) 


Weight of 3 apples 


Ton 


Weight of a buffalo 


^ 


Capacity 

[.J 


Fluid ounce (fl oz) 


Amount of water in 2 tablespoons 




r-^'-'- 


Cup (c) 


Capacity of a standard measuring cup 


< 


Gallon (gal) 


Capacity of a large milk jug 



EXAMPLE 



^ 



Choosing the Appropriate Customary Unit 

Choose the most appropriate customary unit for each 
measurement. Justify your answer. 

A the length of a rug 

Feet — the lengtli of a rug is about the length of several sheets of 
paper. 

B the weight of a magazine 

Ounces — the weight of a magazine is about the weight of 
several slices of bread. 



C the capacity of an aquarium 

Gallons — the capacity of an aquarium is about the capacity of 
I several large milk jugs. 



232 Chapter 4 Proportional Relationships 



l/jiliii; Lesson Tutorials Oniins mv.hrw.com 



The following table shows some common equivalent customan,' units. 
You can use equivalent measures to convert uitits of measure. 



Length 


Weight 


Capacity 


12 inches (in.) = 1 foot (ft) 


15 ounces (oz) = 1 pound 


(lb) 


8 fluid 


ounces (fl oz) = 1 cup (o 


3 feet = 1 yard (yd) 


2,000 pounds = 1 ton 






2 cups = 1 pint (pt) 


5,280 feet = 1 mile (mi) 








2 pints = 1 quart (qt) 


1,760 yards = 1 mile (mi) 








4 quarts = 1 gallon (gal) 



EXAMPLE [2] Converting Customary Units 



Convert 19 c to fluid ounces. 

Method 1: Use a proportion. 

Write a proportion using a 
ratio of equivalent measures. 

fluid ounces — >■ 8 _ .v 
cups — t. 1 19 

8 • 19 = 1 • X 

152 = X 



Method 2: Multiply by 1. 

Multiply by a ratio equal 

to 1, and divide out the units. 

in „ _ 19 if V 8floz 

_ I9-8floz 

1 



Nineteen cups is equal to 152 fluid ounces. 



EXAMPLE [3J Adding or Subtracting Mixed Units of Measure 



A carpenter has a wooden post that is 4 ft long. She cuts 17 in. off 
the end of the post. What is the length of the remaining post? 

First convert 4 ft to inches. 



inches 
feet 



\2 _x 
1 4 



Write a proportion using 1 ft = 12 in. 



X = 48 in. 
The carpenter cuts off 17 in., so subtract 17 in. 
4 ft - 17 in. = 48 in. - 17 in. 
= 31 in. 
Write the answer in feet and inches. 



31 in. X 



12 in. 12 



Multiply by a ratio equal to 1. 



= 2^ ft, or 2 ft 7 in. 



Tfimk and Discuss 

1. Describe an object that you would weigh in ounces. 

2. Explain how to convert yards to feet and feet to yards. 



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4-5 Customary Measurements 233 



i 



4-5 



dL 



Homework Help Online go.hrw.com, 



keyword MblliEHiM ® 
Exercises 1-18, 19, 23, 25, 29, 
31,35,37 



GUIDED PRACTICE 



See Example 1 Choose the most appropriate customary unit for each measurement. Justify 
your answer. 

1 . the width of a sidewalk 2. the amount of water in a pool 

3. the weight of a truck 4. the distance across Lake Erie 



See Example 2 Convert each measure. 

5. 12 gal to quarts 

7. 72 oz to pounds 



6. 8 mi to feet 

8. 3.5 c to fluid ounces 



See Example 3 9. A pitcher contains 4 c of pancake batter. A cook pours out 5 fl oz of the 
L batter to make a pancake. How much batter remains in the pitcher? 



INDEPENDENT PRACTICE 



See Example 1 Choose the most appropriate customary unit for each measurement. Justify 
your answer. 

10. the weight of a watermelon 
L 12. the capacity of a soup bowl 



1 1 . the wingspan of a sparrow 
13. the height of an office building 



See Example 2 Convert each measure. 

14. 28 pt to quarts 

L 16. 5.4 tons to pounds 



15. 15,840 ft to miles 
17. e^ft to inches 



See Example 3 18. Asculptor has a 3 lb block of clay. He adds 24 oz of clay to the block in order 
to make a sculpture. What is the total weight of the clay before he begins 
sculpting? 



Extra Practice 



See page EP12. 



Helpful Hint 



For more on 
units of time 
see Skills Bank 
p. SB8. 



PRACTICE AND PROBLEM SOLVING 



Compare. Write <, >, or =. 

19. 6 yd 12 ft 20. 80 oz 

22. 5 tons 12,0001b 23. 8 gal 
25. 10,000 ft 2 mi 26. 20 pt 



5 1b 
30 qt 
40 c 



21. 18 in. 3 ft 
24. 6.5 c ,^^ 52 fl oz 
27. 1 gal 18 c 



28. Grayson has 3 music lessons each week. Each lesson is 45 minutes long. 
How many total hours will he spend in music lessons in 1 year? 

29. Earth Science The average depth of the Pacific Ocean is 12,925 feet. 
How deep is this in miles, rounded to the nearest tenth of a mile? 



234 Chapter 4 Proportional Relationships 



.* O* *-f . Order each set of measures from least to greatest. 




30. 8 ft; 2 yd; 60 in. 

32. l^ ton; 8,000 oz; 430 lb 

34. 63floz; 7 c; 1.5 qt 



31. 5qt;2gal; 12 pt; 8 c 

33. 2.5 mi; 12,000 ft; 5,000 yd 

35. 9.5 vd; 32.5 ft; 380 in. 



<^ 



The winning 
pumpl<in at the 
34th annual 
Punnpl<in Weigh- 
Off in Half IVloon 
Bay, California, 
weighed 1,524 
pounds! 



Agriculture In one year, the 
United States produced nearly 
895 million pounds of 
pumpkins. How many ounces 
were produced by the state with 
the lowest production shown in 
the table? 



State Pumpkins (million pou 



California 


180 


Illinois 


364 


New York 


114 


Pennsylvania 


109 




37. 



Multi-Step A marathon is a race 
that is 26 miles 385 yards long. What 
is the length of a marathon in yards? 



38. Estimation In 2007, $1 was approximately equal to 1.052 Canadian dollars. 
About how many Canadian dollars equaled $25? 

39. Critical Thinking Explain why it makes sense to divide when you 
convert a measurement to a larger unit. 

@ 40. What's the Error? A student converted 480 ft to inches as follows. 
What did the student do wrong? Wliat is the correct answer? 

1ft _ X 
12 in. 480 ft 

M 41. Write About It Explain how to convert 1.2 tons to ounces. 



^ 42. Challenge A dollar bill is approximately 6 in. long. A radio station gives 
away a prize consisting of a mile-long string of dollar bills. What is the 
approximate value of the prize? 



Test Prep and Spiral Review 



43. Multiple Choice Which measure is the same as 32 quarts? 
CA) 64pt (X) 128 gal CT) 16 c 



CS:> 512floz 



44. Multiple Choice ludy has 3 yards of ribbon. She cuts off 16 inches of the 
ribbon to wrap a package. How much ribbon does she have left? 



CD 1 ft 8 in. 



CE) 4 ft 8 in. 



CH) 7 ft 8 in. 



CD 10 ft 4 in. 



45. A store sells a television for $486.50. That price is 3.5 times what the store 
paid. What was the store's cost? (Lesson 3-5) 



Determine whether the ratios are proportional. (Lesson 4-3) 



46. 



20 8_ 
45' 18 



47. 



6 5 
5' 6 



48. 



11 JL 

44' 28 



49. 



9 27 
6' 20 



4-5 Customary Measurements 235 



4-6 



Metric Measurements 



The Micro Flying Robot II is the world's 
lightest helicopter. Produced in Japan 
in 2004, the robot is 85 millimeters tall 
and has a mass of 8.6 grams. 

You can use the following benchmarks 
to help you understand millimeters, 
grams, and other metric units. 




"^.:. 



If #r^ 



IMiIlifiji' 



For more on metric 
units, see Sl<ills Banl< 
p. SB7. 



^ 



rf^'^ 





Metric Unit 


Benchmark 


Length 


Millimeter (mm) 


Thickness of a dime 


Centimeter (cm) 


Width of your little finger 


IVIeter (m) 


Width of a doorway 


Kilometer (km) 


Length of 10 football fields 


Mass 


Milligram (mg) 


Mass of a grain of sand 


Gram (g) 


Mass of a small paperclip 


Kilogram (kg) 


Mass of a textbook 


Capacity 


Milliliter (ml) 


Amount of liquid in an eyedropper 


Liter (L) 


Amount of water in a large water bottle 


Kiloliter (kL) 


Capacity of 2 large refrigerators 



EXAMPLE (l I Choosing the Appropriate Metric Unit 

Choose the most appropriate metric unit for each measurement. 
Justify your answer. 

A The length of a car 

Meters — the length of a car is about the width of several 
doorways. 



B The mass of a skateboard 

Kilograms — the mass of a skateboard is about the mass 
of several textbooks. 

C The recommended dose of a cough syrup 

Milliliters — one dose of cough syrup is about the amount 
of liquid in several eyedroppers. 



236 Chapter 4 Proportional Relationships 



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Jii---^ 



;t|rp|jTi-njrvVo *■ ' - ? The table shows how metric units are based on powers of 10. 



Prefixes: 
Milli- means 
"thousandth" 
Centi- means 
"hundredth" 
Kilo- means 
"thousand" 



— U 



10^ = 1,000 


10^ = 100 


10^ = 10 


10° = 1 


^ = 0.1 


^=0.01 

10^ 


^-0.001 
10^ 


Thousands 


Hundreds 


Tens 


Ones 


Tenths 


Hundredths 


Thousandths 


Kilo- 


Hecto- 


Deca- 


Base unit 


Deci- 


Centi- 


Milli- 



To convert metric units, multiply or divide by a power of 10. Multiply 
to convert to a smaller unit and divide to convert to a larger unit. 



EXAMPLE 



(B 



Converting Metric Units 

Convert each measure. 

A 510 cm to meters 

510 cm = (510 H- 100) m 100 cm = 7 m, so divide by 100. 

= 5.1 m Since 100 = 10^, move the decimal point 

2 places left: 5 1 0. 

B 2.3 L to milliliters 

2.3 L = (2.3 X 1,000) mL 1 L = 1,000 mL, so multiply by 1,000. 

= 2,300 mL Since 1,000 = 10\ move the decimal point 

3 places right: 2.300 



EXAMPLE [3 J Using Unit Conversion to Make Comparisons 



Mai and Brian are measuring the mass of rocks in their earth 
science class. Mai's rock has a mass of 480 g. Brian's rock has 
a mass of 0.05 kg. Whose rock has the greater mass? How much 
greater is its mass? 

480 _ 1,000 



1 



Write a proportion. 

The cross products are equal. 
480 g = 0.48 kg 



480 = l,OOO.v 

0.48 = .V 

Since 0.48 kg > 0.05 kg, Mai's rock has the greater mass. 

0.48 - 0.05 = 0.43 Subtract to find how/ much greater the 

mass of Mai's rock is. 

The mass of Mai's rock is 0.43 kg greater than the mass of Brian's rock. 



Think and Discuss 

1. Tell how the metric system relates to the base- 10 number system. 

2. Explain why it makes sense to multiply when you convert to a 
smaller unit. 



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4-6 Metric Measurennents 237 



4-6 



[•rniiii 

Homework Help Online go.hrw.com, 



keyword ■MMWEgiM 
Exercises 1-18, 23, 25, 29, 35, 
37,39 



GUIDED PRACTICE 



See Example 1 Choose the most appropriate metric unit for each measurement. 
Justify your answer. 

1. The mass of a pumpkin 

3. The length of an eagle's beak 



2. The amount of water in a pond 
4. The mass of a penny 



See Example 2 Convert each measure. 

5. 12 kg to grams 

7. 0.7 mm to centimeters 



6. 4.3 m to centimeters 
8. 3,200 niL to liters 



See Example 3 9. On Sunday, Li ran 0.8 km. On Monday, she ran 720 m. On which day did 
L Li run farther? How much farther? 



INDEPENDENT PRACTICE 



See Example l Choose the most appropriate metric unit for each measurement. 
Justify your answer. 

10. The capacity of a teacup 
12. The height of a palm tree 



11. The mass of 10 grains of salt 
13. The distance between your eyes 



See Example 2 Convert each measure. 

14. 0.067 Lto milliliters 



16. 900 mg to grams 



15. 1.4 m to kilometers 
17. 355 cm to millimeters 



See Example 



3 18. Carmen pours 75 mL of water into a beaker. Nick pours 0.75 L of water 
into a different beaker. Wlio has the greater amount of water? How much 
greater? 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EP12. 



Convert each measure. 

19. 1.995 m= cm 20. 0.00004 kg = g 21. 2,050 kL= L 

22. 0.002 mL = ^ L 23. 3.7 mm = cm 24. 61.8 g= mg 

Compare. Write <, >, or =. 

25.0.1cm 1mm 26. 25 g 3,000 mg 27. 340 mg 0.4 g 

28. 0.05 kL 5L 29. 0.3 mL 0.005 L 30. 1.3 kg 1,300 g 

31 . Art The Mona Lisa by Leonardo da Vinci is 77 cm tall. Stony Night by 
Vincent Van Gogh is 0.73 m tall. Wliich is the taller painting? How much 
taller is it? 



238 Chapter 4 Proportional Relationships 




Bats consume up 
to 25% of their 
mass at each 
feeding. 



Write each set of measures in order from least to greatest. 

32. 0.005 kL; 4.1 L; 6,300 mL 33. 1.5 m; 1,200 mm; 130 cm 

34. 4,000 mg; 50 kg; 70 g 35. 9.03 g; 0.0008 kg; 1,000 mg 

36. Measurement Use a ruler to measure the 
line segment at right in centimeters. Then give 
the length of the segment in millimeters 
and meters. 

Life Science The table gives information about several species of Vesper, or 
Evening, bats. Use the table for Exercises 37 and 38. 

37. Which bat has the 
greatest mass? 

38. Wliich bat has a longer 
vvingspread, the Red Bat 
or the Big Brown Bat? 
How much longer is its 
wingspread? 

39. Critical Thinking One milliliter of water has a mass of 1 gram. What is 
the mass of a liter of water? 

.■* 40. What's the Error? A student converted 45 grams to milligrams as 
shown below. Explain the student's error. 

45 g = (45 H- 1,000) mg = 0.045 mg 

41. Write About It Explain how to decide whether milligrams, grams, or 
kilograms are tlie most appropriate unit for measuring the mass of an object. 

42. Challenge A decimeter is j^ of a meter. Explain how to convert 
millimeters to decimeters. 



U.S. Vesper Bats 


Name 


Wingspread 


Mass 


Red Bat 


0.3 m 


10.9 g 


Silver-Haired Bat 


28.7 cm 


8,500 mg 


Big Brown Bat 


317 mm 


0.01 kg 



Test Prep and Spiral Review 



43. Multiple Choice Which of these is the same as 0.4 grams? 
Ca;> 0.0004 mg (X' 0.004 mg CD 400 mg 



CE) 4,000 mg 



44. Short Response Wliich has a greater capacity, a measuring cup that 
holds 250 milliliters or a measuring cup that holds 0.5 liters? lustify your 
answer. 



Find each value. (Lesson 1-2) 
45. 9- 46. 12° 



47. 2' 



48. 7' 



49. 3" 



Use cross products to solve each proportion. (Lesson 4-4) 



50. 



80 _ 1000 
20 



X 



51. 



5.24 



28 
2 



52. 



p_ _ in 

25 15 



53. 



2.4 



4-6 Metric Measurements 239 



4-7 



iifietisioiial Analysis 



Vocabulary 

unit conversion 
factor 



A unit conversion factor is a fraction in whicli the numerator and 
denominator represent the same quantity in different units. 

For example, ^ — — is a unit conversion factor. Because 1 mi = 

^ 1 nil 

5,280 ft, the conversion factor can be simplified to 1. 



SlBOft 

Imi 



§280 ft 



S,180 



- =1 

ft * 



Interactivities Online ► You can use a unit conversion factor to change, or convert, 

measurements from one unit to another. Choosing an appropriate 
conversion factor is called diiiieiisioiial analysis. 



EXAMPLE [1] Using Conversion Factors to Solve Problems 



Helpful Hint 



In Example 1A, 

"1 km" appears to 
divide out, leaving 
"degrees per meter," 
which are the units 
asked for. Use this 
strategy of "dividing 
out" units when 
converting rates. 



As you go deeper underground, the earth's temperature 
increases. In some places, it may increase by 25 °C per kilometer. 
Find this rate in degrees per meter. 

Convert the rate 25 °C per kilometer to degrees per meter. 

To convert the second quantity in a rate, 
multiply by a conversion factor with that unit 
in the first quantity. 



25 °C 



I km 



1 kni 1000 m 



25 °C 
1000 ni 

0.025 °C 



Divide out like units. 



C . krrj . 



1 m 



Divide 25 C by WOO m. 

The rate is 0.025°C per meter. 

B In the United States in 2003, the average person drank about 
22 gallons of milk. Find this rate in quarts per month. 

Convert the rate 22 gallons per year to quarts per month. 

?? era! 4 qt 1 yr ^ , . , , ■ r 

& • r • -^ To convert, multiply by conversion factors 

with those units. 



lyr 

22 • 4 qt 
12 mo 
88 qt 
12 mo 
7.3 qt 
1 mo 



I gal 12 mo 



Divide out like units. 

Multiply. 

Simplify. 



J^ 






J/f _ qt 
mo ~ mo 



The rate is about 7.3 quarts per month. 



240 Chapter 4 Proportional Relationships 



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EXAMPLE [2] Converting Between Metric and Customary Units 



One inch is 2.54 centimeters. A bookmark has a length of 

18 centimeters. What is the length of the bookmark in inches, 

rounded to the nearest inch? 



Write a proportion using 
1 in. = 2.54 cm. 

Tiie cross products are equal. 

Multiply. 

Divide each side by 2.54. 

Round to the nearest whole number. 



inches 


— ► 


1 _ 
2.54 


A 

18 




centimeters 






1 


• 18 = 

18 = 

18 _ 
2.54 


2.54- 

2.54.Y 

2.54.Y 
2.54 


X 



7 =- X 
The bookmark is about 7 inches long 



EXAMPLE [3J Sports Application 



A football player runs from his team's 9-yard line to his team's 
44-yard line in 7 seconds. Find the player's average speed in yards 
per second. Use dimensional analysis to check the reasonableness 
of your answer. 

Average speed = 



total distance 



35 yards -^ 7 



total time 
_ 35 yards 
7 seconds 
5 yards 



The player runs 44 
in 7 seconds. 



9 = 35 yards 



Divide to find yards per second. 



7 seconds ^ 7 1 second 

The player's average speed is 5 yards per second. 

Convert yd/s to mi/h to see if the answer is reasonable. 

1 mi . 3 ft _ 3 mi _ 1 mi 
5280 ft' 1yd 5280 yd 1760 yd 

5 yd 1 mi 
Is 1760 yd 

_ ^ . 1 mi . 3600 jT 
1 X 1 760 iid- 1 li 



. 3 mi 
' 5280 yd 

3600 s 
Ih 



Convert miles to yards. 

Set up the conversion factors. 

Divide out like units. 



_ 5 • 1 mi -3600 _ 



10.2 mi/h 



Multiply. Then simplify. 



1 • 1760- Ih 

The player's average speed is approximately 10.2 mi/h, which is a 
reasonable speed for a football player to run a short distance. 



Think and Discuss 

1. Tell whether you get an equivalent rate when you multiply a rate 
by a conversion factor. Explain. 

2. Compare the process of converting feet to inches with the process 
of converting feet per minute to inches per second. 



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4-7 Dimensional Analysis 241 



4-7 



^I 



tj 



Homework Help Online go.hrw.com, 



keyword MBteiHEBiM ® 

Exercises 1-8, 9, n, 13, 15 



See Example 1 

See Example 
See Example 



GUIDED PRACTICE 



1. The maxmimum speed of the TupolevTu-144 airliner is 694 m/s. Find this 
rate in kilometers per second. 

2. All's car uses 12 gallons of gas each week. Find this rate in quarts per year. 

3. One lap around the Daytona Speedway is 2.5 miles. To the nearest hundredth, 
how many kilometers is this? {Hint: 1 mi = 1.609 km) 

4. Martin begins driving to work at 8:15 a.m. He drives 18 miles and arrives at 
his office at 8:39 a.m. Find Martin's average speed in miles per minute. Use 
dimensional analysis to check the reasonableness of your answer. 



See Example 1 



L 

See Example 2 

L 



INDEPENDENT PRACTICE 



5. Lydia wrote 4:^ pages of her science report in one hour. Wliat was her writing 
rate in pages per minute? 

6. An Olympic athlete can run 1 10 yards in 10 seconds. How fast in miles per 
hour can the athlete run? 

7. One lap around the Talladega Speedway is about 4.3 km. To the nearest tenth, 
how many miles is one lap around the speedway? (Hint: 1 mi = 1.609 km) 



See Example 3 8. There are markers every 1000 feet along the side of a road. While driving, 

Sonya passes marker number 8 at 3:10 p.m. and marker number 20 at 3:14 p.m. 
Find Sonya's average speed in feet per minute. Use dimensional analysis to 
check the reasonableness of your answer. 



Extra Practice 



■^XIS Fprg^fflBS! 



See page EP12. 



Use conversion factors to find each of the following. 

9. concert tickets sold in an hour at a rate of 6 tickets sold per minute 

10. miles jogged in 1 hour at an average rate of 8.5 feet per second 

11. calls made in a 3 day telephone fund-raiser at a rate of 10 calls per hour 

12. Estimation In England, a commonly used unit of measure is the stone. One 
stone is equivalent to 14 pounds. lonathan weighs 95 pounds. About how 
many stones does he weigh? Round to the nearest tenth of a stone. 

13. One pound approximately equals 2.2 kilograms. Water weighs about 62.4 lb 
per cubic foot. About how much does water weigh in kilograms per cubic 
foot? Round to the nearest tenth. 

14. Ellie added 600 liters of water into a pool in one hour. One liter approximately 
equals 1.0567 quarts. How many quarts of water per minute did she add? 
Round to the nearest tenth. 



242 Chapter 4 Proportional Relationships 



tmm 



Life Science 



15. Life Science The Outer Bay exhibit at the Monterey Bay Aquarium holds 
about 1,000,000 gallons of sea water. How many days would it take to fill the 
exhibit at a rate of 1 gallon per second? 



16. 




< 



When running at 
top speed, chee- 
tahs take about 
3.5 strides per 
second. However, 
a cheetah can 
maintain this 
speed for a dis- 
tance of only 
200-300 yards. 



18. 
19. 



Money Fencing costs $3.75 per foot. Bryan wants to enclose his rectangular 
garden, which measures 6 yards by 4 yards. How much will fencing for the 
garden cost? 

Life Science A cheetah can run as fast as 70 miles per hour. To the nearest 
himdredth, what is the cheetah's speed in kilometers per minute? 

Transportation Your car gets 32 miles per gallon of gasoline. Gasoline costs 
$3 per gallon. How many kilometers can you travel on $30? 

Choose a Strategy Which unit conversion factor should you use to convert 
56 square feet to square yards? 



3sqft 
1 sq yd 



6sqft 
1 sq yd 



9 sq ft 
1 sqyd 



d. 



12sqft 
1 sqyd 



20. 
21. 



5.6 ky 



What's the Error? To convert 5.6 kg to pounds, a student wrote ' .. " 
Wliat error did the student make? 



2.2 1b 



Write About It Give an example when you would use customary instead 
of metric measurements, or describe a situation when you would use metric 
instead of customar>' measurements. 



^ 22. Challenge Convert each measure. {Hint: 1 oz - 28.35 g) 

a. 8 oz = g c. ]S38.45g= oz 

b. 538.65 g= lb d. 1.5625 lb = g 



m 



Test Prep and Spiral Review 



23. Multiple Choice A company rents boats for $9 per hour. How much per 
minute is this? 

CX) $0.15 CD $0.25 CD $0.54 CD $1.05 

24. Multiple Choice How many square yards are in 27 square feet? 
CD 3 square yards CE) 81 square yards 
CD 9 square yards CD 243 square yards 

25. Short Response Show how to convert 1.5 quarts per pound to liters per 
kilogram. Round each step to the nearest hundredth. {Hint: IL ~ 1.06 qt, 

1 kg == 2.2 lb) 

Evaluate each expression for the given value of the variable. (Lesson 1-6) 

26. 2.V - 3 for .v = - 1 27. 3a + I for a = 3 28. 3c^ - 1 for c = -3 



Multiply. Write each answer in simplest form. (Lesson 3-9) 



29. 12 



30. 



(4) 



31. 3 



3 2 



32. I-10-7J 

b 



4-7 Dimensional Analysis 243 



Ready To Go On? 



CHAPTER 

4 

SECTION 4B 

Quiz for Lessons 4-5 Through 4-7 

(vj 4-5 ] Customary Measurements 
Convert each measure. 

1. 7 lb to ounces 2. 15 qt to pints 

4. 20 fl oz to cups 5. 39 ft to yards 



,#*9 Learn It Online 



I 



Resources Online go.hrw.com. 



Tml MS10RTGO4B 



3. 3 mi to feet 
6. 7,000 lb to tons 



er 



7. Mara and Andrew are baking cornbread to serve 30 people. They pour 3 cups of milk 
into the batter and then add 18 more fluid ounces. How much milk did they use? 

8. Gabrielle has 3 gal of paint. She tises 9 qt to paint her bedroom. How much paint 
does she have left? 



4-6 ] Metric Measurements 
Convert each measure. 

9. 17.3 kg to grams 10. 540 mL to liters 

12. 172 L to kiloliters 13. 0.36 km to meters 



1 1 . 0.46 cm to millimeters 
14. 54.4 mg to grams 

15. Cat ran in the 400-meter dash and the 800-meter run. Hilo ran in the 2-kilometer 
cross-country race. All together, who ran the farthest. Cat or Hilo? How much 
farther? 

16. Luis and Sara collected rainwater over three days. Luis collected 7.6 liters of 
rainwater, and Sara collected 7,060 milliliters. Who collected more rainwater, 
Luis or Sara? How much more? 



GT 



4-7 ] Dimensional Analysis 

17. A yellow jacket can tly 4.5 meters in 9 seconds. What is this rate in meters per 
minute? 

18. The average U.S. citizen throws avv'ay about 1,606 lb of trash each year. Find this 
rate in pounds per month, to the nearest tenth. 

19. One gallon is about 3.79 liters. A car has a 55-liter gas tank. What is the capacity of 
the tank in gallons, rounded to the nearest tenth of a gallon? 

20. A 1 -pound weight has a mass of about 0.45 kilogram. What is the mass in 
kilograms of a sculpture that weighs 570 pounds? Round your answer to the 
nearest tenth of a kilogram. 

21. A football player runs from his team's 12-yard line to his team's 36-yard line in 

6 seconds. Find the player's average speed in yards per second. Use dimensional 
analysis to check the reasonableness of your answer. 



244 Chapter 4 Proportional Relationships 




Focus on Problem Solving 






Solve 

• Choose an operation: multiplication or division 

When you are converting units, tliint; about wliether the number 
in the answer will be greater than or less than the number given 
in the question. This will help you decide whether to multiply or 
divide when changing the units. 




Tell whether you would multiply or divide by the conversion factor 
to solve each problem. Then solve the problem. 



Q A pontoon built to look like a duck was 
part of a 2007 project. The giant yellow 
duck floated the Loire River in France. Its 
dimensions were 26 x 20 x 32 meters. Find 
the dimensions of the duck in feet. Round to 
the nearest hundredth. (Hint: 1 m = 3.28 ft) 

The length of a rectangle is 8 cm, and its 
width is 5 cm less than its length. A larger 
rectangle with dimensions that are 
proportional to those of the first has a 
length of 24 cm. Wliat is the width of the 
larger rectangle in meters? 




Q One of the world's largest cheeseburgers 
was made in Thailand. The cheeseburger 
weighed 73.6 pounds. It was 23.5 inches in 
diameter and 13.75 inches in height. 

a. Find the weight of the cheeseburger in 
kilograms. {Hint: 1 lb = 2.2 kg) 

b. Find its dimensions in centimeters. 
{Hint: 1 in. = 2.54 cm) 

O Some of the ingredients for the cheeseburger 
are listed in the table. Find the missing 
measures. Round to the nearest hundredth, 
if necessan'. 



Cheeseburger 
Ingredients 


Size 


Beef 


25 kg = lb 


Mustard 


^J cups = mL 
(1 cup = 236.59 mL) 


Ketchup 


1 cup = fluid oz 



Focus on Problem Solving 245 



LASl^ Make Similar Figures 



Use with Lesson 4-8 



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^»** LabResourcesOnlmego.hrw.com, 

|BMMS10Lab4 gr 



Similar figures are figures that have the same shape but not 
necessarily the same size. You can make similar rectangles by 
increasing or decreasing both dimensions of a rectangle while keeping 
the ratios of the side lengths proportional. Modeling similar rectangles 
using square tiles can help you solve proportions. 



Activity 



A rectangle made of square tiles measures 5 tiles long and 2 tiles wide. 
What is the length of a similar rectangle whose width is 6 tiles? 



Use tiles to make a 5 x 2 rectangle. 



2- 
2- 
2- 







5 






r ^ 





















































































Add tiles to increase the width of the rectangle 
to 6 tiles. 

Notice that there are now 3 sets of 2 tiles along 
the width of the rectangle because 2x3 = 6. 



The width of the new rectangle is three times greater than the width of the 
original rectangle. To keep the ratios of the side measures proportional, the 
length must also be three times greater than the length of the original 
rectangle. 



/' Y Y ^ 























































































































































































5x3=15 

Add tiles to increase the length 
of the rectangle to 15 tiles. 



The length of the similar rectangle is 15 tiles. 



246 Chapter 4 Proportional Relationships 



i^»j>ti>ng:Tlj ff1ii— Jill illl I I i— |-T-~-^''>'"»*.""Tiw^'iMW^ -^WTi^--.->»t.-^-- 



To check your answer, you can use ratios. 



15 



































5 












































6 











































































































































2lA 

6 15 

3 3 



Write ratios using the corresponding side lengtlis. 
Simplify each ratio. 



Use square tiles to model similar figures with the given dimensions. 
Then find the missing dimension of each similar rectangle. 

a. The original rectangle is 4 tiles wide by 3 tiles long. 
The similar rectangle is 8 tiles wide by .v tiles long. 

b. The original rectangle is 8 tiles wide by 10 tiles long. 
The similar rectangle is .v tiles wide by 15 tiles long. 

c. The original rectangle is 3 tiles wide by 7 tiles long. 
The similar rectangle is 9 tiles wide by .v tiles long. 



Think and Discuss 



1. Sarah wants to increase the size of her rectangular backyard patio. Why 
must she change both dimensions of the patio to create a patio similar to 
the original? 

2. In a backyard, a rectangular plot of land that is 5 yd x 8 yd is used to grow tomatoes. 
The homeowner wants to decrease this plot to 4 yd x 6 yd. Will the new 

plot be similar to the original? Wliy or why not? 



Try This 



1 . A rectangle is 3 meters long and 1 1 meters wide. What is the width of a similar 
rectangle whose length is 9 meters? 

2. A rectangle is 6 feet long and 12 feet wide. What is the length of a similar 
rectangle whose width is 4 feet? 



Use square tiles to model similar rectangles to solve each proportion. 



3 4 _ 8 

=*■ 5 X 

7.1 = 1. 



4. 5^ A 



9_-P 
12 4 



_6_ 
18 



9. ^= 9 



15 



fi 1 - 4 



10. 



12 



4-8 Hands-On Lab 247 



Similar Figures and 
Proportions 



7.3.5 Identify, describe and construct similarity relationships and solve problems 
involving similarity by using proportional reasoning. 



Vocabulary 

similar 

corresponding sides 
corresponding angles 



Similar figures are figures that have the same shape but not 
necessarily the same size. The symbol ~ means "is similar to. 




Corresponding angles of two or more similar polygons are in the 
same relative position. Corresponding sides of two or more similar 
polygons are in the same relative position. When naming similar 
figures, list the corresponding angles in the same order. For the 
triangles above, AABC ~ ADEF. 



EXAMPLE 



/'•e=- 







A side of a figure 
can be named by its 
endpoints with a bar 
above, such as AB. 
Without the bar, the 
letters indicate the 
length of the side. 



J5HMyMldaGMBi& 



Two figures are similar if 

• the measures of their corresponding angles are equal. 

• the ratios of the lengths of their corresponding sides are 
proportional. 



Determining Whether Two Triangles Are Similar 

Tell whether the triangles are similar. 

The corresponding angles of the 5 

figures have equal measures. 



DE corresponds to QR. 
EF corresponds to RS. 
DF corresponds to QS. 



36 in. 




8 in 



106-^ 



DE 1 EF 1 DF 
QR RS QS 

7_L a_Li2 

21 24 36 

1 _ 1 _ 1 
3 3 3 



21 in. 



Write ratios using the corresponding sides. 
Substitute the lengths of the sides. 
Simplify each ratio. 



Since the measures of the corresponding angles are equal and the ratios 
of the corresponding sides are equivalent, the triangles are similar. 



248 Chapter 4 Proportional Relationships 



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Helpful Hint 



For more on similar 
triangles, see page 
5B20 in the Skills 
Bank. 



With triangles, if the corresponding side lengths are all proportional, 
then the corresponding angles /;?;(srhave equal measures. With 
figures that have four or more sides, if the corresponding side lengths 
are all proportional, then the corresponding angles may or may not 
have equal angle measures. 





5 cm 

T 





10 cm 




J 




L 


1 




r 



10 cm 



R 

5 cm 
S 



A 

4 cm 
D 



8 cm 



ABCD and QRST 
are similar. 



B 

4 cm 
C 




ABCD and WXYZ 
are not similar. 



10 cm 



10 cm 



5 cm 



EXAMPLE 



(B 



Determining Whether Two Four-Sided Figures Are Similar 

Tell whether the figures are similar. 

lOft ^ 135° 

/wVjl/v 




135" 90" 



90" 



1 ft 6 ft 

45 



90 
90" 



4 ft 



20 ft 



H 



L 8ft O 



The corresponding angles of the figures have equal measures. 
Write each set of corresponding sides as a ratio. 



FF — 

JIT. EF corresponds to LM. 



■^ FG corresponds to MN. 



GH 

NO 



GH corresponds to NO. 



EH 
LO 



EH corresponds to LO. 



Determine whether the ratios of the lengths of the corresponding 
sides are proportional. 



EF 1 FG 1 GH L EH 
LM MN NO LO 

15 ^ 10 ^ 10 
6 4 4 



20 
8 

5 _ 5 _ 5 



Write ratios using the corresponding sides. 
Substitute tfie lengths of the sides. 
Write the ratios in simplest form. 



Since the measures of the corresponding angles are equal and the 
ratios of the corresponding sides are equivalent, EFGH ~ LMNO. 



Think and Discuss 

1. Identify the corresponding angles of AJKL and AUTS. 

2. Explain whether all rectangles are similar. Give specific examples 
to justify your answer. 



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4-8 Similar Figures and Proportions 249 



4-8 



i-J 



(•Jiiiiii 

Homework Help Online go.hrw.com, 



keyword ■mbimebiM ® 
Exercises 1-8, 11, 23 



GUIDED PRACTICE 



See Example 1 Tell whether the triangles are similar. 

1. ,n ,fi 2. 

30° f 

,12 m ^ "^7 \/im 

^(104°\ 10^-/ ^^" 




R 38° 



l/K 44° 



15 in. 




2 m 



46° 



3 in.P\7 in. 


/ \28 in. 


120°^^..\ 22° 


7-/105° \ 
'^<^ Y31° 


5^ 


20m^-\,^ 



See Example 2 Tell whether the figures are similar. 



3. 


50 m 




/ic m 4. 


7 cm 




80 m 


90° 90° 
90° 90° 


80 m 








5 cm /l 40° 90° 
/ 40° 90° 




72 m 


90° 90° 
90° 90° 


72 m 


3.5 cm 


11 cm 

5c 


11 cm 


m/l40° 90° 
/40° 90° 




50 m 






45 m 




15 cm 



3.5 cm 



INDEPENDENT PRACTICE 



See Example 1 Tell whether the triangles are similar. 

5. aQ ,.„ 6. 

18cm/ \l8cm 
70°-/ V70° 



12 cm 




56° D 



28 cm 




56° 



40 in. 36 in 

41° 




30 in. 



24 in. 
-83° 



See Example 2 Tell whether the figures are similar. 

7. 14ft ^^^ 8. 



14ft 



90° 90^ 
90° 90' 



14ft 





23 ft 


14ft 


90° 90° 


23 ft 






90° 90° 



3 m 140° 



23 ft 



4 m 



23 ft 



^ /1 20° 60° 
2 "Y60_J120°/ 2 

4 m 




140° 3 m 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EP13. 



9. Tell whether the parallelogram and 
trapezoid could be similar. Explain your 

answer. 



/1 20° 
/60° 



60°y 



120°/ 



/1 20° 120°\ 
■4o° 60° 



250 Chapter 4 Proportional Relationships 



10. Kia wants similar prints in small and large sizes of a favorite photo. The 
photo lab sells prints in these sizes: 3 in. x 5 in., 4 in. x 6 in., 8 in. x 18 in., 
9 in. X 20 in., and 16 in. x 24 in. Wliich could she order to get similar prints? 



Tell whether the triangles are similar. 
11. , 36 12. 




C 28 m D 



32 m 




34 24 m 



96° 



35° 



49° IX 
H 



42 m 



12ft 



A 


5 ft fi 


Ah 

c 


D 



® 



e 12ft c 
The figure shows a 12 ft by 15 ft rectangle divided into four 
rectangular parts. Explain whether the rectangles in each 
pair are similar. 

13. rectangle .4 and the original rectangle 

14. rectangle C and rectangle B 

1 5. the original rectangle and rectangle D 

Critical Thinking For Exercises 16-19, justify your answers using words or 
drawings. 

16. Are all squares similar? 17. Are all parallelograms similar? 
18. Are all rectangles similar? 19. Are all right triangles similar? 

20. Choose a Strategy What number gives the same result when 
multiplied by 6 as it does when 6 is added to it? 

21. Write About It Tell how to decide whether two figures are similar. 

' 22. Challenge Two triangles are similar. The ratio of the lengths of the 
corresponding sides is |. The length of one side of the larger triangle is 
40 feet. What is the length of the corresponding side of the smaller triangle? 



15 ft 



i 



Test Prep and Spiral Review 



23. Multiple Choice Luis wants to make a deck that is similar to one that is 10 feet 
long and 8 feet wide. Luis's deck must be 18 feet long. What must its width be? 



<S) 20 feet 



CEy 16 feet 



(c:> 14.4 feet 



CD) 22.5 feet 



24. Short Response A real dollar bill measures 2.61 inches by 6.14 inches. A play 
dollar bill measures 3.61 inches by 7.14 inches. Is the play money similar to the 
real money? Explain your answer. 



Multiply. Write each answer in simplest form. (Lesson 3-9) 



25. - 



3 



14 



26. 



(-5) 



4 "■ "8 

28. Tell whether 5:3 or 12:7 is a greater ratio. (Lesson 4-1) 



27. 



4 ^8 ^5 



4-8 Similar Figures and Proportions 251 




Using Similar 
Figures 



J3 



7.3.5 Identify, describe and construct similarity relationships and solve problems 
involving similarity by using proportional reasoning. 

Native Americans of the Nortliwest 
carved totem poles out of tree trunks. 
These poles could stand up to 80 feet 
tall. Totem poles include carvings of 
animal figures, such as bears and 
eagles, which symbolize traits of the 
family or clan who built them. 



Vocabulary 

indirect 
measurement 




Measuring the heights of tall 
objects, like some totem poles, 
cannot be done by using a ruler 
or yardstick. Instead, you can use 
indirect measiireine)it. 

Interactivities Online ^ Indirect measurement is a method of using proportions to find an 
unknown length or distance in similar figures. 

Finding Unknown Measures in Similar Figures 
AABC ~ AJKL. Find the unknown measures. 

K 



EXAMPLE 1 



B ]03 
8 cm /^12 cm 



48^^^^"^ 29 



28 cm 




Step 1 Find .v. 

AB_BC 
IK KL 

3. - lA 

28 X 
8 • .V = 28 • 12 

8.V = 336 

8a; _ 336 
8 8 

A- = 42 
AX is 42 centimeters. 

Step 2 Find y. 

Z A' corresponds to ^B. 

V= 103° 



Write a proportion using corresponding sides. 

Substitute ttte lengtiis of tiie sides. 

Find ttie cross products. 

IVIultiply. 

Divide eacli side by 8. 



Corresponding angles of similar 
triangles tiave equal angle measures. 



252 Chapter 4 Proportional Relationships 



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EXAMPLE [2] Measurement Application 



A volleyball court is a rectangle that is similar in shape to an 
Olympic-sized pool. Find the width of the pool. 



T 

9 m 

i 



H 



18m H 



50 m- 



Let ((' = the width of the pool. 

ig _ g Write a proportion using corresponding side 

lengths. 

Find the cross products. 



50 "' 
18- ;('= 50 -9 



18»' = 450 

18»' _ 450 
18 18 

w = 25 



Multiply. 

Divide each side by 18. 



The pool is 25 meters wide. 



EXAMPLE 



[3 



Estimating with Indirect Measurement 



Estimate the height of the totem pole 
shown at right. 



/( _ 15.5 
5 3.75 

h IK 




Write a proportion. 
Use compatible 


5 4 




numbers to estimate. 


^«4 

3 




Simplify. 


5-4«5-4 

3 




Multiply each side by 5 


/;«20 






The totem pt 


)le 


is about 20 feet tall. 




K3.75ft>l 



Think and Discuss 

1. Write another proportion that could be used to find the value of .v 
in Example 1. 

2. Name two objects that it would make sense to measure using 
indirect measurement. 



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4-9 Using Similar Figures 253 



m^!i^W^iiMli^j^ltM4'i^Mt}i^tM^ 




,ii^73333 




b-> 



Homework Help Online go.hrw.com, 



keyword HSQESB ® 
Exercises 1-8, 9, IS 



GUIDED PRACTICE 



See Example 1 AXYZ— APQR in each pair. Find the unknown measures. 



40 

'^<C/9cm 
8 cm ,' „ 

^89 p 



20 cm 




30 cm 




Q58° 

30 m / \40 m 



11 



45° 



35 m 



See Example 2 3. The rectangular gardens at right 
are similar in shape. How wide is 
the smaller garden? 



See Example 3 4. A water tower casts a shadow that 
is 21 ft long. A tree casts a shadow 
that is 8 ft long. Estimate the 
height of the water tower. 



42 ft 




K-Sft-H 



INDEPENDENT PRACTICE 



See Example 1 AABC~ ADEFin each pair. Find the unknown measures 
5 S 6. ^ 87° 



12 in. 



9 in. 



£84° 




40° 



^ 14 in. ^ 



See Example 2 7. The movie still and its 

projected image at right are 
similar. What is the height of 
the projected image to the 
nearest hundredth of an inch? 

See Example 3 8. A cactus casts a shadow that is 
14 ft 7 in. long. A gate nearby 
casts a shadow that is 5 ft long. 
Estimate the height of the cactus. 



/ \l2.96ft S^ 

f 4^^ \ 8ft 

'^^^^^ V-29° 
64 14.4ft^^^C\ 





254 Chapter 4 Proportional Relationships 



PRACTICE AND PROBLEM SOLVING 



Extra Practice 



See page EP13. 



9. A building with a height of 14 m casts a sliadovv tliat is 16 m long while a 
taller building casts a 24 m long shadow. What is the height of the taller 
building? 



10. Two common envelope sizes are 3^^ in, 
11. 



Are these envelopes similar? Explain. 



X 6| in. and 4 in. 



X 9^ in. 



Art An art class has painted 
a mural composed of brightly 
colored geometric shapes. All of 
the right triangles in the design 
are similar to the red right 
triangle. Find the heights of the 
three other right triangles in the 
mural. Round your answers to 
the nearest tenth. 



12. Write a Problem 
©13 




Write a 
problem that can be solved using indirect measurement. 



Write About It Assume you know the side 
lengths of one triangle and the length of one side 
of a second similar triangle. Explain how to use 
the properties of similar figures to find the 
unknown lengths in the second triangle. 

^ 14. Challenge AABE- AACD. What is the value 
of y in the diagram? 



-fy 



D (8, 5) 




£ 



Test Prep and Spiral Review 



15. Multiple Choice Find the unknown length in the similar figures. 
CS) 10 cm CC) 15 cm 



CD 12 cm 



CE' 18 cm 



15 cm 





11.25 cm 



9 cm 



16. Gridded Response A building casts a 16-foot shadow. A 6-foot man 
standing next to the building casts a 2.5-foot shadow. What is the height, 
in feet, of the building? 



Write each phrase as an algebraic expression. (Lesson 1-7) 

17. the product of 18 and I' 18. 5 less than a number 



19. 12 divided by; 



Choose the most appropriate customary unit for each measurement. 
Justify your answer. (Lesson 4 5) 

20. weight of a cell phone 21 . height of a cat 22. capacity of a gas tank 



4-9 Using Similar Figures 255 



\1 



4-1 



m Scale Drawings and 
r Scale Models 



7.3.5 Identify, describe and construct similarity relationships and solve problems 
involving similarity by using proportional reasoning. 

The drawing at right shows a scale 
drawing of the Guggenheim Museum 
in New York. A scale drawing is a 
proportional two-dimensional 
drawing of an object. Its dimensions 
are related to the dimensions of 
the actual object by a ratio called 
the scale factor . For example, if a 
drawing of a building has a scale 
factor of ^, this means that 
each dimension of the drawing is 
^ of the corresponding dimension 
of the actual building. 



Vocabulary 

scale drawing 
scale factor 
scale model 
scale 



Interactivities Onli ne ► 




A scale model is a proportional 
three-dimensional model of an object. 
A scale is the ratio between two sets 
of measurements. Scales can use the 
same units or different units. Both scale 
drawings and scale models can be smaller 
or larger than the objects they represent. 



EXAMPLE [T] Finding a Scale Factor 

Identify the scale factor. 



Caution! 



7///// 

A scale factor is always 
the ratio of the 
model's dimensions to 
the actual object's 
dimensions. 





Race Car 


Model 


Length (In.) 


132 


11 


Height (In.) 


66 


5.5 



You can use the lengths or heights to find the scale factor, 
model length _ 1 1 _ 1 



race car length 132 1 
model height _ 5.5 _ 1 
66 ~ ~ 



Write a ratio. Ttien simplify. 



12 



race car height 

The scale factor is pj. This is reasonable because -j^ the length of the 
race car is 13.2 in. The length of the model is 11 in., which is less than 
13.2 in., and pj is less than j^. 



256 Chapter 4 Proportional Relationships 



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EXAMPLE r2J Using Scale Factors to Find Unknown Lengths 



A photograph of Rene Magritte's 

painting The Schoolmaster has dimensions 

5.4 cm and 4 cm. The scale factor is ^j^. 

Find the size of the actual painting. 

T^, . , photo 1 

Think: -^ — -. — = ^ 
painting 15 



5.4 _ 1 
f 15 

^ = 5.4 • 15 
f = 81 cm 



Write a proportion to 
find tlie length i . 

Find tlie cross products. 
Multiply. 




w 15 



Write a proportion to find the width w. 

W — 4 • 15 Find the cross products, 

w - 60 cm Multiply. 

The painting is 81 cm long and 60 cm wide. 



EXAMPLE 



[3 



Measurement Application 

On a map of Florida, the distance between Hialeah and Tampa is 
10.5 cm. The map scale is 3 cm:128 km. What is the actual distance 
d between these two cities? 



actual distance 


128 


J 3 _ 10.5 
1 128 d 


Write a proportion. 


3-rf= 128- 10.5 


Find the cross products. 


2d^ 1,344 




3d 1,344 
3 3 


Divide both sides by 3. 


rj = 448 km 





The distance between the cities is 448 1cm. 



Think and Discuss 

1. Explain how you can tell whether a model with a scale factor of | 
is larger or smaller than the original object. 

2. Describe how to find the scale factor if an antenna is 60 feet long 
and a scale drawing shows the length as 1 foot long. 



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4-70 Scale Drawings and Scale Models 257 



;'-iit'.:.«jiivit:>aviiEWti;^Hbi^-»itf*uirtijtria*vrit^^ 





See Example 1 



GUIDED PRACTICE 



Identify the scale factor. 
1. 



Grizzly Bear 


Model 


Height (in.) 84 


6 



Moray Eel 


Model 


Length (ft) 5 


H 



See Example 
See Example 



2. 



3. In a photograph, a sculpture is 4.2 cin tall and 2.5 cm wide. The scale 
factor is -j^. Find the size of the actual sculpture. 

4. Ms. lackson is driving from South Bend to Indianapolis. She measures a 
distance of 4.3 cm between the cities on her Indiana road map. 

The map scale is 1 cm:48 km. What is the actual distance between 
these two cities? 



INDEPENDENT PRACTICE 





Eagle 


Model 


Wingspan (in.) 


90 


6 



See Example 1 Identify the scale factor. 
5. 

See Example 2 
See Example 3 



Dolphin 


Model 


Length (cm) 


260 


13 



7. On a scale drawing, a tree is 6^ inches tall. The scale factor is :^. Find the 
height of the actual tree. 

8. Measurement On a road map of Virginia, the distance from Alexandria 
to Roanoke is 7.6 cm. The map scale is 2 cm:80 km. What is the actual 
distance between these two cities? 



PRACTICE AND PROBLEM SOLVING 



Extra Practice 



See page EP13. 



The scale factor of each model is 1:12. Find the missing dimensions. 



10. 



11. 



Item 


Actual Dimensions 


Model Dimensions 


Lamp 


Height: 


Height: 1^ in. 


Couch 


Height: 32 in. 
Length: 69 in. 


Height: 
Length: 


Table 


Height: 
Width: 
Length: 


Height: 6.25 cm 
Width: 11.75 cm 
Length: 20 cm 



12. An artist transferred a rectangular design 13 cm long and 6 cm wide to a 
similar canvas 260 cm long and 120 cm wide. What is the scale factor? 

13. Critical Thinking A countertop is 18 ft long. How long is it on a scale 
drawing with the scale 1 in:3 yd? 

14. Write About It A scale for a scale drawing is 10 cm:l mm. Which will 
be larger, the actual object or the scale drawing? Explain. 



258 Chapter 4 Proportional Relationships 



Use the map for Exercises 15-16. 

15. In 1863, Confederate troops 
marched from Chambersburg 
to Gettysburg in search of 
badly needed shoes. Use the 
ruler and the scale of the 
map to estimate how far the 
Confederate soldiers, many 
of whom were barefoot, 
marched. 

16. Before the Civil War, the Mason-Drxon Line was considered the 
dividing line between the North and the South. Gettysburg is 
about 8.1 miles north of the Mason-Dixon Line. How far apart in 
inches are Gettysburg and the Mason-Dixon Line on the map? 



17. 



18. 




Mason 



Multi-Step Toby is making a scale model of the battlefield 
at Fredericksburg. The area he wants to model measures about 
11 mi by 7.5 mi. He plans to put the model on a 3.25 ft by 3.25 ft 
square table. On each side of the model he wants to leave at 
least 3 in. between the model and tlie table edges. What is the 
largest scale he can use? 



^a Challenge A map of Vicksburg, Mississippi, has a 

scale of "1 mile to the inch." The map has been reduced so 

that 5 inches on the original map appears as 1.5 inches on 

the reduced map. The distance between two points on the President Abraham Lincoln, Major 

reduced map is 1 .75 inches. What is die actual distance in miles? a"'^" Pmkerton, and General John a. 

^ McCleland, October 1862. 




i 



Test Prep and Spiral Review 



19. Multiple Choice On a scale model with a scale of ^, the height of a shed 
is 7 inches. What is the approximate height of the actual shed? 



Ca:. 2 feet 



CD 9 feet 



'X) 58 feet 



CSj 112 feet 



20. Gridded Response On a map, the scale is 3 centimeters: 120 kilometers. The 
distance between two cities on the map is 6.8 centimeters. What is the distance 
between the actual cities in kilometers? 



Order the numbers from least to greatest. (Lesson 2-11) 

21. |, 0.41,0.054 



22. |, 0.2, 



1.2 



23. 0.7 



7 7 



24. 0.3, -^, 0.32 



9' 11 ' 6' 

Divide. Estimate to check whether each answer is reasonable. (Lesson 3-4) 
25. 0.32 ^ 5 26. 78.57 -^ 9 27. 40.5 ^ 15 28. 29.68 ^ 28 



4-70 5ca/e Drawings and Scale Models 259 



LABF and Models 



Use with Lesson 4-10 



<•** Lat 



Learn It Online 

Lab Resources Online go.hrw.com 



lff!B|y^S10Lab4 



Scale drawings and scale models are used in mapmaking, construction, 
and other trades. You can create scale drawings and models using 
graph paper. If you measure carefully and convert your measurements 
correctly, your scale drawings and models will be similar to the actual 
objects they represent. 



Activity 1 



Make a scale drawing of a classroom and items with the following 
dimensions. 



Classroom 


6 Student Desks 


Teacher's Desk 


Aquarium 


1 2 ft X 20 ft 


2 ft X 3 ft 


2 ft X 6 ft 


5 ft X 2 ft 



You can use graph paper for your drawing. When making a scale drawing, 
you can use any scale you wish. For this activity, use a scale in which 2 squares 
represent 1 foot. To convert each measurement, multiply the number of feet by 2. 

This means that the room measures 24 squares (2 • 12 ft) by 40 squares (2 • 20 ft). 
Convert the other measurements in the table using the same scale. 



Classroom 


6 Student Desks 


Teacher's Desk 


Aquarium 


24 sq X 40 sq 


4 sq X 6 sq 


4 sq X 12 sq 


1 sq X 4 sq 



Q Now sketch the room and items on graph paper. Place the items anywhere in 
the room you wish. 



Thinic and Discuss 



1. Write ratios to compare the widths and lengths of the actual classroom and 
the drawing. Can you make a proportion with your ratios? Explain. 

2. Describe how your drawing would change if you used a scale in which 
1 square represents 2 feet. 



Try This 



1. Measure the dimensions of your classroom as well as some items in the room. 
Then make a scale drawing. Explain the scale you used. 



260 Chapter 4 Proportional Relationships 



Activity 2 



Make a scale model of a school gym whose floor is 20 meters x 32 meters 
and whose walls are 12 meters tall. 

O You can use graph paper for your model. For this activity, use a scale in 
which 1 square represents 2 meters. To convert each measurement, divide 
the number of meters by 2. 

Q The two longer sides of the gym floor are 16 squares (32 m -^ 2). 

The other two sides are 10 squares (20 m -^ 2). The walls are 6 squares 
(12 m -=- 2) tall. 





Floor Length 


Floor Width 


Wall Height 


Actual 


20 m 


32 m 


12 m 


Model 


10 squares 


16 squares 


6 squares 



Q Sketch the walls on graph paper as shown. Then cut them out and tape 
them together to make an open rectangular box to represent the gym. 






.-::r 



Thinlc and Discuss 



1. A different gym has a floor that is 120 feet x 75 feet and a height of 45 feet. 
A model of the gym has a height of 9 squares. What are the dimensions of 
the model's floor? What scale was used to create this model? 



Try This 



1. Make a scale model of the building shown. 
Explain the scale you used to create your model. 



24 m 




4-70 Hands-On Lab 261 



CHAPTER 




Ready To Go On? 



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IB*!^] MS1(im(i04C tGo| 



SECTION 4C 



Quiz for Lessons 4-8 Through 4-10 

^) 4-8 ] Similar Figures and Proportions 

1 . Tell whether the triangles are similar. 




35 cm 



10 cm 



2. Tell whether the figures are similar. 
84 ft 



\53^ 

25 ft 



14ft 



127° 



'25 ft \53° 

5 ft \ 127° 



537 
127°/ 5 ft 



48 ft 



8ft 



Qf) 4-9 ] Using Similar Figures 



AABC ~ AXYZ in each pair. Find the unknown measures. 
^- ^ 25.5° 



10m 




13 m 



X 15 



m 



32.5 m 

64.5° 

Z 




36 in. 



5. Reynaldo drew a rectangular design that was 6 in. wide and 8 in. long. He used a 
copy machine to enlarge the rectangular design so that the width was 10 in. 
What was the length of the enlarged design? 

6. Redon is 6 ft 2 in. tall, and his shadow is 4 ft 1 in. long. At the same time, a 
building casts a shadow that is 19 ft 10 in. long. Estimate the height of the 
building. 



Q} 4-10] Scale Drawings and Scale Models 



7. An actor is 6 ft tall. On a billboard for a new movie, the actor's picture is 
enlarged so that his height is 16.8 ft. What is the scale factor? 

8. On a scale drawing, a driveway is 6 in. long. The scale factor is -^. Find the 
length of the actual driveway. 

9. A map of Texas has a scale of 1 in:65 mi. The distance from Dallas to 

San Antonio is 260 mi. What is the distance in inches between these two cities 
on the map? 



262 Chapter 4 Proportional Relationships 




C N N E C T I 



CHAPTER 



4 



SBSv 



Paul BunyaH Statues According to legend, Paul 
Bunyan was a giant lumberjack whose footsteps created 
Minnesota's ten thousand lakes. Statues honoring this 
mythical figure can be found throughout the state. One of 
the largest, in Brainerd, stands 26 feet tall and can greet 
you by name! 

1. A tourist who is 1.8 m tall stands next to the statue 
of Paul Bunyan in Bemidji, MN. He measures 
the length of his shadow and the shadow cast by 
the statue. The measurements are shown in the 
figure. What is the height of the statue? 

2. Show how to use dimensional analysis to convert 
the height of the statue to feet. Round to the 
nearest foot. {Hint. 1 m - 3.28 ft) 

3. The Bemidji statue includes Paul Bunyan's 
companion, Babe, the Blue Ox. The statue's horns 
are 14 feet across. The statue was made using 
the dimensions of an actual ox and a scale of 3 : 1. 
What was the length of the horns of the actual ox? 

4. The kneeling Paul Bunyan statue in Akeley, MN, is 25 feet tall. 
The ratio of the statue's height to its width is 17:11. What is the 
width of the statue to the nearest tenth of a foot? 

5. A souvenir of the Akeley statue is made using the 
scale 2 in:5 ft. Wliat is the height of the souvenir? 



MINNESOTA 





\ 1.8 m 




1.35 m 0.45 m 




SaMejiMe 



Water Works 



You have three glasses: a 3-ounce glass, a 5-ounce glass, 
and an 8-ounce glass. The 8-ounce glass is full of water, 
and the other two glasses are empty. By pouring water 
from one glass to another, how can you get exactly 6 
ounces of water in one of the glasses? The step-by-step 
solution is described below. 

Pour the water from the 8 oz glass into the 5 oz glass. 
Q Pour the water from the 5 oz glass into the 3 oz glass. 
Q Pour the water from the 3 oz glass into the 8 oz glass. 

You now have 6 ounces of water in the 8-ounce glass. 

Start again, but this time try to get exactly 4 ounces of 
water in one glass. {Hint: Find a way to get 1 otmce of 
water. Start by pouring water into the 3-ounce glass.) 



Next, using 3-ounce, 8-ounce, and 1 1 -ounce glasses, try to get 
exacdy 9 ounces of water in one glass. Start widi the 11 -ounce 
glass full of water. {Hi)it: Start by pouring water into the 8-ounce glass.] 

Look at the sizes of the glasses in each problem. The volume of 
the third glass is the sum of the volumes of the first two glasses: 
3 + 5 = 8 and 3 + 8 = 11. Using any amounts for the two smaller 
glasses, and starting with the largest glass full, you can get any 
multiple of the smaller glass's volume. Try it and see. 




Concentration 



Each card in a deck of cards has a ratio on one 
side. Place each card face down. Each player 
or team takes a turn flipping over two cards. 
If the ratios on the cards are equivalent, the 
player or team can keep the pair, if not, the 
next player or team flips two cards. After every card 
has been turned over, the player 
or team with the most pairs wins. 

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264 Chapter 4 Proportional Relationships 




Materials 

• 2 paper plates 

• scissors 

• markers 



\% %v 




PROJECT 



Paper Plate 
Proportions 



Serve up some proportions on this book made from 
paper plates. 

Q Fold one of the paper plates in half. Cut out a 
narrow rectangle along the folded edge. The 
rectangle should be as long as the diameter of 
plate's inner circle. When you open the plate, you 
will have a narrow window in the center. Figure A 

Q Fold the second paper plate in half and then 
unfold it. Cut slits on both sides of the crease 
beginning from the edge of the plate to the inner 
circle. Figure B 

O Roll up the plate with the slits so that the two slits 
touch each other. Then slide this plate into the 
narrow window in the other plate. Figure C 

O Wlien the rolled-up plate is halfway through the 
window, unroll it so that the slits fit on the sides of 
the window. Figure D 

Close the book so that all the plates are folded 
in half. 




Taking Note of the Math 

Write the number and name of the 
chapter on the cover of the book. 
Then review the chapter, using the 
inside pages to take notes on ratios, 
rates, proportions, and similar figures. 



• RatuH, QAt UQAd to SolLf ; i 





'H: 




A RftTio OimpfiHES 

1 NunMbtKS ft CBTfc It 
THftT HPuE OlFfe^EJlT 




flu. ", 
Dtnor. 
THE n 
RflV &£ 

Pat IN fl 



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±Tfi\r\ 



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It's in the Bag! 



265 



CHAPTER 



4 




Vocabulary 

corresponding angles 248 

corresponding sides 248 

cross product 226 

equivalent ratios 222 

indirect measurement 252 



proportion 222 

rate 2I8 

ratio 214 

scale 256 

scale drawing 256 



scale factor 256 

scale model 256 

similar 248 

unit conversion factor 240 

unit rate 21 8 



Complete the sentences below with vocabulary words from the list above. 

1 . ? figures have the same shape but not necessarily the same size. 

2. A(n) ? is a comparison of two numbers, and acn) ? is a ratio 
that compares two quantities measured in different units. 

3. The ratio used to enlarge or reduce similar figures is a(n) __]___. 



EXAMPLES 

4-l3 Ratios (pp. 214-217) 

ij ■ Write the ratio of 2 servings of bread to 
4 servings of vegetables in all three forms. 
Write your answers in simplest form. 

I = i Write the ratio 2 to 4 in 

simplest form. 
\, 1 to 2, 1:2 



EXERCISES 



There are 3 red, 7 blue, and 5 yellow balloons. 

4. Write the ratio of blue balloons to total 
balloons in all three forms. Write your 
answer in simplest form. 

5. Tell whether the ratio of red to blue 
balloons or the ratio of yellow balloons 
to total balloons is greater. 



4-2 j Rates (pp. 21 8-221) 

■ Find each unit price. Then decide which 
I has the lowest price per ounce. 



S2.70 
5 oz 


or ?f 2 
12 oz 


$2.70 


_ $0.54 



5 oz 



and 



$4.32 _ $0.36 
12 oz oz 



Since 0.36 < 0.54, |^ has the lowest 



12 oz 



price per ounce. 



Find each unit rate. 

6. 540 ft in 90s 



7. 436 mi in 4 hr 



Find each unit price. Then decide which 
is the better buy. 



8. 
10. 



$56 



or 



$32.05 



$160 $315 



25 gal"' 15 gal " 5g 9g 

Beatriz earned $197.50 for 25 hours of 
work. How much money did she earn 
per hour? 



266 Chapter 4 Proportional Relationships 



EXAMPLES 



EXERCISES 



4-3] Identifying and Writing Proportions (pp 222-225) 



Determine if ^ and - are proportional. 



_5_ 
12 

3 _ 1 
9 3 

12 ^ 3 



72 



is already in simplest form. 



Simplify 



3 



The ratios are not proportional. 



Determine if tlie ratios are proportional. 

15 20 13 21 18 

25' 30 14' 12 



11. 



12. 



Find a ratio equivalent to the given ratio. 
Then use the ratios to write a proportion. 



14. 



12 



15. 



45 
50 



16. 



15 



4-4 j Solving Proportions (pp 226 229) 
■ Use cross products to solve k = 21 • 



Use cross products to solve each proportion. 



P _ \0 
8 12 






17. 


4 _ 11 
6 3 


18 


p- 12 = 8- 10 
Up = 80 




Multiply the cross 
products. 


19. 


b _8 
1.5 3 


20 


12/^ _ 80 
12 12 




Divide each side by 12. 


21. 


2 _ 1 

y 5 


22 


20 
p = f . or 


«i 











n 


_ 

■ 15 


16 
11 


_ 96 

.V 


/ 
T 


_ 70 



4-5] Customary Measurements (pp 232-235) 

■ Convert 5 mi to feet. Convert each measure. 

feet 



> 5.280 _ X 

miles — *■ 1 5 

X = 5,280 • 5 = 26,400 ft 



23. 32floztopt 24. 1.5 T to lb 

25. Manda has 4 yards of fabric. She cuts 

off 29 inclies. What is the length of the 

remaining fabric? 



4-6] Metric Measurements (pp. 236-239) 
■ Convert 63 m to centimeters. 

100 cm = 1 m 



L 



63 m = (63 X 100) cm 
= 6,300 cm 



Convert each measure. 



26. 18LtomL 
28. 5.3 km to m 



27. 720 mg to g 
29. 0.6 cm to mm 



4-7 j Dimensional Analysis (pp. 240-243) 

■ Amil can run 12 kilometers in 1 hour. How 
I many meters can he run at this pace in 
I 1 minute? 

1,000 m 



km to m: = 



1 kni 



h to min: = 



ih 
60 mill 



2 1trfi . 1,000 m ^ l>f ^ 12- 1,000 m ^ 200 m 
\M 1 iyl"! 60 min 60 min 1 min 



Use conversion factors to find each rate. 

30. 162lb/yrtolb/mo 

31. 1,232 ft/min to mi/h 

32. While driving, Abby passed mile 
marker 130 at 3:10 p.m. and mile 
marker 170 at 4:00 p.m. Find Abby's 
average speed in miles per minute. 



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Study Guide: Review 267 



EXAMPLES 



EXERCISES 



4-8 ] Similar Figures and Proportions (pp. 248-251 



Tell whether the figures are similar. 

The corresponding angles of the figures 
have equal measures. 



5.3.5. 


3 


30 18 30" 


18 


1_ i_ 1 . 


_ 1 



3 cm 



130° 5 cm 50° 
130°/ 3 cm 



5 cm 



30 cm 



6 6 6 6 
The ratios of the 
corresponding sides .,g ^^ /m 
are equivalent. The /50° 1 30° 
figures are similar. 30 cm 



Tell whether the figures are similar. 

33. 8ft 48 ft 

\46° 54°7 \45 

6ft\ /6 ft 6 ft , 

\150° y \150 

3 ft 110° 



18 cm 





4-9] Using Similar Figures (pp 252-255) 

■ A ABC ~ ALMN. Find the unknown 
measures. 



AB _ AC 

LM LN 

8 _ 11 

r 44 

8 • 44 = f • 11 

352 = llr 

352 _ Uf 
11 11 

32 in. = r 

A Af corresponds to /.C. 
X = 46° 




44 in. 



AJKL ~ ADEF. Find the unknown 
measures. 



35. 



18ft, 
J 



K 

a -• 



18 ft 
L 




36. 



37. 



25 ft 



A rectangular photo frame is 24 cm 
long and 9 cm wide. A frame that is 
similar in shape is 3 cm wide. Find the 
length of the frame. 

A tree casts a 30^ ft shadow at the 
time of day when a 2 ft stake casts a 
7| ft shadow. Estimate the height of 
the tree. 



4-10] Scale Drawings and Scale Models (pp. 256-259) 



A model boat is 4 Inches long. The scale 38. 
factor is ^. How long is the actual boat? 



39. 



model _ 1 
boat 24 




4 _ 1 

n 24 


Write a proportion. 


4 • 24 = 7Z • 1 


Find the cross products. 


96 = « 


Solvp. 


The boat is 96 inches long. 



The Wright brothers' Flyer had a 
484-inch wingspan. Carla bought a 
model of the plane with a scale factor 
of ^. What is the model's wingspan? 

The distance from Austin to Houston 
on a map is 4.3 inches. The map scale 
is 1 inch:38 miles. Wliat is the actual 
distance? 



268 Chapter 4 Proportional Relationships 



Chapter Test 



1. Stan found 12 pennies, 15 nickels, 7 dimes, and 5 quarters. Tell whether 
the ratio of pennies to quarters or the ratio of nickels to dimes is greater. 

2. Lenny sold 576 tacos in 48 hours. What was Lenny's average rate of taco sales? 

3. A store sells a 5 lb box of detergent for $5.25 and a 10 lb box of detergent 
for $9.75. Which size box has the lowest price per pound? 

Find a ratio equivalent to each ratio. Then use the ratios to write a proportion. 




4 ^ - 5 - 

30 ,^ 9 

Use cross products to solve each proportion. 



6. 



54 



Q_9__/H Ql = i8 in3_2i 

o. j2 6 2 6 7 t 



11. 



10 



5 _ 10 



12. A certain salsa is made with 6 parts tomato and 2 parts bell pepper. To correctly make 
the recipe, how many cups of tomato should be combined with 1.5 cups of bell pepper? 



15. 6.12 km to m 
18. 4.25L/htomL/h 



Convert each measure or rate. 

13. 13,200 ft to mi 14. 3.5 lb to oz 

16. 57LtokL 17. 828 Ib/yr to lb/mo 

19. Some world-class race walkers can walk 9 miles per hour. What is this rate in feet per 
minute? 

20. One pound is about 2.2 kilograms, lefferson's dog weighs 40 lb. Wliat is the mass of his 
dog in kilograms? 



Tell whether the figures are similar. 

F 



21. 99 C 

9ft>r\5ft 





22 ft 

A WyZ~ AMNO in each pair. Find the unknown measures 
23. M^ 24. Y, 



24 m 



101^ 

6crn A\J0.8cm 

85 Q V 
125 





62 



N 



J1 m 



- 44^- 



74 f^ :0 m 



25. A scale model of a building is 8 in. by 12 in. The scale is 1 in: 15 ft. Wliat 
are the dimensions of the actual building? 

26. The distance from Portland to Seaside is 75 mi. What is the distance in inches 
between the two towns on a map whose scale is l| in:25 mi? 



Chapter 4 Test 269 



CHAPTER 



4 



ra ISTEP+ 
^ Test Prep 



4r?, 



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Applied Skills Assessment 

Constructed Response 

1. Jana begati the month with $102.50 
in her checking account. During the 
month, she deposited $8.50 that she 
earned from baby-sitting, withdrew 
$9.75 to buy a CD, deposited $5.00 that 
her aunt gave her, and withdrew $6.50 
for a movie ticket. Using compatible 
numbers, write and evaluate an 
expression to estimate the balance in 
Jana's account at the end of the month. 

2. A lamppost casts a shadow that is 
18 feet long. At the same time of 
day, Alyce casts a shadow that is 4.2 
feet long. Alyce is 5.3 feet tail. Draw 
a picture of the situation. Set up and 
solve a proportion to find the height 
of the lamppost to the nearest foot. 
Show your work. 

Extended Response 

3. Riley is drawing a map of the state 
of Virginia. From east to west, the 
greatest distance across the state is 
about 430 miles. From north to south, 
the greatest distance is about 

200 miles. 

a. Riley is using a map scale of 1 inch: 
24 miles. Find the length of the map 
from east to west and the length 
from north to south. Round your 
answers to the nearest tenth. 

b. The length between two cities on 
Riley's map is 9 inches. What is the 
actual distance between the cities in 
miles? 



c. About how many minutes will it 
take for an airplane traveling at a 
speed of 520 miles per hour to fly 
from east to west across the widest 
part of Virginia? Show your work. 

Multiple-Choice Assessment 

4. What is the unknown length b in 
similar triangles ABC and DEFl 

D 




18.4 ft 



A. 7.2 feet 

B. 6 feet 



C. 4 feet 

D. 5.6 feet 



5. The total length of the Golden Gate 
Bridge in San Francisco, California, is 
8,981 feet. A car is traveling at a 
speed of 45 miles per hour. How many 
minutes will it take the car to cross the 
bridge? 

A. 0.04 minute C. 1.7 minutes 

B. 1.28 minutes D. 2.27 minutes 

6. For which equation is x = ~ the 
solution? 

A. 5x - ^ = 



B. -Ix + ^ = 



C. ^x - 2 = 



D. -5x + J- = 



270 Chapter 4 Proportional Relationships 



7. A hot air balloon descends 38.5 meters 
In 22 seconds. If the balloon continues 
to descend at this rate, how long will it 
take to descend 125 meters? 



A. 25.25 seconds 

B. 86.5 seconds 



C. 71.43 seconds 

D. 218.75 seconds 



8. Which value completes the table of 
equivalent ratios? 



Microphones 


3 9 15 36 


Karaoiie 
Machines 


1 


3 


? 


12 



A. 5 

B. 7 



C. 8 

D. 9 



9. On a baseball field, the distance from 
home plate to the pitcher's mound is 
60^ feet. The distance from home plate 
to second base is about 1277j feet. 
What is the difference between the 



two distances? 

A. 61 1 feet 

B. 66 1 feet 

b 



C. 66^ feet 

D. 66|, feet 



10. Which word phrase best describes the 
expression n - 6? 

A. 6 more than a number 

B. A number less than 6 

C. 6 minus a number 

D. A number decreased by 6 

11. A football weighs about J^ kilogram. 
A coach has 15 footballs in a large bag. 
Which is the best description of the total 
weight of the footballs? 

A. Not quite 3 kilograms 

B. A little more than 2 kilograms 

C. Almost 1 kilogram 

D. Between 1 and 2 kilograms 



12. The scale on a map is 1 centimeter: 
70 kilometers. The distance between 
two cities on the map is 8.2 centimeters. 
Which is the best estimate of the actual 
distance? 

A. 85 kilometers 

B. 471 kilometers 

C. 117 kilometers 

D. 574 kilometers 



13. 



On a scale drawing, a cell phone tower 
is 1.25 feet tall. The scale factor is ~. 
What is the height of the actual cell 
phone tower? 

A. 37.5 feet C 148 feet 

B. 120 feet D. 187.5 feet 



# 



When a diagram or graph is not 
provided, quickly sketch one to clarify 
the information provided in the test 
Item. 



Gridded Response 

14. The Liberty Bell, a symbol of freedom 
in the United States, weighs 2,080 
pounds. How many tons does the 
Liberty Bell weigh? 

15. Find the quotient of -51.03 and -8.1. 

16. A scale drawing of a rectangular 
garden has a length of 4 inches and 
a width of 2.5 inches. The scale is 

1 inch:3 feet. What is the perimeter of 
the actual garden in feet? 

17. A florist is preparing bouquets of 
flowers for an exhibit. The florist has 
84 tulips and 55 daisies. Each bouquet 
will have the same number of tulips 
and the same number of daisies. How 
many bouquets can the florist make 
for this exhibit? 



Cumulative Assessment, Chapters 1-4 271 



CHAPTER 



5 



I! n' 



ii 



ctio 







5A 


Tables and Graphs 




5-1 


The Coordinate Plane 




5-2 


Interpreting Graphs 




5-3 


Functions, Tables, and 
Graphs 




5-4 


Sequences 




5B 


Linear Functions 




LAB 


Explore Linear 
Functions 




5-5 


Graphing Linear 
Functions 




EXT 


Nonlinear Functions 




5-6 


Slope and Rates of 
Change 


7.3.6 


LAB 


Generate Formulas to 
Convert Units 




5-7 


Slope-Intercept Form 


7.2.6 


5-8 


Direct Variation 


7.2.7 


EXT 


Inverse Variation 






Why Learn this? 



You can use linear equations to repre- 
sent how far a sailboat moving at a con- 
stant rate has traveled after a certain 
amount of time. 

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apter 



Graph linear relationships 
and identify the slope of 
the line. 

Identify proportional 
relationships {y = kx). 



X=^ 



- ♦ 






J«*d-V 




>y (( 






i.V 



Are You Ready? 



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0^ Vocabulary 



Choose the best term from the list to complete each sentence. 

1. A(n) ? is a number that represents a part of 
a whole. 

2. A closed figure with three sides is called acn) ? . 

3. Two fractions are ? if they represent the same 
number. 

4. One way to compare two fractions is to first find 
acn) ? . 



common 
denominator 

equivalent 

fraction 

quadrilateral 

triangle 



Complete these exercises to review skills you will need for this Chapter. 

(^ Write Equivalent Fractions 

Find two fractions that are equivalent to each fraction. 



c 2 c 7 7 25 

5 11 100 




8. 


4 
6 


'• ^ '"■ i "• % 




12. 


150 
325 


Compare Fractions 








Compare. Write < or >. 








13 5 2 14 3 2 15 — 
'^- 6 3 '^-8 5 '^- 11 


1 

4 


16. 


5 11 
8 12 


.17 8 12 1R 5 7 1Q ** 

1/. 9 yg i»- n 21 ^^- To 


3 

7 


20. 


3 2 

4 9 


Solve Multiplication Equations 








Solve each equation. 








21. 3a = 12 22. 15f =75 23. 2y = 


14 


24. 


7m = 84 


25. 25c=125 26. 16/= 320 27. \\n 


= 121 


28. 


53}'= 318 



Qj Multiply Fractions 



Solve. Write each answer in simplest form. 

29. I-I 30. 114 31 



33. 



3 7 

1 5 
5 9 



34. 



I i 
8 3 



35. 



4 18 
9 ' 24 

25 30 
100 90 



32. 



36. 



J_ _50_ 

56 200 

46 3 

91 6 




Graphs and Functions 273 



CHAPTER 



5 



Study 



Where You've Been 

Previously, you 

• graphed ordered pairs of non- 
negative rational numbers on a 
coordinate plane. 

• used tables to generate formulas 
representing relationships. 

• formulated equations from 
problem situations. 



In This Chapter 



You will study 

• plotting and identifying 
ordered pairs of integers on a 
coordinate plane. 

• graphing to demonstrate 
relationships between data sets. 

• describing the relationship 
between the terms in a 
sequence and their positions in 
a sequence. 

• formulating problem situations 
when given a simple equation. 



Where You're Going 

You can use the skills 
learned in this chapter 

• to sketch or interpret a graph 
that shows how a measurement 
such as distance, speed, cost, or 
temperature changes over time. 

e to interpret patterns and make 
predictions in science, business, 
and personal finance. 



Key 

Vocabulary /Vocabulario 



coordinate plane 


piano cartesiano 


function 


funclon 


linear equation 


ecuaclon lineal 


linear function 


funcion lineal 


ordered pair 


par ordenado 


origin 


orlgen 


quadrant 


cuadrante 


sequence 


suceslon 


X-axis 


ejex 



y-axis 



ejey 



Vocabulary Connections 

To become familiar with some of the 
vocabulary terms in the chapter, consider the 
following. You may refer to the chapter, the 
glossary, or a dictionary if you like. 

1. A sequence is an ordered list of numbers, 
such as 2, 4, 6, and 8. Can you make up a 
sequence with a pattern and describe the 
pattern? 

2. The word "linear" comes from the word 
line. Wliat do you think the graph of a 
linear equation looks like? 

3. An origin is the point at which something 
begins. Can you describe where to begin 
when you plot a point on a coordinate 
plane? Can you guess why the point where 
the X-axis and y-axis cross is called the 
origin ? 

4. Quadrupeds are animals with four feet, 
and a quadrilateral is a four-sided figure. 
A coordinate plane has sections called 
quadrants . Wliat does this word imply 
about the number of sections in a 
coordinate plane? 



274 Chapter 5 




Reading x 

and WrlMita 
MathX -^ 




Writing Strategy: 

Write a Convincing Argument 

A convincing argument or explanation should 
include the following: 

• The problem restated in your own words 

• A short response 

• Evidence to support the response 

• A summary' statement 





Write About It 

Explain how to find the 
next three integers in 
the pattern -43, -40 
~37, -34, . 



Identify the goal. 

Explain how to find the next three integers in the pattern 
-43, -40, -37. -34 



Provide a short response. 

As the pattern continues, the integers increase in value. Find the amount 
of increase from one integer to the next. Then add that amount to the last 
integer in the pattern. Follow this step two more times to get the next three 
integers in the pattern. 



Provide evidence to support your response. 

-43 -40 -37 -34 

+ 3 +3 +3 +3 +3 +3 

-34 + 3 = -31 -31 + 3 = -28 -28 + 3 = -25 
The next three integers are -31, -28, and -25. 

Summarize your argument. 

To find the next three integers in the pattern -43, 



Find the amount of 
increase from one 
integer to the next. 

The pattern is to add 
3 to each integer to 
get the next integer. 



-40, -37, -34, 



find 



the amount that is added to each integer to get the next integer in the pattern. 



TVy This 



Write a convincing argument using the method above. 

1. Explain how to find the next three integers in the pattern 0, -2, -4, -6, . 

2. Explain how to find the seventh integer in the pattern — 18, - 13, —8, -3, 



Graphs and Functions 275 




Vocabulary 

coordinate plane 
X-axis, y-axis 
origin 
quadrant 
ordered pair 



A coordinate plane is 
a plane containing a 
horizontal number line, 
the .v-axis , and a vertical 
number line, the y-axis . 
The intersection of these 
axes is called the origin . 

Tlie axes divide the 
coordinate plane into four 
regions called quadrants, 
which are numbered I, II, 
III, and IV. 



4'y 



Quadrant II 



H 1 1 i 1- 



-6-5-4-3-2 



Quadrant III 



-1? 
-2 
-3 
-4 
-5 
-6 



y 



y-axis 



Quadrant I 



x-axis 



H 1 h 



\l 2 3 4 5 6 
Origin __^_^ 

Quadrant IV 



X 



EXAMPLE [l J Identifying Quadrants on a Coordinate Plane 



Identify the quadrant that 
contains each point. 
A P 

P lies in Quadrant II. 



Q lies in Quadrant IV. 



C R 



R lies on the .v-axis, between 
Quadrants II and III. 



Ay 



P» 



R 

-• — h 
-3-2 



3 
2 
1 



-1? 
-2 
-3 

-4 



X 



12 3 4 



An ordered pair is a pair of numbers 
that can be used to locate a point on 
a coordinate plane. The two numbers 
that form the ordered pair are called 
coordinates. The origin is identified by 
the ordered pair (0,0). 

Ordered pair 

(3,2) 



AT-coordinate 

Units right 
or left from 



y-coordinate 

Units up 
or down from 



AV 



2 units up 
3 units 



-4-3-2-lP 

4-T- -3- 
-^-1 4- 



right 



12 3 4 



276 Chapter 5 Graphs and Functions 



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EXAMPLE [2] Plotting Points on a Coordinate Plane 



Plot each point on a coordinate 
plane. 

A G(2, 5) 

Scart at the origin. Move 2 units 
right and 5 units up. 

B N{-3,-4) 

Siart at the origin. IVIove 3 units 
left and 4 units down. 

C P(0, 0) 

n r li ac the Origin. 



^y 



4 

2 

1 + 



< — t — I — i — I — I 

_4_3_2-j|0 

-2 
- -3 

• -4 

A/(-3, -4) 



' 6 (2, 5) 



P (0, 0) X 

— I — I — I — I > 



12 3 4 



EXAMPLE [3] Identifying Points on a Coordinate Plane 

Give the coordinates of each point. 



A / 

Start at the origin. Point J is 
3 units right anc i. 

The coordinates of/ are (3, -2). 



Start at the origin. Point K is 

2 units left and 4 units ui.' 

The coordinates of /Care (-2, 4). 



C L 



Ki 



i^y 



4 
3 
2 

1 



-4-3-2-1P 
-1 

-2 

-3 

— -4 



X 



12 3 4 
• J 



Start at the origin. Point L is 3 units left on the x-axis. 
The coordinates of L are (-3, 0). 



Think and Discuss 

1. Explain whether point (4, 5) is the same as point (5, 4). 

2. Name the x-coordinate of a point on the y-axis. Name the 
y-coordinate of a point on the x-axis. 

3. Suppose the equator represents the x-axis on a map of Earth and 
a line called the prime meridian, which passes through England, 
represents the y-axis. Starting at the origin, which of these directions 
— east, west, north, and south — are positive? Which are negative? 



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5-1 The Coordinate Plane 277 



5-1 



3.. 



3 



GUIDED PRACTICE 



[•Jiiiiii-i 

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Exercises 1-26, 27, 29, 33 



See Example 1 Identify the quadrant that contains each point. 
^. A 2. B 



^y 



3. C 



4. D 



See Example 2 Plot each point on a coordinate plane. 

5. £(-1,2) 6. N(2, -4) 

_ 7. H[-3, -4) 8. 7(5,0) 

See Example 3 Give the coordinates of each point. 
I 9. / 10. P 

I 11. S 12. M 



: 1 1 u 






-1-4- 



S 

■< 1 — •- 



(3f 

'-2- 
1 



-5-4-3-2-1 P 

, ] ; 3 



-4 
-5 



M X 

4 — • »■ 



12 3 4 5 



i — U- 



7» 



INDEPENDEN.T.BRACTICE 



See Example 1 Identify the quadrant that contains each point. 
13. F 14. ; 



15. a: 



16. E 



See Example 2 Plot each point on a coordinate plane. 

17. A(-l,l) 18. A/f2, -2) 

; 19. W{-5, -5) 20. G(0, -3) 

See Example 3 Give the coordinates of each point. 
I 21. Q 22. V 

23. R 24. P 

25. S 26. L 



Ay 



.-l.-_-L9 ' 



-5-4-3-2-JlO 

-2 

^^_4__ _3r 



-4^P 
-5- 






-*t 



H 1 1 1 h- 

12 3 4 5 
•K , 



Vm 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EP14. 



Graph each set of ordered pairs. Then connect the points, identify the figure 
created, and name the quadrants in which it is located. 

27. (-8, 1); (4,3); (-3,6) 28. (-8, -2); (-1, -2); (-1, 3); (-8, 3) 

Identify the quadrant of each point described below. 

29. The A-coordinate and the y-coordinate are both negative. 

30. The A-coordinate is negative and the y-coordinate is positive. 

31. The A-coordinate is positive and the y-coordinate is negative. 

32. Wliat point is 5 units left and 2 units down from point (1,2)? 



278 Chapter 5 Graphs and Functions 




When the wind 
speed of a 
tropical storm 
reaches 74 mi/h, 
it is classified as 
a hurricane. 



33. What point is 9 units right and 3 units up from point (3, 4]? 

34. VVliat point is 4 units left and and 7 units up from point (-2, -4)? 

35. What point is 10 units right and and 1 unit down from point (-10, 1)? 

36. Critical Thinking After being moved 6 units right and 4 units down, a 
point is located at (6, 1 ). What were the original coordinates of the point? 

-^2 Weather The map shows 
the path of fiurricane Rita. 
Estimate to the nearest 
integer the coordinates of 
the storm for each of the 
times below. 



, 












-~T^ 


1 


" 


._„p 


^-.- 






— 




-^ 


— 


n 




1 1 












.' 


1 






hurricane nixa 
' September 2005 


M 










^ 


4t 


















/^ 


i 




















: 1 ' 






1 


1 


3. Rita becomes 
a tropical 
depression 






\ ' i 






1 




j 


1 














' 




1 


JL 






t 






3"o '■ 






r; 




- 


1 




i 


~* , 


! 
















^ 7i\zri 






1 
















'-'- ! / 


1 

akes 


^ 


! 




J 
















' 2. Rita m 


j5.^ 








m 




sr^ 


J 






landfall in 
4 the United 
-| States. 


1 


Pis^_ 






_ 1 .."W. 










3«:«1 


! 


! ^••*. — ..! 


■""i 


\ ^ =' 






1 




4^ + 


.J? ; 


-i4 


•^» 


-)■■- 
t" 










1 








1. Rita becomes a rl Mulr-l i j 














hurricane. 


'--— 


^■tei. ;, 




.]„, 




■ 


I-- - j^2IZ 


' 


20°^ 
_7i;° ; 


Q«;<> ' ! Jjo 


n« I 


b'5° 


-80° ' 










■TAJ 




^ 




>\-r 



a. when Rita first became 
a hurricane 

b. when Rita made landfall 
in the United States 

c. when Rita weakened to 
a tropical depression 

p 38. What's the Error? To plot (-12, 1), a student started at (0, 0) and 

moved 12 units right and 1 unit down. What did the student do wrong? 

39. Write About It Why is order important when graphing an ordered pair 
on a coordinate plane? 

^ 40. Challenge Armand and Kayla started jogging from the same point. 
Armand jogged 4 miles south and 6 miles east. Kayla jogged west and 
4 miles south. If they were 1 1 miles apart when they stopped, how far 
west did Kayla jog? 



Test Prep and Spiral Review 



41. Multiple Choice Which of the following points lie 
within the circle graphed at right? 



CS) (2,6) 



CX' (-4, 4) (©(0,-4) CD) (-6, 6) 



42. Multiple Choice Which point on the x-axis is the 
same distance from the origin as (0, -3)? 



CD (0,3) 



® (3,0) 



(K) (3,-3) CD (-3,3) 



Find each sum. (Lesson 2 2) 

43. -17+11 44. 29 + 



♦ J' 




45. 40 +(-64) 



Divide. Write each answer in simplest form. (Lesson 3-10) 



47. 8 + 1 



1 



48. 



6^ 
15 



49 2- + 1- 



46. 



50. 



-55 +(-32) 



5 . 3 



8 ■ 4 



5-7 The Coordinate Plane 279 



5-2 



Interpreting 
Graphs 



You can use a graph to show 
the relationship between speed 
and time, time and distance, or 
speed and distance. 

The graph at right shows the 
varying speeds at which Emma 
exercises her horse. The horse 
walks at a constant speed for the 
first 10 minutes. Its speed 
increases over the next 7 minutes, 
and then it gallops at a constant 
rate for 20 minutes. Then it slows 
down over the next 3 minutes 
and then walks at a constant pace 
for 10 minutes. 




20 30 40 50 
Time (min) 



EXAMPLE 







Relating Graphs to Situations 

Jenny leaves home and drives to the beach. She stays at the 
beach all day before driving back home. Which graph best shows 
the situation? 



Graph A 




Graph B 




Time 



Time 




Graph C 



Time 



As Jenny drives to the beach, her distance from home increases. 
While she is at the beach, her distance from home is constant. As 
she drives home, her distance from home decreases. The answer 
is graph B. 



280 Chapter 5 Graphs and Functions 



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EXAMPLE 



(3 



PROBLEM 



SOLVING 



PROBLEM SOLVING APPLICATION 

Maili and Katrina traveled 10 miles from Maili's house to the 
movie theater. They watched a movie, and then they traveled 
5 miles farther to a restaurant to eat lunch. After eating they 
returned to Maili's house. Sketch a graph to show the distance 
from Maili's house compared to time. Use your graph to find the 
total distance traveled. 

^ Understand the Problem 

The answer will be the total distance that Katrina and Maili traveled. 
List the important information: 

• The friends traveled 10 miles from Maili's house to the theater. 

• They traveled an additional 5 miles and then ate lunch. 

• They returned to Maili's house. 

Make a Plan 

Sketch a graph that represents the situation. Then use the graph to 
find the total distance Katrina and Maili traveled. 

•U Solve 

The distance from Maili's house 
increases from to 10 miles 
when the friends travel to the 
theater. The distance does not 
change while the friends watch 
the movie and eat lunch. The 
distance increases from 10 to 15 
miles when they go to the 
restaurant. The distance 
decreases from 15 to miles 
when they return home. 



■=■ 20 



E 

0) 

o 



E 
o 



c 

IT! 



Ate lunch 



Maili and Katrina traveled a 
total of 30 miles. 




Went to 
Maili's 



Time 



Q Look Back 

The theater is 10 miles away, so the friends must have traveled 
twice that distance just to go to the theater and return. The answer, 
30 miles, is reasonable since it is greater than 20 miles. 



Thmk and Discuss 

1. Explain the meaning of a horizontal segment on a graph that 
compares distance to time. 

2. Describe a real-world situation that could be represented by a 
graph that has connected lines or curves. 



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5-2 Interpreting Graphs 281 



■L'-^ 

H^^^ 



' v;!^l^l5!«i^^sy;ri^W&iS^iGi*ijii^.;*-W4"J;i^^ 



^i^?i;'J33^ 




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S Homework Help Online go.hrw.com, 



keyword WMMsfliM ® 
Exercises 1-4, 7, 9 



See Example 2 



GUIDED PRACTICE 



See Example 1 1. 



The temperature of an ice cube increases until it starts to melt. While it 
melts, its temperature stays constant. Which graph best shows the situation? 



Graph A 



Graph B 



Graph C 



Time 



Time 




Time 



Mike and Claudia rode a bus 15 miles from home to a wildlife park. They 
waited in line to ride a train, which took them on a 3-mile ride around the 
park. After the train ride, they ate lunch, and then they rode the bus home. 
Sketch a graph to show the distance from their home compared to time. Use 
your graph to find the total distance traveled. 



See Example 1 



See Example 2 



INDEPENDENT PRACTICE 



3. The ink in a printer is used until the ink cartridge is empty. The cartridge is 
refilled, and the ink is used up again. Which graph best shows the situation? 



Graph A 



Graph B 



Time 




Graph C 



Time 




Time 



On her way from home to the grocery store, a 6-mile trip, Veronica stopped 
at a gas station to buy gas. After filling her tank, she continued to the 
grocery store. She then returned home after shopping. Sketch a graph to 
show the distance from Veronica's home compared to time. Use your graph 
to find the total distance traveled. 



Extra Practice 



See page EP14. 



PRACTICE AND PROBLEM SOLVING 



5. Describe a situation that fits the graph 
at right. 

6. Lynn jogged for 2.5 miles. Then she walked 
a little while before stopping to stretch. 
Sketch a graph to show Lynn's speed 
compared to time. 



Time 



On his way to the library, Jeff runs two blocks and then walks three more 
blocks. Sketch a graph to show the distance Jeff travels compared to time. 



282 Chapter 5 Graphs and Functions 



High School Enrolltn 




^ 20 

C 

o 

1 

- 10 

1 5 

o 

w 

c 
uj 




^ • ! ? ^ 






i i i : : 


1990 1995 2000 2005 2010 

Year 



9. 



11. 



Critical Thinking The graph at right 
shows high school enroUment, including 
future projections. 

a. Describe what is happening 
in the graph. 

b. Does it make sense to connect the 
points in the graph? Explain. 

c. Graphs that are not connected are 
called discrete. Describe another 
situation where the graph that shows 
the situation would be discrete. 

Choose a Strategy Three bananas were given to two mothers who were 
with their daughters. Each person had a banana to eat. How is that possible? 

Write About it A driver sets his car's cruise control to 55 mi/h. Describe 
a graph that shows the car's speed compared to time. Then describe a 
second graph that shows the distance traveled compared to time. 

Challenge The graph at right shows the temperature 
of an oven after the oven is turned on. Explain what the 
graph shows. 




Time 



Test Prep and Spiral Review 



12. Multiple Choice How does speed compare to time in the 
graph at right? 



CS) It increases. 
CD It decreases. 



CE) It stays the same. 
CE It fluctuates. 



■D 

a. 

in 



Time 



13. Short Response Keisha takes a big drink from a bottle of water. She sets the 
bottle down to tie her shoe and then picks up the bottle to take a small sip of 
water. Sketch a graph to show the amount of water in the bottle over time. 



Find each absolute value. (Lesson 2-1) 

14. |9| 15. |-3| 16. |-15| 



17. 



18. 



Find the greatest common factor. (Lesson 2-7) 

19. 12,45 20. 33,110 21. 6,81 22. 24,36 

Divide. Estimate to check whether each answer is reasonable. (Lesson 3-4) 
23. 48.6 -^ 6 24. 31.5 H- (-5) 

25. -8.32 -^4 26. -74.1 h- 6 



5-2 Interpreting Graplis 283 



5-3 



Functions, Tab 



Vocabulary 

function 

input 

output 



WBBCJfcV 




(csmxi^ixsci 




gAFETY 




PEvice 


^^'^ 


FOB 


^$ 


WMJCINS 


'■W 


ON 


/■ 


ICY 


f 


WMCMBNTS. 


el^ 







When you slip 

ON ICE, YOUC 
FOOT K1CK5 
PAPPLECA), 
LOWEClNS FINSEEfB) 
SKJAPPWe TURTLE ( C) 
EXTENDS NECK 
TO BITE FIN6ER, 
OPENING 
ICE TONSS ( D) AND 
DROPPING PILLOW (E) 
THUS ALLOWING ' 

VOU TO PALL 

ON SOMETHINS 

SOFT/ 



When you slip on ice, your foot kicks paddle (A), lowering finger (B), snapping turtle (Q extends neck to bite finger, 
opening ice tongs (D) and dropping pillow (E), thus allowing you to fall on something soft. 



Rube Goldberg, a famous cartoonist, invented machines that perform 
ordinary taslcs in extraordinary ways. Each machine operates according 
to a aile, or a set of steps, to produce a particular output. 

In mathematics, a function operates according to a rule to produce 
exactly one output value for each input value. The input is the value 
Interactivities Online ► substituted into the function. The output is the value that results from the 
substitution of a given input into the fimction. 



A function can be represented by a rule 
written in words, such as "double the 
number and then add nine to the result," 
or by an equation with two variables. One 
variable represents the input, and the 
other represents the output. 

You can use a table to organize and 
display the input and output values of a 
function. 



Function Rule 



y-Zx+9 

t f 

Output Input 
variable variable 



EXAMPLE Ilj Completing a Function Table 

Find the output for each input. 
A y = 4x - 2 





Input 


Rule 


Output 


X 


4x- 2 


y 


-1 


4(-1)-2 


-6 





4(0) - 2 


-2 


3 


4(3) -2 


10 



Substitute - 1 for x. Then simplify. 
Substitute for x. Tlien simplify. 
Substitute 3 for x. Then simplify. 



284 Chapter 5 Graphs and Functions 



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Find the output for each input. 
B y = 6x- 



An ordered pair is a 
pair of numbers that 
represents a point on 
a graph. 



Input 


Rule 


Output 


X 


6x^ 


y 


-5 


6(-5)^ 


150 





6(0)^ 





5 


6(5)^ 


150 



Substitute -5 for X. Then simplify. 
Substitute for x. Tlien simplify. 
Substitute 5 for x. Then simplify. 



EXAMPLE 



You can also use a graph to represent a function. The corresponding 
input and output values together form unique ordered pairs. 

[21 Graphing Functions Using Ordered Pairs 

Make a function table, and graph the resulting ordered pairs. 



When writing an 
ordered pair, write 
the input value first 
and then the output 
value. 



A y=2.V 



Input 


Rule 


Output 


Ordered Pair 


X 


2x 


y 


{X'V) 


-2 


2(-2) 


-4 


(-2, -4) 


-1 


2(-1) 


-2 


(-1,-2) 





2(0) 





(0,0) 


1 


2(1) 


2 


(1,2) 


2 


2(2) 


4 


(2,4) 



B y = x- 



Input 


Rule 


Output 


Ordered Pair 


X 


x' 


y 


(x-y) 


-2 


i-2)' 


4 


(-2, 4) 


-1 


(-1)^ 


1 


(-1,1) 





(0)^ 





(0,0) 


1 


(1)^ 


1 


(1,1) 


2 


(2)2 


4 


(2,4) 



*y 



-- •(!, 



< — I — I — I — i — li — I — f 



-4 O 

(-1, -2). ■ 

(-2, -4) . -4- 



(2,4) 
2) 



(0,0) 



xy 



(-2, 4) 



(-1, 1)' 



H 1 i h- 



o 

-2- 



' (2, 4) 



•(1, 1) 



H 1 1 H 



X 



(0,0) 



Think and Discuss 

1. Describe how a function works like a machine. 

2. Give an example of a rule that takes an input value of 4 and 
produces an output value of 10. 



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5-3 Functions, Tables, and Graphs 285 



5-3 




•p! HomeworkHelpOnllnego.hrw.com, " 
IgoI 



keyword ■mBiiiiiaM 
Exercises 1-10, 11, 17 



GUIDED PRACTICE 



See Example 1 Find the output for each input. 

1. y = 2x+\ 2. y= -x + 3 



3. y = 2x^ 



Input 


Rule 


Output 


X 


2x+ 1 


y 


-3 













1 







Input 


Rule 


Output 


X 


-x + 3 


y 


-2 













2 




B 



Input 


Rule 


Output 


X 


2x^ 


y 


-5 






1 




lli 


3 




H 



See Example 2 Make a function table, and graph the resulting ordered pairs. 

4. ]' = 3.V - 2 5. r = .V- + 2 



Input 


Rule 


Output 


Ordered Pair 


X 


3x -2 


y 


(x,y) 


-1 

















1 








2 









Input 


Rule 


Output 


Ordered Pair 


X 


x^ + 2 


y 


(x,y) 


-1 















■l: 


1 






^-, 


2 









jiiyiMiJitAuy iim- 



See Example 1 Find the output for each input. 

6. y = -2.V 7. y = Sx + 2 



8. v = 3.y' 



Input 


Rule 


Output 


X 


-2x 


y 


-2 






-0 






4 







Input 


Rule 


Output 


X 


3x + 2 


y 


-3 






-1 




;; ^ 


2 







Input 


Rule 


Output 


X 


3x^ 


y 


-10 






-6 




B ', 


-2 







See Example 2 Make a function table, and graph the resulting ordered pairs. 

9. y = .v-=- 2 10. y = X- - 4 



Input 


Rule 


Output 


Ordered Pair 


X 


X ^2 


y 


(x,y) 


-1 

















1 








2 









Input 


Rule 


Output 


Ordered Pair 


X 


x^ -4 


y 


(x,y) 


-1 















11 


1 






^ 


2 






n 



286 Chapter 5 Graphs and Functions 



PRACTICE AND PROBLEM SOLVING 



Extra Practice 



See page EP14. 



Mil 



c 
o 

n: 

[5. 



11. Weather The Northeast gets an average 
of 11.66 inches of rain in the summer. 

a. Write an equation that can be used to 
find y, the difference in rainfall between 
the average amount of simimer rainfall 
and .V, a given year's summer rainfall. 

b. Make a function table using each year's 
summer rainfall data. 

12. Physical Science The equation 
F —tc + 32 gives the Fahrenheit 
temperature F for a given Celsius 
temperature C. Make a function table for 
the values C = -20°, -5°, 0°, 20°, and 100°. 

@ 13. What's the Error? What is the error in the 
function table at right? 

14. Write About It Explain how to make a function 
table for v = 2.v + 11. 

§9 15. Challenge Mountain Rental charges a $25 deposit 
plus $10 per hour to rent a bicycle. Write an 
equation that gives the cost y to rent a bike 
for X hours. Then write the ordered pairs for 
x = ^,5, and 8|. 



Selected Dry Summers 
in the Northeast 



8.73 



8.44 




iE 8.66 



1913 1930 1957 1995 1999 

Year 



Source USA Today. August 17, 2001 



X 


y - -X - 5 


y 


-2 


y - -(-2) - 5 


-7 


-1 


y= -H)-5 


-6 





y- -(0) - 5 


-5 


1 


y= -{!)- 5 


-6 


2 


y= -(2)- 5 


-7 



Test Prep and Spiral Review 



16. Multiple Choice Which table shows correct input and output values 
for the function v = -2.v + 3? 



(S) 



X 


y 


-1 


-1 









CBJ 



X 


y 


-3 


-2 


-2 


-1 



CD 



X 


y 


-5 


-7 


-1 


1 



CE 



17. Multiple Choice Which function matches the function table? 

CD _v = .V + 3 CH) y = 5.V + 1 

CS) V = x'- + 7 (X) V = x^ + 3 



Simplify. (Lesson 2-3) 
18. 43 - (-18) 



X 


y 


-3 


9 


-1 


5 



X 


1 2 


y 


3 4 11 



19. 3- (-2) - (5+ 1) 



Solve. Write each answer in simplest form. (Lesson 3-11) 

- 22 4c = ^ 

7 ^^- ^~ 5 9' 



21. iA-=6 



23. |y = 3 



20. -4-8- (-3) 



24 ^x=^- 
10 8 



5-3 Functions, Tables, and Graphs 287 



5-4 



Sequences 



Vocabulary 

sequence 

term 

arithmetic sequence 

common difference 

geometric sequence 



v^Vf^ 



f^~ 



i> 



'is?r>^ 



Many natural things, such as the 
arrangement of seeds in the head 
of a sunflower, follow the pattern 
of sequences. 

A sequence is an ordered list of numbers. 
Each number in a sequence is called a term . 
When the sequence follows a pattern, the 
terms in the sequence are the output values 
of a function, and the value of each term 
depends on its position in the sequence. 

You can use a variable, such as /;, to represent a number's position 
in a sequence. 



y<' 



X\ 



\ 



n (position in the sequence) 



1 



y (value of term) 



+ 2 +2 +2 

In an arithmetic sequence , the terms of the sequence differ by the same 
nonzero number. This difference is called the common difference . In 
a geometric sequence , each term is multiplied by the same amount to 
get the next term in the sequence. 



EXAMPLE [ 1 1 Identifying Patterns in Sequences 



Tell whether each sequence of y-values is arithmetic or geometric. 
Then find y when n - S. 



n 


1 


2 


3 


4 


5 


y 


-12 


-5 


2 


9 





In the sequence 7 is added to each term. 

9 + 7 = 16 Add 7 to the fourth term. 

The sequence is arithmetic. When u = 5, y = 16. 



n 


1 


2 


3 


1 

4 


5 


y 


4 


-12 


36 


I -108 





In the sequence each term is multiplied by —3. 
-108 • (-3) = 324 Multiply the fourth term by -3. 
The sequence is geometric. When n = 5, y = 324. 



288 Chapter 5 Graphs and Functions 



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EXAMPLE [2] Identifying Functions in Sequences 

Write a function that describes each sequence. 



A 2,4,6,8.... 

Make a function table. 



n 


Rule 


y 


1 


1 -2 


2 


2 


2 -2 


4 


3 


3 -2 


6 


4 


4 • 2 


8 



Multiply n by 2. 
The function y = 2ii 
describes this sequence. 



4,5,6,7,... 

Make a function table. 



n 


Rule 


y 


1 


1 + 3 


4 


2 


2 + 3 


5 


3 


3 + 3 


6 


4 


4 + 3 


7 



/^dd 3 to n. 

The function v = » + 3 

describes this sequence. 



EXAMPLE 



C3 



Using Functions to Extend Sequences 

Sara has one week to read a book. She plans to increase the 
number of chapters that she reads each day. Her plan is to read 
3 chapters on Sunday, 5 on Monday, 7 on Tuesday, and 9 on 
Wednesday. Write a function that describes the sequence. 
Then use the function to predict how many chapters Sara will 
read on Saturday. 

Write the number of chapters she reads each day: 3, 5,7,9,... 
Make a function table. 



Multiply n by 2. Then add 1. 



n 


Rule 


y 


1 


1-2+1 


3 


2 


2-2+1 


5 


3 


3-2 + 1 


7 


4 


4-2 + 1 


9 



3/ = 2;/ + 1 



Write the function. 



Saturday corresponds to // = 7. When » = 7, y 
Sara plans to read 15 chapters on Saturday. 



+ 1 = 15. 



Think and Discuss 

1. Give an example of a sequence involving addition, and give the 
rule you used. 

2. Describe how to find a pattern in the sequence 1, 4, 16, 64, ... . 



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5-4 Sequences 289 



5-4 



)s-JJi^■^>ii»t»iJ^immllhimr^MiK':AVJKAi^i»ri^ 



ZJ 





Homework Help Online go.hrw.com, 



keyword ■BHW.-gM ® 
Exercises 1-16, 21, 25 



GUIDED PRACTICE 



See Example 1 Tell whether each sequence of y- values is arithmetic or geometric. Then find 
j/when n - 5. 

1. 



n 


1 


2 


3 


4 


5 


y 


-4 


9 


22 


35 





n 


1 


2 


3 


4 1 5 


y 


8 


4 


2 


1 
1 



See Example 2 



See Example 3 



Write a function that describes each sequence. 

3. 3, 6, 9, 12, . . . 4. 3, 4, 5, 6, . . . 5. 0, 1, 2, 3, 



6. 5, 10.15,20, 



7. In March, Waterworks recorded $195 in swimsuit sales. The store 
recorded $390 in sales in April, $585 in May, and $780 in lune. Write a 
function that describes the sequence. Then use the function to predict 
the store's swimsuit sales in luly. 



INDEPENDENT PRACTICE 



See Example 1 Tell whether each sequence of y-values is arithmetic or geometric. Then find 
y when n = 5. 

8. 



n 


1 


2 


3 


4 


5 


y 


13 


26 


52 


104 





n 


1 ' 2 


3 


4 


5 


y 


14 30 


46 


62 



See Example 2 Write a function that describes each sequence. 



See Example 3 



10. 5,6, 7,8 

13. 20,40,60,80, 



11. 7, 14,21,28, 
14. ^, 1,|2, ... 



12. -2, -1,0, 1,.. . 
15. 1.5,2.5,3.5,4.5, 



16. The number of seats in the first row of a concert hall is 6. The second 

row has 9 seats, the third row has 12 seats, and the fourth row has 15 seats. 
Write a function to describe the sequence. Then use the function to 
predict the number of seats in the eighth row. 



Extra Practice 



See page EP14. 



PRACTICE AND PROBLEM SOLVING 



Write a rule for each sequence in words. Then find the next three terms. 



17. 35,70, 105, 140, 
20. -1,0, 1,2,... 



18. 0.7, 1.7,2.7,3.7, . . 



21 i ^ 1 ^ 

3' 3' ' 3 



19. 



3 5 7 9 



2' 2' 2' 2' ' ' ' 
22. 6, 11, 16,21, . . 



Write a function that describes each sequence. Use the function to find the 
tenth term in the sequence. 

23. 0.5, 1.5, 2.5, 3.5, ... 24. 0, 2, 4, 6, . . . 25. 5, 8, 11, 14, . . . 



26. 3,8, 13, 18,... 



27. 1,3,5,7,... 



28. 6, 10, 14, 18, 



290 Chapter 5 Graphs and Functions 



Computer programmers use fimctions to create designs 
known ?iS fractals. A fractal is a self-similar pattern, wliicli 
means that each part of the pattern is similar to the whole 
pattern. Fractals are created by repeating a set of steps, 
called iterations. 

29. Below is part of a famous fractal called the Cantor set. 

In each iteration, part of a line segment is removed, resulting in twice as 
many segments as before. The table lists the number of line segments that 
result from the iterations shown. Find a function that describes the sequence 






1 


2 


2 


4 


3 


8 



30. Multi-Step These are the first three iterations of the Sierpinski triangle. In each iteriition, 
a certain number of smaller triangles are cut out of the larger triangle. 




V 
V V 



V V 

V V 

V V V V 



Iteration 1 Iteration 2 Iteration 3 

1 triangle removed 3 more triangles removed 9 more triangles removed 
Create a table to list the number of yellow triangles that exist after 
each iteration. Then find a function that describes the sequence. 

31. ^^Challenge Find a function that describes the number of 
triangles removed in each iteration of the Sierpinski triangle. 



m 



Test Prep and Spiral Review 



32. Multiple Choice Which function describes the sequence 1, 4, 7, 10, ... ? 

CE) y = 3ii CE) y= n + 3 CD y ^ 'in - 2 CS.' y = 2n 

33. Extended Response Create a sequence, and then write a function that 
describes it. Use the function to find the ninth term in the sequence. 



Find each value. (Lesson 12) 



34. V: 



35. 10' 



Find each product. (Lesson 2-4) 

38. -16 • 2 39. -40 • (-5) 



36. 7* 



40. 4 • (-11) 



37. 9^ 



41. -5 • (-21) 



5-4 Sequences 291 




To Go On? 



Quiz for Lessons 5-1 Through 5-4 

5-1] The Coordinate Plane 



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^^ ResourcesOnlinego.hrw.com, 
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Plot each point on a coordinate plane. Then identify the quadrant that 
contains each point. 



1. IV(1,5) 



2. X(5, -3) 



3. r(-i, 



4. Z(-8, 2) 



■; 5-2 ] Interpreting Graphs 

5. Raj climbs to the top of a cliff. He descends a little bit to another cliff, and 
then he begins to climb again. Which graph best shows the situation? 



I Graph A 




Graph B 




A Graph C 




Time 



Time 



Time 



6. Ty walks 1 mile to the mall. An hour later, he walks \ mile farther to a park 
and eats lunch. Then he walks home. Sketch a graph to show the distance Ty 
traveled compared to time. Use your graph to find the total distance traveled. 



Qy 5-3 ) Functions, Tables, and Graphs 

Make a function table, and graph the resulting ordered pairs. 

7. v=-6x 8. v=4.v-3 9. y - 4x'' 10. -2a- + 4 



Q^ 5-4 j Sequences 

Tell whether the sequence of y-values is arithmetic or geometric. 
Then find y when n = 5. 



11. 



n 


1 


2 


3 


4 


5 


y 


-2 


7 


16 


25 





12. 



n 


1 2 


3 


4 


5 


y 


-5 


15 


-45 


135 





Write a function that describes each sequence. Use the function to 
find the eleventh term in the sequence. 

13.1,2,3,4 14.4.8,12,16 15.11,21,31,41 16.1,4,9,16, 



292 Chapter 5 Graphs and Functions 



' iJV'^fJ ' 




Focus on Problem Solving 








Understand the Problem 

• Sequence and prioritize information 

When you are reading a math problem, putting events in order, or 
in sequence, can help you understand the problem better. It helps 
to prioritize the information when you put it in order. To prioritize, 
you decide which of the information in your list is most important. 
The most important information has highest priority. 



H- 



Use the information in the list or table to answer 



Q The list at right shows all of the things that Roderick 
has to do on Saturday. He starts the day without any 
money. 

a. Which two activities on Roderick's list must be done 
before any of the other activities? Do these two 
activities have higher or lower priority? 

b. Is there more than one way that he can order his 
activities? Explain. 

c. List the order in which Roderick's activities could 
occur on Saturday. 

O Tara and her family will visit Ocean World Park from 
9:30 to 4:00. They want to see the waterskiing show 
at 10:00. Each show in the park is 50 minutes long. 
The time they choose to eat lunch will depend on 
the schedule they choose for seeing the shows. 

a. Which of the information given in the 
paragraph above has the highest priority? 
Which has the lowest priority? 

b. List the order in which they can 
see all of the shows, including 
the time they will see each. 

c. At what time should 
they plan to have lunch? 



each question. 

# 




^ay\6<>''0v\ 



Explore Linear Functions 



Use with Lesson 5-5 



£?. 



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keyword MHItlBia.-! ® 



When the graph of a function is a line or a set of points that lie on a line, 
the function is linear. You can use patterns to explore linear functions. 



Activity 



O The perimeter of a 1 -inch-long square tile is 
4 inches. Place 2 tiles together side by side. 
The perimeter of this figure is 6 inches. 

a. Complete the table at right by adding tiles 
side by side and finding the perimeter of 
each new figure. 

b. If .V equals the number of tiles, what is the 
difference between consecutive .v- values? 
If y equals the perimeter, what is the 
difference between consecutive y-values? 
How do these differences compare? 

c. Graph the ordered pairs from your table 
on a coordinate plane. Is the graph linear? 
What does the table indicate about 

this function? 



1 in. 



1 in. 



1 in. 2 in. 

1 in. 1 in. 



1 in. 



2 in. 



Number of 
Tiles 


Perimeter (in.) 


1 


4 


2 


6 


3 




4 




5 





Draw the pattern at right and complete the next 
two sets of dots in the pattern. 

a. Complete the table at right. Let x equal 

the number of dots in the top row of each set. 
Let y equal the total number of dots in the set. 

b. What is the difference between consecutive 
X- values? What is the difference between 
consecutive y- values? How do these 
differences compare? 

c. Graph the ordered pairs on a coordinate 
plane. Is the graph linear? What does the 
table indicate about this function? 



I • • • • 
• • • 



X y 

2 3 

3 

4 

5 

6 



294 Chapter 5 Graphs and Functions 



s»^ ".. ■ -^ 



Q Use square tiles to model rectangles with the following 
dimensions: 2 x 1, 2 x 2, 2 x 3, 2 x 4, and 2x5. 
The first three rectangles are shown. 

a. Find the perimeter and area of each rectangle. 
Complete the table at right. Let .v equal perimeter 
and y equal area. (To find the area of a rectangle, 
multiply its length by its width. The areas of the first 
two rectangles are shown in the table.) 

b. What is the difference between consecutive .v- values? 
What is the difference between consecutive y- values? 
How do these differences compare? 

c. Using what you have observed in Q and Q, tell 
whether the relationship between .v and y in the table 
is linear. 

d. Graph the ordered pairs from your table on a coordinate plane. 
Does the shape of your graph agree with your answer to c? 



'B 



2 



Rectangle 


Perimeter 

X 


Area 

y 


2 X 1 




2 


2x2 


P 


4 


2x3 


■" " 




2X4 






2 X 5 







Think and Discuss 



1. How can you tell by looking at a function table whether the graph of the 
function is a line? 



2. Is y = A" a linear function? Explain your answer. 



Try This 



1. Use square tiles to model each of the patterns shown below. 

2. Model the next two sets in each pattern using square tiles. 

3. Complete each table. 

4. Graph the ordered pairs in each table, and then tell whether the 
function is linear. 



Pattern 1 



D 



Pattern 2 



Pattern 3 

D a: 



Number of 
Tiles X 


Perimeter 

y 


EJ 


4 


■^ 


8 




12 


i : 


■^^r: 


f^ 





Perimeter 

X 


Area 

y 


8 




12 




16 








mm 





Perimeter 

X 


Area 

y 


4 




6 




8 




r::'" 









5-5 Hands-On Lab 295 



Vocabulary 

linear equation 
linear function 



The graph below shows how far a kayak 
travels down a river if the kayak is 
moving at a rate of 2 miles per hour. 
The graph is linear because all of the 
points fall on a line. It is part of the 
graph of a linear equation. 

A linear equation is an equation ^ 

whose graph is a line. The solutions 
of a linear equation are the points 
that make up its graph. Linear 
equations and linear graphs can be 
different representations of linear functions. 
A linear function is a fimction whose graph 
is a nonvertical line. 

Only two points are needed to draw the graph 
of a linear function. However, graphing a third 
point serves as a check. You can use a fimction 
table to find each ordered pair. 




hy 

/ X 

i^ — i — I — I — \ >- 



2 4 

Hours 



EXAMPLE jlj Graphing Linear Functions 

Graph the linear function y = 2x + 1. 



Ay 




Input 


Rule 


Output 


Ordered Pair 


X 


2x+ 1 


y 


(x,y) 


-1 


2(-1) + 1 


-1 


(-1,-1) 





2(0) + 1 


1 


(0,1) 


1 


2(1)+ 1 


3 


(1,3) 



Place eacli ordered pair on the 
coordinate grid. Then connect the 
points to form a line. 



296 Cliapter 5 Graphs and Functions 



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EXAMPLE 







Physical Science Application 

For every degree that temperature 
increases on the Celsius scale, 
the temperature increases by 
1.8 degrees on the Fahrenheit 
scale. When the temperature is °C, 
it is 32 °F. Write a linear function 
that describes the relationship 
between the Celsius and 
Fahrenheit scales. Then make a 
graph to show the relationship. 

Let.v represent the input, which is the temperature in 
degrees Celsius. Let y represent the output, which is the 
temperature in degrees Fahrenheit. 

The function is v = l-S.v + 32. 

Make a function tahle. hiclude a column for the rule. 




The solutions to a 
function lie on the 
line. 



Input 


Rule 


Output 


X 


1.8x + 32 


y 





1.8 (0) + 32 


32 


15 
30 


1.8(15) + 32 


59 


1.8(30) + 32 


86 



Multiply the input by 1.8 
and then add 32. 



Grapii the ordered pairs (0, 32), (15, 59), 
and (30, 86) from your table. Connect 
the points to form a line. 

Check 

Substitute the ordered pairs into 

the function y = l.Sx + 32. 

32 I 1.8(0) + 32 59 1 1.8(15) + 32 
32 ^ 32 • 59 1 59 • 




10 20 30 40 50 
Temperature (C) 



Since each output y depends on the input x, y is called the dependent 
variable and .v is called the independent variable. 



Ttimk and Discuss 

1. Describe how a linear equation is related to a linear graph. 

2. Explain how to use a graph to find the output value of a linear 
function for a given input value. 



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5-5 Graphing Linear Functions 297 



5-5 



3 



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keyword ■SSQBgB ® 
Exercises 1-8, 9, 11 



GUIDED PRACTICE 



See Example 1 Graph each linear function. 

1. y = .v + 3 



See Example 2 



Input 


Rule 


Output 


Ordered 
Pair 


X 


x+ 3 


y 


(x,y) 


-2 

















2 









2. y = 2.V - 2 



Input 


Rule 


Output 


Ordered 
Pair 


X 


2x-2 


y 


(x,y) 


-1 

















1 









3. A water tanker is used to fill a community pool. The tanker pumps 
750 gallons of water per hour. Write a linear function that describes the 
amount of water in the pool over time. Then make a graph to show the 
amount of water in the pool over the first 6 hours. 



INDEPENDENT PRACTICE 



See Example 1 Graph each linear function. 

4. Y = -A- - 2 



Input 


Rule 


Output 


Ordered 
Pair 


X 


-X - 2 


y 


(x,y) 











1 








2 









6. y = 3.V - 1 



Input 


Rule 


Output 


Ordered 
Pair 


X 


3x- 1 


y 


(x,y) 


-4 

















4 









5. V = A- - 1 



Input 


Rule 


Output 


Ordered 
Pair 


X 


X - 1 


y 


(x,y) 


3 








4 








5 









7. v= 2x + 3 



Input 


Rule 


Output 


Ordered 
Pair 


X 


2x + 3 


y 


(x,y) 


-2 








-1 


















See Example 2 8. Physical Science The temperature of a liquid is increasing at the rate 
of 3 °C per hour. When Joe begins measuring the temperature, it is 40 °C. 
Write a linear function that describes the temperature of the liquid 
over time. Then make a graph to show the temperature over the 
first 12 hours. 



298 Chapter 5 Graphs and Functions 



PRACTICE AND PROBLEM SOLVING 



Extra Practice 



See page EP15. 






10. 



Environment 




The Mauna Loa 
Observatory is 
located on Mauna 
Loa volcano, the 
largest volcano 
on Earth. Its most 
recent eruption 
occurred in 1984. 



12. 



Earth Science The water level in a weU is 100 m. Water is seeping into the 
well and raising the water level by 10 cm per year. Water is also draining out 
of the well at a rate of 2 m per year. What will the water level be in 10 years? 

Multi-Step Graph the function v = — 2.v + 1. If the ordered pair (.v, -5) lies 
on the graph of the function, what is the value of .v? Use your graph to find 
the answer. 



Carbon Dioxide in the Atmosphere 

Mauna Loa, Hawaii 



Environment The Mauna Loa 
Observatory in Hawaii has been 
monitoring carbon dioxide levels 
in the atmosphere since 1957. 

a. The graph is approximately 
linear. About how many parts 
per million (ppm) were added 
each 10-year period? 

b. Given the parts per million in 2000 shown on the graph, about how 
rnany parts per million do you predict there will be in 2020? 

Wliat's the Question? Tron used the equation y = 100 + 25.v to track 
his savings y after .v months. If the answer is $250, what is the question? 




13. Write About It Explain how to graph j' = 
14. 



.::.v 



Challenge Certain bacteria divide ever^' 30 minutes. You can use the function 
)' = 2* to find the number of bacteria after each half-hour period, where .v is 
the number of half-hour periods. Make a table of values for.v = 1, 2, 3, 4, and 5. 
Graph the points. How does the graph differ from those you have seen so far 
in this lesson? 




Test Prep and Spiral Review 



15. Multiple Choice The graph of which linear function passes through the origin? 



CS) y = x + 2 CD y = 3.v CD y = 

16. Short Response Simon graphed the linear function 
y = -.V -I- 3 at right. Explain his error, and graph 

y = -.V -I- 3 correctly on a coordinate grid. 

17. Tell a stor\' that fits the graph. (Lessorn 5-2) 



Time 
Write a function that describes each sequence. (Lesson 5-4) 

18. 15, 10, 5,0 19. -4, -2,0,2 



CS:) y = 2.x + 4 
Ay 



■1.2). 



:«7 



H — 7f — I- 

-a -2 



X' 



(1.4) 
(0,3) 



20. 0.2, 1.2,2.2,3.2, 



5-5 Graphing Linear Functions 299 



LESSON 5 5 ■ Nonlinear 
EXTEiusipw I Functions 



Vocabulary 

nonlinear function 



I EXAMPLE 



As you inflate a balloon, its volume 
increases. The table at right shows 
the increase in volume of a round 
balloon as its radius changes. Do 
you think a graph of the data would 
or would not be a straight line? 
You can make a graph to find out. 



I 300 
= 200 
> 100 





12 3 4 
Radius (in.) 



Radius (in.) 


Volume (in^) 


1 


4.19 


2 


33.52 


3 


113.13 


4 


268.16 


5 


523.75 



A nonlinear function is a function whose graph is not a straight line. 

MM Identifying Graphs of Nonlinear Functions 

Tell whether the graph is linear or nonlinear. 





The graph is not a straight 
line, so it is nonlinear. 



The graph is a straight 
line, so it is linear. 



Helpfulfiyi 



Exponential and 
quadratic functions 
are nonlinear. For 
information on these 
relationships, see 
pp. SB18-SB19 in the 
Skills Bank. 



You can use a function table to determine whetlier ordered pairs describe 
a linear or a nonlinear relationship. 

For a function that has a linear relationship, when the difference between 
each successive input value is constant, the difference between each 
corresponding output value is constant. 

For a function that has a nonlinear relationship, when the difference 
between each successive input value is constant, the difference between 
each corresponding output value varies. 



300 Chapter 5 Graphs and Functions 



EXAMPLE [2] Identifying Nonlinear Relationships in Function Tables 



Tell whether the function represented in each table has a 
linear or nonlinear relationship. 



difference 



Input 


Output 


1 


4 


2 


6 


3 


10 



difference — 1 
difference = 1 

t 

The difference is constant. 

The function represented in the table has a nonlinear 
relationship. 



2 
4 



difference 

t 

r/ie difference varies. 





Input 


Output 


< 
< 


3 


4 


6 


8 


9 


12 



difference = 3 
difference = 3 

t 
Tiie difference is constant 

The function represented in the table has a linear 
relationship. 



difference = 4 
difference - 4 

t 

Tiie difference is constant. 



EXTENSION 



Exercises 



Tell whether the graph is linear or nonlinear. 






Tell whether the function represented in each table has a linear 
or nonlinear relationship. 

4. 



Input 


Output 


5. 


Input 


Output 


6. 


Input 


Output 


2 


5 


1 


6 


4 


25 


4 


7 


2 


9 


8 


36 


6 


9 


3 


14 


12 


49 



Lesson 5-5 Extension 301 



5-6 



B 




Slope and Rat 
of Change 



7.3.6 Solve simple problems mvolving distance, speed and time. 

Understand concepts ot speed and average speed. Understand... 

Baldwin Street, located in 
Dunedin, New Zealand, is 
considered one of the world's 
steepest streets. The slope of 
the street is about }s. 

The slope of a line is a 
measure of its steepness 
and is the ratio of rise to run: 



Vocabulary 

slope 

rate of change 

0the relationship 
between distance, time 
and speed- Find speed, 
distance or time given the 
other two quantities Write 
speed in different units 
(km/h, m/s, cm/s, mi/hr. 
ftysec). 



EXAMPLE 




slope 



rise _ vertical change 

run ~ hori^^ifji^jJ ■x.k 



If a line rises from left to right, 
its slope is positive. If a line falls 
from left to right, its slope is 
negative. 



Ay 



■• Rise,f- 



Run 



H 1 1 1 1 \ >■ 



{ij Identifying the Slope of the Line 

Tell whether the slope is positive or negative. Then find the slope. 




The line falls from left to right. 

f 

The slope is negative. 



slope = ^^^ 



run 

_ 4 



-2 

= -2 



The rise is 4. 
The run is -2. 




The line rises from left to right. 
The slope is positive. 



slope = fif 



The rise is 2. 
The run is 3. 



302 Chapter 5 Graphs and Functions 



[vlJiiLi] Lesson Tutorials Online mv.hrw.com 



You can graph a line if you know its slope and one of its points. 



EXAMPLE [Zj Using Slope and a Point to Graph a Line 

Use the given slope and point to graph each line. 



Slope of a line can be 
represented as a unit 
rate. For example, — - 

4 

can be thought of as 
a rise of — - to a run 
ofl. 



A -^;(-3,2) 

slope = fifl = r|_ or -^ 

From point (-3, 2), move 3 units 
down and 4 units right, or move 
3 units up and 4 units left. Mark 
the points, and draw a line 
through the two points. 



B 3; (-1,-2) 



3 = 



slope = 



Write the slope 
as a fraction. 



rise _ 3 



From point (—1, -2), move 3 units 
up and 1 unit right. Mark the points, 
and draw a line through the 
two points. 




The ratio of two quantities that change, such as slope, is a rate of change . 

A constant rate of change describes changes of the same amount during 
equal intervals. Linear functions have a constant rate of change. The 
graph of a constant rate of change is a line. 

A variable rate ofcliange describes changes of a different amount during 
equal intervals. The graph of a variable rate of change is not a line. 



EXAMPLE 



[ 3 J Identifying Rates of Change in Graphs 

Tell whether each graph shows a constant or variable rate of change. 



*y 



*y 




The graph is a line, so the 
rate of change is constant. 



-<- — I — ) — t — I- 

-4 -2 O 




X 

H > 



The graph is not a line, so the 
rate of change is variable. 



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5-6 Slope and Rates of Change 303 



EXAMPLE [Vj Using Rate of Change to Solve Problems 



The graph shows the distance a bicyclist 
travels over time. Does the bicyclist travel 
at a constant or variable speed? How fast 
does the bicyclist travel? 

The graph is a line, so the bicyclist is 
traveling at a constant speed. 

The amount of distance is the rise, and 
the amount of time is the run. You can 
find the speed by finding the slope. 



slope (speed) = 



rise (distance) _ 15 



run (time) 



1 



L 



The bicyclist travels at 15 miles per hour. 



60 

50 

I 40 

w 

g 30 



1 


1 j 


1 ! 


1 
j 


^ \ r\ ru' 


I 


i ■ 


i 


/ 


j:j„ l_l^ I 




: A 








15; 


/ ' 


i 


^'V 


9 


i.! 


I 


1 1 


1 


151 / M ! ! 


y 1 ! ; j i 


y ! ! i i 1 


/ 1 




1 j 
1 1 


1 


/ i 


h 


1 t 


— —>■ 



1 2 3 
Time (hr) 



Think and Discuss 

1. Describe a line with a negative slope. 

2. Compare constant and variable rates of change. 

3. Give an example of a real-world situation involving a rate of change. 




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keyword ■BEHiMM 
Exercises 1-20, 21, 25, 29 



GUIDtl5?«ACTlCE 



See Example 1 Tell whether the slope is positive ornegative. Then find the slope. 





See Example 2 Use the given slope and point to graph each line. 

L 3. 3; (4, -2) 4. -2; (-3, -2) 5. -|; (0, 5) 



6. f; (-1,1) 



304 Chapter 5 Graphs and Functions 



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See Example 3 Tell whether each graph shows a constant or variable rate of change. 



Ay 



8. 



*y 



H 1 1 ► 



/ X 

H 1 > 



See Example 4 10. The graph shows the distance a trout ^ 

swims over time. Does the trout swim at a> ^g 

a constant or variable speed? How fast S 20 

does the trout swim? 5 




12 3 4 

Time (hr) 



INDERJNPJNT PRACTICE 



See Example 1 Tell whether the slope is positive or negative. Then find the slope. 



11. 



*y 



2- 



-4 -2 o 

(-1,-1) . 

— (0,-4); 



12. 



^y 



X 

-I — I — ■*■ 



< 1 1 1 1 H 



X 

H >■ 



\ 



-4 -2 
^(-2. -1) 

■''(-4, -2) 



See Example 2 Use the given slope and point to graph each line. 

L 13. -1; (-1,4) 14. 4; (-1,-3) 15. |; (3, -1) 



16. f, (0,5) 



See Example 3 Tell whether each graph shows a constant or variable rate of change. 



17. 




18. 




19. 



-f V 




X 

H > 



See Example 4 20. The graph shows the amount of rain 
that falls over time. Does the rain 
fall at a constant or variable rate? 
How much rain falls per hour? 




2 4 6 8 
Time (hr) 



PRACTICE AND PROBLEM SOLVING 



Extra Practice 



See page EP15. 



21 . Multi-Step A Hne has a slope of 5 and passes through the points (4, 3) 
and (2, y). What is the value of y? 

22. A line passes through the origin and has a slope of -=^. Through which 
quadrants does the line pass? 



5-6 Slope and Rates of Change 305 






Agriculture 




This water tower 
can be seen in 
Poteet, Texas, 
where the Poteet 
Strawberry 
Festival- is held 
every April, Known 
as the "Strawberry 
Capital of Texas," 
Poteet produces 
40% of Texas' 
strawberries. 



52 




Graph the Une containing the two points, and then find the slope. 

23. (-2, 13), (1,4) 24. (-2, -6), (2, 2) 25. (-2, -3), (2, 3) 26.(2, -3), (3, -5) 

27. Explain whether you think it would be more difficult to run up a hill with 
a slope of I or a hill with a slope of |. 

•^Q Agriculture The graph at right shows e 

^ the cost per pound of buying strawberries. 

a. Is the cost per pound a constant or variable rate? ^ 

b. Find the slope of the line. 

c. Remember, a unit rate is a rate whose denominator is 
one. Using the slope from part b, find the unit rate 2 4 6 
of the line. What does it tell you? Weight (lb) 

29. Critical Thinking A line has a negative slope. Explain how they- values 
of the line change as the .v- values increase. 

30. What's the Error? Kyle graphed a hue, given a slope of -| and the 
point (2, 3). When he used the slope to find the second point, he 
found (5, 7). Wliat error did Kyle make? 

31. Write About It Explain how to graph a line when given the slope and 
one of the points on the line. 

^ 32. Challenge The population of prairie dogs in a park doubles every year. 
Does this population show a constant or variable rate of change? Explain. 



i 



Test Prep and Spiral Review 



33. Multiple Choice To graph a line, Caelyn plotted the point (2, 1) and then 
used the slope —\ to find another point on the line. Which point could be the 
other point on the line that Caelyn found? 



(S) (1,3) 



CD (4,0) 



CT) (1,-1) 



CE) (0,0) 



34. Multiple Choice A line has a positive slope and passes through the 
point (-1,2). Through which quadrant can the line NOT pass? 



CD Quadrant I 



CS) Quadrant II 



3:) Quadrant III CD Quadrant IV 



35. Short Response Explain how you can use three points on a graph to 
determine whether the rate of change is constant or variable. 



Find each value. (Lesson 1-2) 



36. 3' 



37. 



38. 4' 



39. 10' 



Write a rule for each sequence in words. Then find the next three terms 

(Lesson 5-3) 

40. 3.7,3.2,2.7,2.2 



41. -1,0,1,3,... 



42 3 -1 - -- 



306 Chapter 5 Graphs and Functions 



LAB 



Generate Formulas to 
Convert Units 



Use with Lesson 5-6 



Sf. 



Activity 



Publishers, editors, and graphic designers measure 
lengths in picas. Measure each of the following line 
segments to the nearest inch, and record your results 
in the table. 



o 

o 
o 
o 




Learn It Online 

Lab Resources Online go.hrw.com 



Segment 


Length 
(in.) 


Length 
(picas) 


Ratio of 

Picas to 

Inches 


1 




6 




2 




12 




3 




24 




4 




30 




5 




36 





Thinic and Discuss 



1. Make a Conjecture Make a conjecture about the relationship between picas and 
inches. 

2. Use your conjecture to write a formula relating inches /; to picas p. 

3. How many picas wide is a sheet of paper that is 8^ in. wide? 



Try This 



Using inches forx-coordinates and picas fory-coordinates, write ordered 
pairs for the data in the table. Then plot the points and draw a graph. 

1. What shape is the graph? 

2. Use the graph to find the number of picas that is equal to 3 inches. 

3. Use the graph to find the number of inches that is equal to 27 picas. 

4. A designer is laying out a page in a magazine. The dimensions of a 
photo are 18 picas by 15 picas. She doubles the dimensions of the 
photo. What are the new dimensions of the photo in inches? 



5-6 Hands-On Lab 307 



B 



7.2.6 Draw the graph of a line given its slope and one point on the line or two 
points on the line, 

Tom wants to see how far he can drive on one tank of gas in his new 
hybrid car. He starts with a full tank of 12 gallons of gas and averages 
45 miles per gallon. The graph shows the relationship between 
number of gallons of gas and distance traveled. 



Vocabulary s 

.V- intercept ^ 

y- intercept ^ 

slope-intercept form -g 

3 




\(iO 



V » I » I » T 

xoD ■JOO l\00 500 

Pistance Ti-avelecl (mO 



I 

boo 



I 

TOO 



The points where the line intersects the axes 
can help you understand more about the line. 

The .v-intercept of a line is the .v-coordinate of 
the point where the line intersects the .v-axis. 
The y-coordinate of this point is alw^ays 0. 

The y-intercept of a line is the y-coordinate of 
the point where the line intersects the y-axis. 
The -V-coordinate of this point is always 0. 



7^ 

6 

5 

4 

3 

2 

1 -I- 



^,y-intercept - — 








1 i i 




\._J_ : 


-'- 'V..-- 







x-mtercept 

H 1 h-^ii — *- 



12 3 4 5 6 7 



EXAMPLE 




Finding x- and y-lntercepts 

Find the x- and y-intercepts. 



-% 




The line intersects tlie x-axis at (1, 0). 
The x-intercept is 1. 
The line intersects the y-axis at (0, -2). 
The y-intercept is -2. 



The line intersects the x-axis at (4, 0). 
The -v-intercept is 4. 
The line intersects the y-axis at (0, 2). 
The y-intercept is 2. 



308 Chapter 5 Graphs and Functions 



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If you know the slope of a line and the y- intercept, you can write an 
equation that describes the line. Recall from Lesson 5-6 that the slope 
of a line is the ratio of rise to run. 

The linear equation y = mx + bis written in slope-intercept form, 
where ni is the slope and b is the y-intercept of the line. 



'Slope 



y=mx + b 



y-intercept 



EXAMPLE [2 J Graphing by Using Slope and y-lntercept 

Graph each equation. 



j'i3JJi3JJdJJ3Si 



Since the y-intercept 
is 1, tine point (0, 1) is 
a point on the line. 



A y=|.v-hl 
Step 1: Find ni and b. 

y= r^.\+ 1 
Step 2: Plot (0, 1). 



1)1 = 



b^ 1 




Step 3: Use the slope "r. to plot at least 
1 more point on the line. 

Step 4: Draw a line through the points. 

B 2x + y = 2 

Step 1: Find ni and b. 2x + y = 2 is not in 

2x + y = 2 the form y = mx + b, 
—2x —2x so solve for y. 

y = 2 - 2.V 
y = -2a- -I- 2 

m=-2 b = 2 
Step 2: Plot (0,2). 

Step 3: Use the slope -2 to plot at least 1 more point on the line. 
Step 4: Draw a line through the points. 




EXAMPLE [3] Writing an Equation in Slope-Intercept Form 

Write the equation of the line in slope-intercept form. 




))i — Y^ = 4 The line rises from left to right, 
so the slope is positive. 
The line intersects the y-axis 
at (0, -2), so the y-intercept is -2. 
Substitute for m and b. 



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5-7 Slope-intercept Form 309 



EXAMPLE [7] Using Slope-Intercept Form 



' j,iijijLiaUjiicl?f 



A constant rate of 
change describes a 
linear function. 



Rea's house is 350 meters from her friend's house. Rea walks to 
her friend's house at a constant rate of 50 meters per minute. The 
linear equation y = -50x + 350 represents the distance y that Rea 
has left to walk after x minutes. Graph the equation, and then 
identify the x- and y-intercepts and describe their meanings. 



Use the slope and y-intercept to graph 
the equation. 

Plot (0, 350). Use the slope -50 to plot 
the line down to the x-axis. 

The y-intercept is 350. This represents 
the total distance in meters that Rea has 
to walk. 

The .V- intercept is 7. This represents the 
time in minutes it takes Rea to walk the 
350 meters. 



f 400 
^ 350 
1 300 
I 250 
£ 200 

1 150 

2 100 
5 50 





Rea's Walk 



J i 

. , , _J. ; i I I : : ' 

V ' ! ^_J I )_ ! j j 

\J I I ' ■ i 

1 — \ — I — t — I — I — : — I — I — > 



2 3 4 5 6 7 
Time (min) 



8 9 10 



^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^B 


Think and Discuss 






1. Explain how to find the slope and y-intercept of the line y = 


■-2x- 


-4. 


2. Describe how to graph the equation y = - ~x + 6. 








£?. 



Learn It Online 

Homework Help Online go.hrw.com. 



keyword ■BHWiBJ ® 
Exercises 1-22, 25, 31, 35 



GUIDED PRACTICE 



See Example 1 Find the. v- and y-intercepts. 
1. 




; i : ! ; 


; i i ! 






See Example 2 Graph each equation. 



3. y = ^x - 2 



4. y + 4 = -X 



3,. 



5. y=-|x+l 6. y-2x=-5 



310 Chapter 5 Graphs and Functions 



See Example 3 Write the equation of each line in slope-intercept form. 



^— 2 



-* — I — I — ^ 



-2 O 

- 4 



/■ 




See Example 4 



9. Pete walks down a 280 ft hill at a constant rate of 70 ft per minute. The 
linear equation y - -70.v + 280 represents the distance y Pete has to walk. 
Graph the equation, and then identif\' the .v- and I'-intercepts and describe 
their meanings. 



INDEPENDENT PRACTICE 



See Example 1 Find the. v- and y-intercepts. 
10. i- 



2- 



^ 



-2 O 
-2 



11. 



-i — I — I — I — t*- 




See Example 2 Graph each equation. 

12. j' = 5.V + 3 13. y = 



-X - 6 



16. J' = f-V-5 



17. V- 2 = 7.V 



14. y- 4a = -2 
18. y+^x=4 



15. 1= -.v-l- 3 
19. v= 2x- 5 



See Example 3 Write the equation of each line in slope-intercept form. 



20. 



■*i — I — I — f 



2 O 
-2 

-4 



21. 



H 1 1 1 f> 



■*\ 1 1 \ 1- 



-2 



t 



See Example 4 22. Fred slides down a 200 ft water slide at a constant rate of 10 ft per second. 
The linear equation y - - 1 0.v + 200 represents the distance )/ that Fred has 
to slide. Graph the equation, and then identify the x- and y-intercepts and 



Extra Practice 



describe their meanings. 



PRACTICE AND PROBLEM SOLVING 



See page EP15. 



23. An airplane is cruising at an altitude of 35,000 feet. It begins to descend for 
landing at a rate of 700 feet per minute. Write an equation that represents the 
distance y the airplane has left to descend. Find the slope and x- and 
y-intercepts. What does each intercept represent? 



5-7 Slope-Intercept Form 311 



Use the following values to write an equation in slope-intercept form. 

2A.ni = ^,b = 6 25.m = -7,b = 5 26.iu = l,b=-5 



27. ni = 4.b = 2 



28. m = --lb= -2 



29. m = -l,b = 




Write each equation in slope-intercept form. Use the equation to find the slope 
and the x- and y-intercepts. 



30. 2.V + y=8 
34. -X + y= 15 



Watermeal is the 
world's smallest 
flowering plant. 
The average size of 
a plant is 0.6 mm 
long and 0.3 mm 
wide, and they 
have no roots. 
Watermeal grows 
in dense colonies 
on still ponds and 
rivers. 



31. 4y = -3.V - 12 32. -lOy = 20.v - 30 33. x -I- y = 4 

35. y -I- 30 = 15x 36. 8y = 4x - 16 37. x + y = 

■^Q Life Science Shelley buys a house plant from a nursery. When she brings it 
^ home, it is 5 cm high. The plant grows 2 centimeters each day. 

a. Write an equation expressing this relation, where H is the height of 
the plant and d represents the number of days. 

b. Graph the linear function. 

c. Explain the significance of the point where the line meets the y-axis. 
Will the line ever intersect the x-axis? Explain. 

39. Jani receives a gift card to her favorite smoothie shop for $30. Each smoothie 
costs $2.75 with tax. Write an equation to represent the amount )/ she will 
have left on the card after buying x smoothies. Does she have enough money 
on the gift card to buy 1 1 smoothies? Explain. 

40. Critical Thinking Hayden decides to open a savings account using $25 she 
got for her birthday. Each week she deposits $25. Write an equation in slope- 
intercept form to represent the amount of inoney in her bank account. Is there 
an .V- and y-intercept? If so, what are they, and what does each represent? 

41. Make a Conjecture Make a conjecture about the y-intercept of a line of 
the form ]' = ;»x. 

^ 42. What's the Error? For the equation y = -2x -I- 3, a student says the 
y-intercept is —2 and the slope is 3. Identify the student's error. 

43. Write About It Give a real-world example that could represent a line with 
a slope of 2 and ay-intercept of 10. 



^ 44. Challenge What value of » in the equation ?l\ 
a slope of 8? 



2y = 4 would give the line 



Test Prep and Spiral Review 



45. Multiple Choice Which equation does NOT represent a line with an x-intercept of 3? 

CA> y = -2x -I- 6 cb:> y = -|x + \ (X) y = fx - 2 C5) y = 3x - i 

46. Short Response Graph the equation y = -|x -I- 2 . Find thex- and y-intercepts. 

47. A car travels 150 miles in 3 hours. What is the unit rate of speed per hour? (Lesson 4-1) 

48. Tell whether the ratios | and |^ are proportional. (Lesson 4-2) 



312 Chapter 5 Graphs and Functions 



7.2.7 



Identify situations that 
situations, and recogni 



'ocabulary 



irect variation 
onstant of variation 



You can read direct 
variation as "y varies 
directly as x" or "y is 
directly proportional 
to x" or "y varies 
with X." 



-K= mx where the unit rate 
mis the slope of the line. 



involve proportional relationships, draw graphs representing these 
ze that these situations are described by a linear function in the form 

An Eastern box turtle can travel at a 
speed of about 18 feet per minute. 
The chart shows the distance an 
Eastern box turtle can travel when 
moving at a constant speed. 

The distance traveled is found by 
multiplying time by 18. Distance 
and time are directly proportional. 

Direct variation is a linear relationship 
between two variables that can be written in the form )' = kx or A.- = j, 
where A.' ^ 0. The fixed number k in a direct variation equation is the 
constant of variation . 




Time (min) 


112 3 4 


Distance (ft) 


18 36 54 72 



y = kx k = — 



To check whether an ecjuation represents a direct variation, solve 
for y. If the equation can be written as j' = kx, then it represents a 
direct variation. 



EXAMPLE fij Identifying a Direct Variation from an Equation 



Tell whether each equation represents a direct variation. If so, 
identify the constant of variation. 



A 2 y = .V 

2 ~ 2 



Solve the equation for y. Divide botli sides by 2. 



Write f as ^x. 



The equation is in the form y = kx, so the original equation 
2y = .V is a direct variation. The constant of variation is }j. 



y+l = 2x 
y + 1 = 2.Y 
-1 



Solve the equation for y Subtract 1 from both sides. 



-1 



y = 2.V - 1 
The equation is not in the form y = kx, so y + 1 = 2x is not a 
direct variation. 



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5-8 Direct Variation 313 



The equation y = k.x can be solved for the constant of variation, k = y. 
If ^ is the same for all ordered pairs in a set of data, then the data set 
represents a direct variation. To write a direct variation equation for a 
set of data, substitute the value of ^ for k iny = kx. 



EXAMPLE 21 identifying a Direct Variation from a Table 



IJJaJ^JuJJJIjj'i 



In a direct variation 
where k is positive, 
when X increases, y 
also increases; when 
X decreases, y also 
decreases. 



Tell whether each set of data represents a direct variation. If 
so, identify the constant of variation and then write the direct 
variation equation. 



Find \ for each 
ordered pair. 



Weight (lb) 


1 2 3 


Price ($) 


3 6 1 9 



3' _ 3 _ 



3' _ 6 _ 



^=9 = 3 
A 3 -5 



k = 3 for each 
ordered pair 



The data represent a direct variation where k = 3. The equation 
is V = 3.r. 



Constant Speed (mi/h) 10 20 30 




Find \ for each 


Time(h) 3 1.5 1 


ordered pair 


y _ 3 3' _ 1.5 _ 3 3' _ 1 _ 1 
X 10 v 20 40 X 30 30 


k is not the sam 



each ordered pair 
The data do not represent a direct variation. 



The graph of any direct variation is a straight line that passes through the 
origin, (0, 0). The slope of a line of direct variation is the constant of 
variation, A.". 



EXAMPLE {3] identifying a Direct Variation from a Grapli 

Tell whether each graph represents a direct variation. If so, 
identify the constant of variation and then write the direct 
variation equation. 



2- 



•«H 1 f 



HelDf uliHinfe 



In a direct variation, 
the slope, k, 
represents a constant 
rate of change. 



2 01 



The graph is a line 
through (0, 0). This is a 
direct variation. The slope 
of the line is 2, so k = 2. 
The equation is y = 2x. 




The line does not 
pass through (0, 0). 
This is not a direct 
variation. 



314 Chapter 5 Graphs and Functions 



I fvh'j] Lesson Tutorials Online my.hrw.com 



EXAMPLE 



C3 



In this problem the 
variable x represents 
time and y represents 
distance, so 162 will 
be substituted for y. 



Life Science Application 

An Eastern box turtle travels on the ground at a speed of about 
18 feet per minute. 

a. Write a direct variation equation for the distance y an Eastern 
box turtle travels in .v minutes. 

distance = 18 feet per minute times number of minutes Use the 
y = 18 • X formula 

y = kx. 

y= \8x . k = 18 

h. Graph the data. 

Make a table. Since time cannot be 
negative, use nonnegative numbers for x. 



X 


y = 18x 


y 


(x,y) 





y = 18(0) 





(0,0) 


1 


y= 18(1) 


18 


(1, 18) 


2 


y= 18(2) 


36 


(2, 36) 



Use the ordered pairs to plot the points 
on a coordinate plane. Connect the 
points in a straight line. Label the axes. 

Check 

y = IB.v is in slope-intercept form with 
in - 18 and b = 0. The graph shows a 
slope of 18 and a j'-intercept of 0. 




2 3 4 5 
Time (min) 



c. How long does it take an Eastern box turtle to travel 162 feet? 

Find the value of x when y = 162. 

y = 18.V Write the equation for the direct variation. 

162 = 18.T Substitute 162 for y. 

Divide both sides by 18. 

9 = A- 
It will take an Eastern box turtle 9 minutes to travel 162 feet. 



162 _ 18.Y 
18 ~ 18 



Think and Discuss 

1. Explain how to use a table of data to check whether the 
relationship between two variables is a direct variation. 

2. Describe how to recognize a direct variation from an equation, 
from a table, and from a graph. 

3. Discuss why every direct variation equation is a linear equation, 
but not every linear equation is a direct variation equation. 



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5-8 Direct Variation 315 



5-8 






Exercises 



Homework Help Online go.hrw.com, 



keyword MiaiifcgJ ® 
Exercises 1-18, 19, 25, 27 



GUIDED PRACTICE 



See Example 1 Tell whether each equation represents a direct variation. If so, identify the 
I constant of variation. 

L 1. v= 5.V + 8 



2. y = 3.6.V 



3. 8y = 2x 



4. X = 3y + 1 



See Example 2 Tell whether each set of data or graph represents a direct variation. If so, identify 
the constant of variation and then write the direct variation equation. 

6. 



See Example 3 



Number of Boxes 2 


3 


4 


Rolls of Tape Needed 1 


2 


5 



X 2 


4 8 


y 3 


7 15 




H 1 \ H 



See Example 4 



-2: 



9. Physical Science Belinda's garden hose sprays about 4 gallons of water 
each minute. 

a. Write a direct variation equation for the number of gallons y Belinda 
uses during .v minutes of watering her garden. 

b. Graph the data. 

c. How many gallons of water does Belinda use in 20 minutes? 



INDEPENDENT PRACTICE 



See Example 1 Tell whether each equation represents a direct variation. If so, identify the 
constant of variation. 



10. y = 



11. --^^ 

v 3 



12. 3v= 15 - 6x 



13. 3xy = 9.\- 



X 


7,8 9 


y 


0.5 1.2 1 1.5 



Cans of Food 


2 


4 


6 


Dinners Made 


4 


8 


12 



See Example 2 Tell whether each set of data or graph represents a direct variation. If so, identify 
the constant of variation and then write the direct variation equation. 

\ 14. m^ ^ ^ 15. 



See Example 3 16. ci 





10 20 30 40 50 60 70 80 



12 3 4 5 



316 Chapter 5 Graphs and Functions 



See Example 4 18. Physical Science Neil Armstrong's weight on tlie moon was about ^ his 
weight on Eartli. 

a. Write a direct variation equation for the number of pounds y an object 
on the moon weighs if the object weighs .v pounds on Eartli. 

b. Graph the data. 

c. Li would weigh 24 pounds on the moon. What does he weigh on Earth? 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EP15. 





Sea snakes are 
found in warm 
waters ranging 
from the Indian 
Ocean to the 
Pacific. They do 
not have gills 
and must sur- 
face regularly to 
breathe. 



Write an equation for the direct variation that Includes each point. 

19. (7,2) 20. (6,30) 21. (4,8) 22. (17,31) 

23. If y varies directly as .v, and y = 8 when .v = 2, find y when x =10. 

24. Is a direct variation a function? E.xpiain. 

Tell whether each relationship is a direct variation. Explain. 

25. pay per hour and the number of hours worked 

26. pay per hour and the number of hours worked, including a $100 bonus 

^^7f Life Science A sea snake can swim at a rate of 60 meters per minute, fiow far 
can a sea snake swim in half an hour? 

28. Critical Thinking If you double an .v- value in a direct variation equation, 
will the 3'- value double? E.\plain your answer. 3: 

2 
1 



29. What's the Error? Phil says that the graph represents a 
direct variation because it passes through the origin. 
What's the error? 




12 3 4 5 



30. Write About It Compare the graphs of a direct variation equation with a 
slope of 3 and an equation with the same slope and a v-intercept of 2. 

^31. Challenge Explain why the graph of a line that does not pass through the 
origin cannot be a direct variation. 



i 



Test Prep and Spiral Review 



32. Multiple Choice Which equation does NOT show direct variation? 

(S) y= Kiv CD y - 19 = x - 19 <X) 2O3' = .v CE) y = 25 

33. Short Response Ron buys 5 pounds of apples for $3.25. Write a direct variation 
equation for the cost y of .v pounds of apples. Find the cost of 2 1 lbs of apples. 



Add. Write each answer in simplest form. (Lesson 3 8j 



34. U + 3 



3 



35. 7| + 



36. 9i + 6| 



37.4^ + 31 



Plot each point on a coordinate plane. (Lesson 51) 

38. A(-4,l) 39. B(0, 3) 40. C(2, -2) 41. D(-l,4) 



5-8 Direct Variation 317 



LESSON 5-8 



EXTEiusiofll Inverse Variation 



Vocabulary 

inverse variation 



Inverse variation is a relationship between two variables that can be 
written in the form y = ~, or xy = k, where k is a nonzero constant and 
.V ?t 0. 



y = — xy = k 



In an inverse variation, the product of x and )' is constant. 

EXAMPLE MJ Identifying an Inverse Variation 

Tell whether each relationship is an inverse variation. Explain. 

Find the product of xy. 



You can read inverse 
variation as "y 
varies inversely as 
x" or "y is inversely 
proportional to x." 



X 


2 3 4 


y 


12 8 6 



2(12) = 24 3(8) = 24 4(6) = 24 Substitute for x and y. 
The product for xj' is constant, so the relationship is an inverse 
variation with A: = 24. 



Find tlie product of xy. 



X 


5 7 9 


y 80 , 75 70 



5(80) = 400 7(75) = 525 9(70) = 630 

The product for .xy is not constant, so the relationship is not an 

inverse variation. 



EXAIVIPLE |2i Geometry Application 

David is building a rectangular flowerbed. He has soil to cover 
48 square feet. The flowerbed can be 4, 6, or 12 feet long. For 
each length x, find the width of the flowerbed y to use all the 
soil. 

The area A of the flowerbed is a constant k. The length .v times die width 
y must equal the area, 48. The equation xy - 48 is an inverse variation. 
xy = k xy = k xy = k Use xy = /c. 

4y =48 By = 48 12y = 48 Substitute for x and l<. 

y = 12 y—8 y—^ Solve for y. 

David can build a flowerbed that is 4 ft long by 12 ft wide, 6 ft long by 
8 ft wide, or 12 ft long by 4 ft wide. 



318 Chapter 5 Graphs and Functions 



EXAMPLE 



9 



An inverse variation can also be 
identified by its graph. Since k is a 
nonzero constant, xy + 0. Therefore, 
neither .v nor y can equal 0, and no 
solution points will be on the .v-axis 
ory-axis. 



Identifying a Graph of an Inverse Variation 

Tell whether each graph represents an inverse variation. Explain. 




A ^ 



^25 
g 20 

•£ 10 

g 5 
E 
< O 



Wages 



12 3 4 5 
Time (h) 



Identify points on the graph. 
Use the equation xy = k. 
(1)5 = 5, (3)15 = 45, (5)25 = 125 
The values of k are not constant. 
The graph does not represent 
an inverse variation. 



Relay Runners 



12 3 4 5 6 7 
Number of Runners 

Identity points on the graph. 
Use the equation .v)' = k. 
(2)6 = 12, (3)4 = 12, (4)3 = 12 
The values of k are constant. 
The graph represents an 
inverse variation. 



i) 



EXTENSION 



Exercises 



Determine whether each set of data shows inverse or direct variation. 




2. 



12345 012345 

3. If .V and y show inverse variation, and you know that y = 10 when .v = 6, find 
y when X =12. 

4. You are on a trip to a museum that is 120 miles away. You know that if you 
travel 60 miles per hour, you will arrive in 2 hours. How long will the trip take 
if you travel at 30 miles per hour? 

5. Write About It Explain the difference between a direct variation and an 
inverse variation. 

6. Critical Thinking The definition of inverse variation says that k is a nonzero 
constant. What would y = | represent if k were 0? 



Lesson 5-8 Extension 319 



CHAPTER 




Ready To Go On? d:t^«">'0"'"- 



SECTION SB 



Resources Online go.hrw.com, 
IBWB^MsTo RTGOSBlGoj 



Quiz for Lessons 5-5 Through 5-8 

(^ 5-5 ] Graphing Linear Functions 

Graph each hnear function. 

1. y = X - 4 2. y - 2.v - 5 3. y - -x + 7 4. y = - 

5. A freight train travels 50 miles per hour. Write a linear function that 
describes the distance the train travels over time. Then make a graph 
to show the distance the train travels over the first 9 hours. 



2a- + 1 



Q) 5-6 ] Slope and Rates of Change 



Tell whether each graph shows a constant or variable rate of change. 
If constant, find the slope. 



*y 



(-1,4)«4 



(0, ^)\ 

H \ 1- 



-2 



7. 



Ay 



Jk. 



X 



/(2,3) 
/l, 1) 



er 



5-7 ] Slope-Intercept Form 
Write the equation of the line in slope-intercept form. 



•*) 1 1 H 



-2 O 
-2 



10. 



H 1 1 h*- 



: \_.. 




8. 



*y 



(-1,2), 



(2,4) 



X 



11. 




12. A skier skis down a 108-meter ramp at a constant rate of 27 m per second. The linear 
equation )' = -27.v -t- 108 represents tlie distance )/ die skier has left to ski. Graph the 
equation and then identify tlie .v- and ]'-intercepts and describe dieir meanings. 

Q) 5-8 j Direct Variation 

Tell whether each set of data represents a direct variation. If so, identify the constant 
of variation, and then write the direct variation equation. 



13. 



Weight (lb) 


1 


2 3 


Price ($) 


1.50 


3.00 4.50 



14. 



320 Chapter 5 Graphs and Functions 




CONNECTIONS 




The Alabama National Fair where can you see trapeze acts, 
a cheerleading competition, and racing pigs all in one place? 
Since the 1950s, the annual Alabama National Fair has brought 
all of this — and much more — to the Agricultural Center and 
Fairgrounds in Montgomery. 

A teacher is planning to take some of her students to the fair. 

1. The Alabama National Fair has one admission fee for adults and 
a different fee for students. The table can be used to determine 
how much it will cost for the teacher and her students to attend 
the fair. Complete the table. 

2. What is the fair's admission fee for adults? 
What is the fair's admission fee for students? 

3. Suppose x represents the number of 
students that the teacher brings to the 
fair and v represents the total cost. Write a 
function that describes the data in the table. 

4. Use the function you wrote in Problem 3 to 
find the total cost of bringing 14 students to 
the fair. 

5. Make a graph that shows the total cost as a 
function of the number of students. 

6. What is the slope of the line in your graph? 

7. A count\' fair offers admission to a teacher 
and any number of students for $85. For 
what number of students would it be 
less expensive for the teacher to take her 
students to the county fair than the Alabama 
National Fair? 



ALABAMA 



ky 



Montgomery 



ADMIT ONE 


Number of 
Students 


Rule 


Total 
Cost 







$9 


1 


9 + 7(1) 


$16 


2 




$23 


3 


9 + 7(3) 




4 




$37 


6 


9 + 7(6) 




8 




$65 


12 














Real-World Connections 321 



6§asjiMe 

Clothes Encounters 

Five students from the same math class met to 
study for an upcoming test. They sat around a 
circular table with seat 1 and seat 5 next to each 
other. No two students were wearing the same 
color of shirt or the same type of shoes. From the 
clues provided, determine where each student sat, 
each student's shirt color, and what type of shoes 
each student was wearing. 

Q The girls' shoes were sandals, flip-flops, and boots. 

Q Robin, wearing a blue shirt, was sitting next to the 
person wearing the green shirt. She was not sitting 
next to the person wearing the orange shirt. 

Q Lila was sitting between the person wearing sandals 
and the person in the yellow shirt. 

QThe boy who was wearing the tennis shoes was wearing the orange shirt. 

April had on flip-flops and was sitting between Lila and Charles. 

Glenn was wearing loafers, but his shirt was not brown. 

Q Robin sat in seat 1. 

You can use a chart like the one below to organize the information given. 
Put X's in the spaces where the information is false and O's in the spaces 
where the information is true. Some of the information from the first 
two clues has been included on the chart already. You will need to read 
through the clues several times and use logic to complete the chart. 





Lila 












X 
















X 


X 


Robin 












O 


X 


X 


X 


X 








X 


X 


April 












X 
















X 


X 


Charles 












X 




















Glenn 












X 





















322 Chapter 5 Graphs and Functions 




Materials 

• 6 sheets of 
unlined paper 

• scissors 

• markers 



^^ 



PROJECT 



Graphs and 

Functions 

Fold-A-Books 

These handy books will store your notes from each 
lesson of the chapter. 

Directions 

O Fold a sheet of paper in half down the middle. 
Then open the paper and lay it flat so it forms a 
peak. Figure A 

Q Fold the left and right edges to the crease in the 
middle. When you're done, the paper will be folded 
into four sections, accordion- st\'le. Figure B 

Pinch the middle sections together. Use scissors 
to cut a slit down the center of these sections, 
stopping when you get to the folds. Figure C 

Q Hold the paper on either side of the slit. As you 
open the slit, the paper will form a 
four-page book. Figure D 




I 




&f 




Crease the top edges and fold the book 
closed. Repeat all the steps to make five 
more books. 

Taking Note of the Math 

On the cover of each book, write the 
number and name of a lesson from 
the chapter. Use the remaining pages 
to take notes on the lesson. , 



S) 



e'* 





O 


.--^ 


N 


G) 


tammsm 


—a 






^ 





^/^, 



%r.f% 



<Q 



^?4 



9 



<^ 



Q) 



Le, 



G) 



l9 vS, 



|^*n 5-4 



It's in the Bag! 323 



e^we 



}rirp<: 



CHAPTER 






'^'\- . 



Vocabulary 

arithmetic sequence . . . 288 

common difference 288 

constant of variation ...313 

coordinate plane 276 

direct variation 313 

function 284 

geometric sequence 288 

input 284 



linear equation 296 

linear function 296 

ordered pair 276 

origin 276 

output 284 

quadrant 276 

rate of change 303 

sequence 288 



slope 302 

slope-intercept form . . . 309 

term 288 

X-axis 276 

A-intercept 308 

y-axis 276 

y-intercept 308 



Complete the sentences below with vocabulary words from the list above. 

1 . A ( n) ? is an ordered list of numbers. 

2. A(n) ? gives exactly one output for every input. 

3. A(n) ? is a function whose graph is a nonvertical line. 



EXAMPLES 



5-IIJ The Coordinate Plane (pp 276-279) 

I Plot each point on a coordinate plane. 

■ M(-3, 1) 

Start at the origin. 
Move 3 units left 
and 1 unit up. 



M(-3, 1) 



R(3, -4) -" 


-2 


Start at the origin. 


--|-72- 


Move 3 units right H— 


— ^ -4- 


and 4 units down. 


' 


Give the coordinates 





ff(3, -4) 



of each point and 
tell which quadrant 
contains it. 

^(-3, 2); II 
B(2, -3); IV 
C(-2, -3); III 
D(3, 2);I 




EXERCISES 



Plot each point on a coordinate plane. 

4. ^(4,2) 5. B(-4, -2) 

6. C(-2,4) 7. D(2, -4) 

Give the coordinates of each point and tell 
which quadrant contains it. 

Ay 



4 - 



2 



-4 1 1 1- 



Mi 



-2 O. 
-2- 



-• — I — I — I- 



X 



8. / 



9. K 



10. L 



11. M 



324 Chapter 5 Graphs and Functions 



EXAMPLES 



EXERCISES 



j^-23 Interpreting Graphs (pp. 280-283) 

■ Ari visits his grandmother, who lives 
45 miles away. After the visit, he returns 
home, stopping for gas along the way. 
Sketch a graph to show the distance Ari 
traveled compared to time. Use your 
graph to find the total distance traveled. 




Time 

The graph increases from to 45 miles 
and then decreases from 45 to miles. 
The distance does not change while Ari 
visits his grandmother and stops for gas. 
Ari traveled a total of 90 miles. 



12. Amanda walks 1.5 miles to school in 
the morning. After school, she walks 
0.5 mile to the public libraiy. After she 
has chosen her books, she walks 2 miles 
home. Sketch a graph to show the 
distance Amanda traveled compared 

to time. Use your graph to find the 
total distance traveled. 

13. Joel rides his bike to the park, 12 miles 
away, to meet his friends. He then rides 
an additional 6 miles to the grocery 
store and then 18 miles back home. 
Sketch a graph to show the distance loel 
traveled compared to time. Use your 
graph to find the total distance traveled. 



5-3] Functions, Tables, and Graphs (pp 284-287) 



Find the output for each input. 

1' = 3.V + 4 



L 



Input 


Rule 


Output 


X 


3x + 4 


y 


-1 


3(-1) + 4 


1 





3(0) + 4 


4 


2 


3(2) + 4 


10 



Find the output for each input. 

14. i' = -v^ - 1 



Input 


Rule 


Output 


X 


x^-l 


y 


-2 






3 






5 







5-4] Sequences (pp. 288-291) 

■ Write a function that 

describes the sequence. 

Use the function to find the 
eighth term in the sequence. 

3.6,9, 12,... 

Function: y = 3ii 
When n = 8,v = 24. 



n 


Rule 


y 


1 


1 -3 


3 


2 


2 • 3 


6 
9 
12 


3 


3- 3 


4 


4- 3 



Write a function that describes each 
sequence. Use the function to find the 
eighth term in the sequence. 

15. 25, 50, 75. 100 

16. -3, -2, -1,0, ... 

17. -4, -1,2,5,... 

18. 4,6,8, 10, . .. 



'Ailh'j Lesson Tutorials OnlinE mv.hrw.com 



Study Guide: Review 325 



EXAMPLES 



5-5 ] Graphing Linear Functions (pp. 296-299) 
■ Graph the Hnear function y = —x + 2. 



EXERCISES 



Input 


Output 


Ordered 
Pair 


X 


y 


(x,y) 


-1 


3 


(-1,3) 





2 


(0,2) 


2 





(2,0) 



*y 



2V 



-2 



X 

-♦ — f*- 



Graph each Hnear function. 

19. y = 2.V-1 

20. y = -3x 

21. y = A--3 

22. y = 2.x + 4 

23. y = .V - 6 

24. V = 3.V - 9 



5-6 j Slope and Rates of Change (pp 302-306) 

■ Tell whether the graph shows a constant 
or variable rate of change. If constant, 



find the slope. 

The graph is a line, 
so the rate of change 
is constant. 

slope = f^ 



Ay 






-4 



P^+- 






5 



(-2, -2) 
3 



X 



Tell whether each graph shows a constant 
or variable rate of change. If constant, find 
the slope. 



25. 



*y 



2 



26. 



(0, 1) 

— I — I- 



X 



/<-2,-2. 




5-7] Slope-intercept Form (pp. 308-312) 

■ Write the equation of the line in slope- 
intercept form. 

^y Find m and b. 




Ill 



\:b = 



Substitute. 



27. Write the equation 
of the line in 
slope-intercept 
form. 



28. Graph y = -i.v + 4. 



Ay 



./- 



::k 



■rr-^ — I — ^ 



X 



5-8] Direct Variation (pp. 313-317) 

■ Tell whether each equation represents 
a direct variation. If so, identify the 
constant of variation. 



So/i/e tirie equation for y. 
Divide by 3 on both sides. 
The constant of variation is |. 



3j/ = 


.X 


3y_ 


.V 


3 


3 


y = 


i- 



Tell whether the set of data represents a 
direct variation. If so, identify the constant 
of variation and then write the direct 
variation equation. 

29. 30. 



^ I 1 



y I 18 I 36 54 



X 


1 2 3 


y 


4 j 7 


10 



326 Chapter 5 Graphs and Functions 



Chapter Test 



CHAPTER 



5 



Plot each point on a coordinate plane. Then identify the quadrant that 
contains each point. 

1. L(4, -3) 2. M(-5, 2) 3. N(7,l) 4. 0(-7, -2) 

5. Ian jogs 4 miles to the lake and then rests for 30 min before jogging 
home. Sketch a graph to show the distance Ian traveled compared 
to time. Use your graph to find the total distance traveled. 

Write a function that describes each sequence. Use the function to 
find the eleventh term in the sequence. 

6.1,3,5,7... 7.11,21,31,41... 8.0,3,8,15... 

Make a table of values to graph each linear function. 

9. y = 3.v-4 10. j' = .v-8 11. v = 2.v + 7 12. 3'=-.v+l 

Tell whether each graph shows a constant or variable rate of change. 
If constant, find the slope. 



13. 



ly 



14.. 



-4 o 
-4 



14. 



Ay 



i(3, 1) 

H 1 *■ 



(-I.3V 



4 X 

*(1, -2) 



-4 



4 

^(2, -3) 



Write the equation of each line in slope-intercept form. 



15. 




16. 



Ay 




17. Paula walks up a 520-meter hill at a pace of 40 meters per minute. The 
linear equation r = -40.Y -I- 520 represents the distance y that Paula has 
left to walk after .v minutes. Graph the equation, and then identif\' the 
X- and y-intercepts and describe their meanings. 



Tell whether each equation represents a direct variation. If so, identify the 
constant of variation. 



18. 5y = lOx 



19. y-3 = x 



20. X + y = 4 



21. -7x^y 



Chapter 5 Test 327 




Test Tackier 




Extended Response: Understand the Scores 

Extended-response test items usually involve multiple steps and require 
a detailed explanation. The items are scored using a 4-point rubric. A 
complete and correct response is worth 4 points, a partial response is 
worth 2 to 3 points, an incorrect response with no work shown is worth 
1 point, and no response at all is worth points. 



EXAMPLE 



Extended Response A 10-pound bag of apples costs $4. Write and solve a 
proportion to find how much a 15-pound bag of apples would cost at the same 
rate. Explain how the increase in weight is related to the increase in cost. 

Here are examples of how different responses were scored using the scoring 
rubric shown. 



4-point response: 

Let c = fhe cost of fhe 15 Ih hacj. 
W pounds 15 pounds 



15 



The 15 Ih hacj costs $6. 

For every addifional 5 pounds, 
fhe cost increases fey 2 dollars. 



$H 






10 


• c = 


H • 




10c 
10 ~ 


60 
10 




c = 


6 



3-point response: 

Let c — fhe cost of fhe 15 Ih hac 



10^ 



nds 



15 . 



nds 



$H 






c 


10 


• c - 


= H • 


15 




10c 
10 ' 


60 
' 10 






c - 


-6 





The 15 Ih bacj costs $6. 

For every addifional 5 pounds, 
fhe cosf increases fey 6 dollars. 



2-point response: 



Let c — fhe cost 


of fhe 


appi 


es 


/ pounds 




c 








SH 


15 


pounds 




10 ■ 15 = 


H 


c 








150 


He 










H 


H 










31.5 = 


c 











The proportion is set up incorrectly, and 
no explanation is given. 



The proportion is set up and solved 
correctly, and all work is shown, but the 
explanation is incorrect. 

1 -point response: 

375 = c 

The answer is incorrect, no work is shown, 
and no explanation is given. 



328 Chapter 5 Graphs and Functions 




After you complete an extended- 
response test item, double-check that 
you have answered all parts. 



Read each test item and answer 
the questions that follow using the 
scoring rubric below. 



Scoring Rubric 

4 points: The student correctly answers all 
parts of the question, shows all work, and 
provides a complete and correct explanation. 

3 points: The student answers all parts of 
the question, shows all work, and provides a 
complete explanation that demonstrates 
understanding, but the student makes minor 
errors in computation. 

2 points: The student does not answer all 
parts of the question but shows all work and 
provides a complete and correct explanation 
for the parts answered, or the student 
correctly answers all parts of the question but 
does not show all work or does not provide 
an explanation. 

1 point: The student gives incorrect answers 
and shows little or no work or explanation, 
or the student does not follow directions. 

points: The student gives no response. 



Item A 

Extended Response Alex drew a 
model of a birdhouse using a scale 
of 1 inch to 3 inches. On the 
drawing, the house is 6 inches tall. 
Define a variable, and then wTite 
and solve a proportion to find how 
many inches tall the actual 
birdhouse is. 



1. Should the response shown receive a 
score of 4 points? Why or why not? 



/ inch 3 inche 



6 inches h 

1 • h = 3 ■ 6 
h= 18 
The actual birdhouse is 18 inches -tall. 



Item B 

Extended Response Use a table 
to find a rule that describes the 
relationship between the first four 
terms of the sequence 2, 4, 8, 16, . . . 
and their positions in the sequence. 
Then find the next three terms in 
the sequence. 



2. What should you add to the response 
shown, if anything, so that it receives 
full credit? 



n 


; 


2 


3 


H 


■Ru/e 


Z' 


2' 


2' 


V 


y 


2 


H 


8 


16 



E-ach ferm is 2. finnes as qreaf as the 
term before if. The rule is 2". 



Item C 

Extended Response The figures 
are similar. Find the value of .v and 
the sum of the side lengths of one 
of the figures. 



9ft 



4ft 



8ft 




6 ft 



X- 3ft 



3. What needs to be included in 
a response that would receive 
4 points? 

4. Write a response that would 
receive full credit. 



Test Tackier 329 



CHAPTER 



5 



ra ISTEP+ 
^ Test Prep 



j:* 



Learn It Online 

State Test Practice go.hrw.com, 




Applied Skills Assessment 

Constructed Response 

1 . A teacher discussed 1 1 2 of the 1 54 
pages of the textbook. What portion 
of the pages did the teacher discuss? 
Write your answer as a decimal 
rounded to the nearest thousandth 
and as a fraction in simplest form. 

2. A bag of nickels and quarters contains 
four times as many nickels as quarters. 
The total value of the coins in the bag 
is $1.35. 

a. How many nickels are in the bag? 

b. How many quarters are in the bag? 

3. Describe in what order you would 
perform the operations to find the 
value of (4 • 4 - 6)^ + (5 • 7). 

4. A recipe calls for | cup flour and | cup 
butter. Does the recipe require more 
flour or butter? Is this still true if the 
recipe is doubled? Explain how you 
determined your answer. 

Extended Response 

5. A bus travels at an average rate of 
50 miles per hour from Nashville, 
Tennessee, to El Paso, Texas. To find 
the distance y traveled in x hours, use 
the equation y = 50x. 

a. Make a table of ordered pairs using 
the domain x = 1, 2, 3, 4, and 5. 

b. Graph the solutions from the table of 
ordered pairs on a coordinate plane. 

c. Brett leaves Nashville by bus at 
6:00 A.M. He needs to be in El Paso 
by 5:00 a.m. the following day. If 
Nashville is 1,100 miles from El Paso, 
will Brett make it on time? Explain 
how you determined your answer. 

330 Chapter 5 Graphs and Functions 



Multiple-Choice Assessment 

6. The fraction j^ is found between which 
pair of numbers on a number line? 



A. ^ and 1 

B. I|and| 



C. ^and 



11 

24 



D. 4 and I 



7. Which description shows the 

relationship between a term and n, 
its position in the sequence? 



Position Value of Term 


1 


1.25 


2 


3.25 


3 


5.25 


4 


7.25 


n 





A. Add 1.25 to n. 

B. Add 1 to n and multiply by 2. 

C. Multiply n by 1 and add 1.25. 

D. Multiply n by 2 and subtract 0.75. 



8. For which equation is X 
solution? 

A. 2x - 20 = 



B. Ix + 2 = 



-10 the 



C ' 



5^ 



2 = 
D. -2x + 20 = 



9. What is the least common multiple of 
10, 25, and 30? 

A. 5 C. 150 

B. 50 D. 200 

10. Which problem situation matches the 
equation below? 

X + 55 = 92 

A. Liam has 55 tiles but needs a total 
of 92 to complete a project. How 
many more tiles does Liam need? 

B. Cher spent $55 at the market and 
has only $92 left. How much did 
Cher start with? 

C. Byron drove 55 miles each day for 
92 days. How many total miles did 
he drive? 

D. For every 55 students who buy 
"spirit wear," the boosters donate 
$92. How many students have 
bought spirit wear so far? 

11. A recipe that makes 2 cups of 
guacamole dip calls for l| cups of 
mashed avocados. How much avocado 
is needed to make 4 cups of dip with 
this recipe? 

A. 3.25 cups C. 3.75 cups 

B. 3.5 cups D. 4 cups 

12. Which ordered pair is located on the 
X-axis? 



A. (0, -5) 

B. (5, -5) 



C. (-5, 0) 

D. (1, -5) 



13. Which ordered pair is NOT a solution 
of y = 5x - 4? 

A. (2, 6) C. (1, 0) 

B. (0, -4) D. (-1, -9) 



@ 



Work backward from the answer 
choices if you cannot remember 
how to solve a problem. 



14. Carolyn makes between $5.75 and 
$9.50 per hour baby-sitting. Which is 
the best estimate of the total amount 
she makes for 9 hours of baby-sitting? 

A. From $30 to $55 

B. From $55 to $80 

C. From $80 to $105 

D. From $105 to $130 

Gridded Response 

15. Patrick plans to spend the next 28 
days preparing for a weight-lifting 
competition. He plans to spend a total 
of 1 19 hours at the gym. If Patrick is at 
the gym for the same amount of time 
every day, how many hours will he be 
at the gym each day? 

16. Solve the equation -4.3x = -0.215 
for X. 

17. Determine the y-coordinate of the point. 

Ay 



• 4 



■* — I — I — I — h 



-4 -2 O 
-2-. 



f4-- 



H 1 1 1 >- 



2 4 



18. What is the sixth term in the following 
sequence? 



1 ^1 2 2l 
2' '4' '"' ■^4' 



Cumulative Assessment, Chapters 1-5 331 



CHAPTER 




Per 



c ^1 



6A 


Proportions and 
Percents 


6-1 


Percents 


LAB 


Model Percents 


6-2 


Fractions, Decimals, and 
Percents 


6-3 


Estimating with Percents 


6-4 


Percent of a Number 


6-5 


Solving Percent Problems 


6B 


Applying Percents 


6-6 


Percent of Change 


6-7 


Simple Interest 







7.1.9 



7.1.9 



apter 



Worl< with proportions 
involving percents. 

Solve a wide variety of percent 
problems. 



Why Learn THifl 



Percents are commonly used to express 
and compare ratios. For example, about 
70% of the Earth's surface is covered 
in water. 



X*. 



Learn It Online 

Chapter Project Online qo.hrw.com, 



keyword ■BHIlWTil ® 



332 Chapters 




Are You Ready? 



£t. 



Learn It Online 

Resources Online go.hrw.com, 

^i WM' i l M^10 AYR6 KGoJ 



0^ Vocabulary 



Choose the best term from the list to complete each sentence. 

1. A statement that two ratios are equivalent is called 



acn) 



2. To write = as a(n) 



divide tlie numerator bv the 



denominator. 

3. A(n) ? is a comparison by division of two quantities. 

4. The 



? of^is^ 



decimal 

equation 

fraction 

proportion 

ratio 

simplest form 



Complete these exercises to review skills you will need for this chapter. 

Q) Write Fractions as Decimals 

Write each fraction as a decimal. 

5. A 6. 4^ 7. 



10. 



11. 



739 
1,000 

7 



12. 



100 



20 



Write Decimals as Fractions 

Write each decimal as a fraction in simplest form. 
13. 0.05 14. 0.92 15. 0.013 

17. 0.006 18. 0.305 19. 0.0007 

Q) Solve Multiplication Equations 

Solve each equation. 

21. 100/; = 300 22. 38 = 0.4.v 

24. 9 = 72y 25. 0.07;?; = 56 

Q) Solve Proportions 

Solve each proportion. 



16. 0.8 
20. 1.04 



23. \6p= 1,200 
26. 25 = lOOf 



27. 


2 _ A- 

3 12 


30. 


16 _ 4 
28 " 


33. 


8 _ 10 

.V 5 



28- i 



31. 



P 
100 



12 
36 



29. 
32. 



15 



x 

45 



42 _ 14 

12 '; 



34. t = ^ 



24 



35 - 



Percents 333 



CHAPTER 



6 



Study Guide: Preview 



Where You've Been 

Previously, you 

• modeled percents. 

• wrote equivalent fractions, 
decimals, and percents. 

• solved percent problems 
involving discounts, sales tax, 
and tips. 



Key 

Vocabulary /Vocabulario 



■-"^-^-^^^.-v.-.-. 



In This Chapter 



You will study 

• modeling and estimating 
percents. 

• writing equivalent fractions, 
decimals, and percents, 
including percents less than 1 
and greater than 100. 

• solving percent problems 
involving discounts, sales tax, 
tips, profit, percent of change, 
and simple interest. 

• comparing fractions, decimals, 
and percents. 



Where You're Going 

You can use the skills 
learned in this chapter 

9 to find or estimate discounts, 
sales tax, and tips when 
shopping and eating out. 

® to solve problems involving 
banking. 



interest 


interes 


percent 


porcentaje 


percent of change 


porcentaje de cambio 


percent of decrease 


porcentaje de 
disminucion 


percent of increase 


porcentaje de 
incremento 


principal 


capital 


simple interest 


interes simple 



Vocabulary Connections 

To become familiar with some of the 
vocabulary terms in the chapter, consider the 
following. You may refer to the chapter, the 
glossary, or a dictionary if you like. 

1 . The Italian word cento and the French term 
cent mean "hundred." What do you think 
percent means? 

2. The word interest stems from Latin Unter- 
+ esse) and means "to be between" and "to 
make a difference." In business, interest is 
an amount collected or paid for the use of 
money. How can you relate the Latin roots 
and meanings to the business definition of 
interest? 

3. Principal is the amount of money 
deposited or borrowed. Interest builds 
upon the principal. How might common 
definitions oi principal, such as "leader of 
a school" and "a matter of primary 
importance," help you remember this 
business meaning of principal ? 



334 Chapter 6 




Readirijg X 

and WrlMAixi 
Math ^ -^ 




Study Strategy: Use Multiple Representations 

When a new math concept is introduced, the explanation given often 
presents the topic in more than one way. As you study, pay attention 
to any models, tables, lists, graphs, diagrams, symbols, and words used 
to describe a concept. 

In this example, the concept of finding equivalent fractions is represented 
in model, number, and word form. 



Finding Equivalent Fracti( 

Find a fraction equivalent to 

LAJa ^1? oStevj 

2 '^ '"■' 


>ns 

1 

3 ■ 

To model equivalent fractions, 

divide the fraction bars. 

J 

WORDS 


^ MODELS 

1 _ 1 -2 _ 2 
3 3-2 6 

-- — ^ NUMBERS I 


To find a fraction equivalent 
to ^, multiply the numerator 
and the denominator by the 
same number. 


WORDS ' 1 



Try This 



1 . Explain why it could be beneficial to represent a new idea in more 
than one way when taking notes. 

2. Explain how you can use models and numbers to find equivalent 
fractions. Which method do you prefer? Explain. 



Perce nts 335 



B 



B 



7.1.9 Solve problems involuing ratios and proportions. Express one 
quantity as a fraction of another, given their ratio, and vice... 

It is estimated that over 
half the plant and animal 
species on Earth live in 
rain forests. However, 
rain forests cover less 
than 6 out of every 100 
square miles of Earth's 
land. You can write this 
ratio, 6 to 100, as a 
percent, 6%. 



Vocabulary 

percent 



., versa. Find how many 
times one quantity is as 
large as another, given 
their ratio, and vice versa 
Express one quantity as a 
fraction of another given 
the two quantities. Find the 
whole, or one part, when a 
whole IS divided into parts 
in a given ration Solve 
problems involving two pairs 
of equivalent ratios. 




A percent is a ratio of a number to 100. 
The symbol % is used to indicate that 
a number is a percent. 



tI, = 6% 



EXAMPLE fllj Modeling Percents 

Write the percent modeled by each grid. 



Reading Math 



The word percent 
means "per 
hundred." So 5% 
means "6 out of 
100." 



shaded 
total 



MJ = 47% 



shaded 
total 



49 + 9 _ 
100 



m = 5«'^° 



You can write percents as fractions or decimals. 



EXAMPLE [2J Writing Percents as Fractions 

Write 35% as a fraction in simplest form. 



35% = ^ 



J7_ 
20 



Write the percent as a fraction 
with a denominator of 100. 

Simplify. 



So 35% can be written as 



20' 



336 Chapter 6 Percents 



y'i'Snu Lesson Tutorials Online mv.hrw.com 



EXAMPLE 



I 3 J Writing Percents as Decimals 

Write each percent as a decimal. 
A 43% 

Method 1: Use pencil and paper. 

43% = Y^ Write the percent as a fraction. 
= 0.43 Divide 43 by 100. 
B 30.75% 

Method 2: Use mental math. 

30.75% = 0.3075 Moi/e t/ie decimal point two places to the left. 



Think and Discuss 

1. Tell in your own words what percent means. 




tV Hni 



Learn It Online 

Homework Help Online go.hrw.com, 



keyword MMtaiHlM^ ® 
Exercises 1-26, 33, 35 



GUIDED PRACTICE 



See Example 1 Write the percent modeled by each grid 

1. I I I I I M I I I I 2 






See Example 2 Write each percent as a fraction in simplest form. 

L 4. 65% 5. 82% 6. 12% 7. 38% 

See Example 3 Write each percent as a decimal. 

9. 22% 10. 51% 11. 8.07% 12. 1.6% 



8. 75% 



13. 11% 



INDEPENDENT PRACTICE 



See Example 1 Write the percent modeled by each grid. 

14. I I II I I I I M I 15. 






16. 



Vldau Lesson Tutorials Online mv.hrw.com 



6-1 Percents 337 



See Example 2 Write each percent as a fraction in simplest form. 

17. 55% 18. 34% 19. 83% 20. 53% 



21. 81% 



See Example 3 Write each percent as a decimal. 

22. 48% 23. 9.8% 24. 30.2% 



25. 66.3% 



26. 8.39% 



Extra Practice 



See page EP16. 



PRACTICE AND PROBLEM SOLVING 



Write each percent as a fraction in simplest form and as a decimal. 

27. 2.70% 28. 7.6% 29. 44% 30. 3.148% 31. 10.5% 



Compare. Write <, >, or = 
32. ^ 22% 33 



35 



36. 41% 



13 
30 



17 



72% 



37. ^ 85% 



34. 



38. 



50 



22% 
60% 



35 ^ 

20 

39. 15% 



56% 

J_ 
30 



40. Multi-Step A nutrition label states that one serving of tortilla chips 
contains 7 grams of fat and 11% of the recommended daily allowance 
(RDA) of fat. 

a. Write a ratio that represents the percent RDA of fat in one serving of 
tortilla chips. 

b. Use the ratio from part a to write and solve a proportion to determine 
how many grams of fat are in the recommended daily allowance. 

^ 41. Choose a Strategy During class, Brad finished 63% of his homework, 
and Liz completed ^ of her homework. Wlio must finish a greater percent 
of homework at home? 

y^ 42. Write About It Compare ratios and percents. How are they alike? How are 
they different? 

^ 43. Challenge Write each of the following as a percent: 0.4 and 0.03. 



Test Prep and Spiral Review 



44. Multiple Choice Which inequality is a true statement? 



(3) 24% >i 



CT) 0.76 < 76% 



(X) 8%<0.8 



cm F<5% 



45. Short Response Nineteen out of the 25 students on Sean's team sold mugs, 
and 68% of the students on Chi's team sold caps. Which team had a greater 
percent of students participate in the fundraiser? 



Estimate each sum or difference. (Lesson 3-6) 



46 ^-^ 

" 8 7 



47. 6jL + 5^ 



48. 5| 



(-!) 



Plot each point on a coordinate plane. (Lesson 5-1) 

50. A{2,3) 51. B(-l,4) 52. C(-2, -6) 



49. f, + 2| 



53. D(0, -3) 



338 Chapter 6 Percents 



A ^^^bkn»..u<i.'w^/A 




^ 

% 


n(m 


d$'OAi 




\ 


fi 




A 




^^ 


"(Tc , 




Use witi 


REMEMBER 

































































































































































































































Model Percents 



£?. 



Learn It Online 

Lab Resources Online go.hrw.com, 
■BMMS10Lab6aoTl 



I 1% is 1 out of 100. 



8% is 8 out of 100. 53% is 53 out of 100. 



Percents less than 1% represent numbers less than 0.01 , or -^ . 
Percents greater than 100% represent numbers greater than 1. You can 
use 10-by-lO grids to model percents less than 1 or greater than 100. 



Activity 1 



O Use 10-by-lO grids to model 132%. 



Think: 132% means 132 out of 100. 

Shade 100 squares plus 32 squares to model 132%. 



O Use a 10-by-lO grid to model 0.5%. 



Think: One square equals 1%, so ^ of one square equals 0.5%. 
Shade | of one square to model 0.5%o. 



Thinic and Discuss 



1. Explain how to model 36.75% on a 10-by-lO grid. 

2. How can you model 0.7%)? Explain your answer. 



Try This 



Use 10-by-lO grids to model each percent. 
1. 280% 2. 16^% 3. 0.25% 



4. 65% 



5. 140.757o 



6-1 Hands-On Lab 339 







6-2 



Fractions, De 
and Percents 



*lLLLl*ij-: 



7.1.9 Solve problems involving ratios and proportions. Express one quantity as a fraction of another, given their ratio, and vice versa. Find how 
many times one quantity is as large as another, given their ratio, and vice versa. Express one quantity as a fraction of another given the... 

The students at Westview Middle School are collecting cans of food 
for the local food bank. Their goal is to collect 2,000 cans in one 
month. After 10 days, they have 800 cans of food. 



0. . two quanti 
whole, or one 



antities. Find the 
ne part, when a 
whole IS divided into parts 
m a given ration Solve 
problems involving two 
pairs of equivalent ratios. 



1 \ 1 r 1 — \ — \ — \ — \ — \ — \ — \ — \ — \ — \ — \ — 1 





200 


400 


600 


800 


1,000 


1,200 


1,400 


1,600 


1,800 2,00 


lv^v3lt;i.-t-V'"vl^it^lls---!lib*itB**al 1 1 1 1 1 1 1 1 1 1 1 1 





1 

10 


1 

5 


3 
10 


2 
5 


1 

2 
1 1 1 


3 
5 


7 
10 


4 

5 

1 1 


1 \ 1 





0.1 


0.2 


0.3 


0.4 


' 1 1 

0.5 


0.6 


0.7 


1 1 

0.8 


1 ' 1 

0.9 1.0 


t U- — 1 1 1 -^ — \ — ^ — \ — '■ — \ — i — \ — \ — \ — \ — 1 



10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 



Interactivities Online ► The models show that 800 out of 2,000 can be written as t^, ^, 0.4, 

2,000 5 



or 40%. The students have reached 40% of their goal. 



EXAMPLE [Tj Writing Decimals as Percents 

Write 0.2 as a percent. 



EXAIVIPLE 



•J- 



isMijJvasi 



(B 



To divide 4 by 5, use 
long division and 
place a decimal point 
followed by a zero 
after the 4. 

0.8 
5)4.0 



Method 1: Use pencil and paper. 

Write the decimal 
as a fraction with 
a denominator 
of 100. 
Write the 
numerator with 
a percent sign. 



02 = - = -^ 
10 100 



= 20% 



Writing Fractions as Percents 

Write I as a percent. 

Method 1: Use pencil and paper. 



5 ^ ^ 
= 0.8 

= 0.80^ 

= 80% 



Use division to write 
the fraction as a 
decimal. 

Write the decimal 
as a percent. 



Method 2: Use mental math. 



0.25,= 20.0% 
= 20% 



Move the decimal 
point two places 
to the right and 
add a percent sign. 



Method 2: Use mental math 

4-20 _ 



5 • 20 



80 
100 



= 80% 



Write an equivalent 
fraction with a 
denominator of WO. 
Write the numerator 
with a percent sign. 



340 Chapter 6 Percents 



yidau Lesson Tutorials Online my.hrw.com 



EXAMPLE 



53 



Ordering Rational Numbers 

Order 1 1, 0.33, -1.6, 3, 2^ and 70.2% from least to greatest. 

Step 1 Write tlie numbers as decimals mth the same number of 
decimal places. 



1^=1.8 
3 = 3.0 



0.33 « 0.3 

ol — 00 

^5 " 



-1.6= -1.6 
70.2% « 0.7 



Step 2 Graph the numbers on a number line. 

— h-»^ — \ — \ — I •!• I — I •!• I — ¥- 



-2-10123 

Step 3 Compare the decimals. 

-1.6 < 0.3 < 0.7 < 1.8 < 2.2 < 3.0 
From least to greatest, the numbers are; -1.6, 0.33, 70.2%, 1 ^, 2^, 3 



EXAMPLE 



S) 



Choosing a Method of Computation 

Decide whether using pencil and paper, mental math, or a 
calculator is most useful when solving the following problem. 
Then solve. 

In a survey, 55 people were asked whether they prefer cats or dogs. 
Twenty-nine people said they prefer cats. What percent of the people 
surveyed said they prefer cats? 



29 out of 55 



29 
55 



Think: Since 29 ^ 55 does not divide evenly, pencil 
and paper is not a good choice. 
Think: Since the denominator is not a factor of 100, 
mental math is not a good choice. 

Using a calculator is the best method. 

29 Ei 55 BfSB 10.52727272731 

0.^72727273 = 52.72727273% Write the decimal as a percent. 

» 52.7% Round to the nearest tenth of a 

percent. 

About 52.7% of the people surveyed said they prefer cats. 



Think and Discuss 

1. Describe two methods you could use to write | as a percent. 

2. Write the ratio 25:100 as a fraction, as a decimal, and as a percent. 



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6-2 Fractions, Decimals, and Percents 341 



6-2 



il3C?33333 



^iitorniiit 

Homework Help Online go.hrw.com, 



keyword mfMbt»M ® 
Exercises 1-35, 37, 39, 41 



GUIDED PRACTICE 



See Example 1 Write each decimal as a percent. 

L 1. 0.6 2. 0.32 3. 0.544 



4. 0.06 



5. 0.087 



See Example 2 



Write each fraction as a percent. 

6. I 7. ^ 8. 



11 

20 



See Example 3 Order the numbers from least to greatest. 



See Example 4 



11. 0.5,50%, 



11 



7_ 
40 



13. 10%, 1%, 



J_ 
10 



12. 



14. 



15. 72%, ||, 0.6 



-0.9, 90% 



-0.8, ^, 8% 

5 
1 



16. -^,5%, -0.05 



10. 



17. Decide whether using pencil and paper, mental math, or a calculator is 
most useful when solving the following problem. Then solve. 

In a survey, 50 students were asked wliether they prefer pepperoni pizza 
or cheese pizza. Twenty students said they prefer cheese pizza. What 
percent of the students surveyed said they prefer cheese pizza? 



INDEPENDENT PRACTICE 



See Example 1 Write each decimal as a percent. 

L 18. 0.15 19. 0.83 20. 0.325 



21. 0.081 



22. 0.42 



See Example 2 Write each fraction as a percent. 



L 



23. 



24. # 



25. 



26. 



16 



See Example 3 Order the numbers from least to greatest. 



See Example 4 



28. 0.6, 6%, I 

b 

30. |, 30%, 3 
32. 2%, |, l.T 



29. -|, -0.7,7% 
31. -0.1, 1%, -^ 



33. 



1 



-0.01,2% 



27. 



25 



Decide whether using pencil and paper, mental math, or a calculator is most 
useful when solving each of the following problems. Then solve. 

34. In a theme-park survey, 75 visitors were asked whether they prefer the 
Ferris wheel or the roller coaster. Thirty visitors prefer the Ferris wheel. 
What percent of the visitors surveyed said they prefer the Ferris wheel? 

35. In a survey, 65 students were asked whether they prefer television sitcoms 
or dramas. Thirteen students prefer dramas. Wliat percent of the students 
surveyed prefer dramas? 



342 Chapter 6 Percents 



PRACTICE AND PROBLEM SOLVING 



Extra Practice 



See page EP16. 







One of the 
world's largest 
flowers, the Titan 
arum, is native 
to the Sumatran 
rain forests. 
These flowers 
can grow to 
over 5 feet tall; 
the tallest ever 
recorded was 
over 10 feet tall. 



Compare. Write <, >, or =. 

36. 9% 0.9 37. 45% | 38. 0.037 37% 39. ^ 60% 

^•<^ Life Science Rain forests are iiome to 90,000 of the 250,000 identified 
^ plant species in tlie world. What percent of the world's identified plant 
species are found in rain forests? 

41. Multi-Step One-half of the 900 students at Jefferson Middle School are 
boys. One-tenth of the boys are in the band, and one-fifth of those play 
the trimipet. What percent of the students at Jefferson are boys who play 
the trumpet in the band? 

Use the table for Exercises 42-45. 

42. What percent of the championship 
appearances did Dudley win? 

43. Write the schools in order from least 
portion of games won to greatest 
portion of games won. 

44. Which school won 5 out of 6 games? 

45. Estimate the percent of the games 
Wallace-Rose Hill lost. 

46. What's the Error? A student 
wrote ~ as 0.4%. Wliat was the error? 

47. Write About It Describe two ways to change a fraction to a percent. 

48. Challenge A desert area's average rainfall is 12 inches a year. This year 
the area received 15 inches of rain. What percent of the average rainfall 
amount is 15 inches? 



North Carolina Men's Basketball 
Championship Appearances 


School Name 


Portion of 
Games Won 


Cummings 


0.83 


Dudley 


0.6 


North Mecklenburg 


0.3 


Wakefield 


1.0 


Wallace-Rose Hill 


0.6 



Test Prep and Spiral Review 



49. Multiple Choice Which value is NOT equivalent to 45%? 



C^ 



20 



CSj 0.45 



^^ 100 



CS) 0.045 



50. Short Response Melanie's room measures 10 ft by 12 ft. Her rug covers 
90 ft-^. Explain how to determine the percent of floor covered by the rug. 



Make a function table forx = -2, -1,0, 1, and 2. (Lesson 5-3) 

51. y = 5a- + 2 52. y = -2x 53. y = 

54. The actual length of a room is 6 m. The scale factor of a model is 1:15. 
What is the length of the room in the model? (Lesson 4- 1 0) 



-|.v - 4 



6-2 Fractions, Decimals, and Percents 343 








Estimafi 
Percents 







7.1.9 Solve problems involving ratios and proportions. Express one quantity as a fraction of 
another, given their ratio, and vice versa Find how many times one quantity is as .. 

A basketball at Hoops Haven costs $14.99. 
Cam's Sports is offering the same basketball 
at 20% off the regular price of $19.99. 
To find out which store is offering the 
better deal on the basketball, you can 
use estimation. 

The table shows common percents and 

their fraction equivalents. You 

can estimate the percent of a number 

... large as another, given , i ,-, .• e ^- *i » • i 

their ratio, and vice versa by Substituting a fraction that IS close 

Express one quantity as a 
fraction of another given 
the two quantities. Find the 
whole, or one part, when a 
whole IS divided into parts 
in a given ration. Solve 
problems involving two 
pairs of equivalent ratios. 














.^ 




Percent 


10% 


20% 


25% 


331% 


50% 


56|% 


Fraction 


1 

10 


1 
5 


1 

4 


1 
3 


1 
2 


2 
3 



EXAMPLE 







Compatible numbers 
are close to the 
numbers in a 
problem and help 
you use mental math 
to find a solution. 



Using Fractions to Estimate Percents 

Use a fraction to estimate 48% of 79. 



48% of 79 « i • 79 



1 

2 

40 



80 



ihinK: 48" is about 50% and 
50% is equivalent to |. 

Ciiange 79 to a compatible number. 

Multiply. 



48% of 79 is about 40. 



EXAMPLE 12 



Consumer Math Application 

Cam's Sports is offering 20% off a basketball that costs $19.99. 
The same basketball costs $14.99 at Hoops Haven. Which store 
offers the better deal? 

First find the discount on the basketball at Cam's Sports. 

20% of $19.99 = \ • $19.99 Think: 20% is equivalent to |. 

$20 Change $19.99 to a compatible number. 

$4 Multiply. 



«1 
5 



The discount is approximately $4. Since $20 - $4 = $16, 
the $14.99 basketball at Hoops Haven is the better deal. 



344 Chapter 6 Percents 



Mbd Lesson Tutorials OnllnE my.hrw.com 



Another way to estimate percents is to find 1% or 10% of a number. 
You can do this by moving the decimal point in the number. 



1% of 45: 45.0 

= 0.45 

To find 7% of a number, 
move the decimal point 
two places to the left. 



10% of 45: 45.0 

= 4.5 
To find 10% of a number, 
move the decimal point 
one place to the left. 



EXAMPLE [T) Estimating with Simple Percents 

Use 1% or 10% to estimate the percent of each number. 



3%) of 59 

59 is about 60, so find 3% of 60. 

I%of60 = 60.0 = 0.60 

3% of 60 = 3 -0.60 = 1.8 

3%of59isabout 1.8. 



3% equals 3 • 1%. 



B 18% of 45 

18% is about 20%, so find 20% of 45. 



10% of 45 = 45.0 = 4.5 
20% of 45 = 2 • 4.5 = 9.0 

18%of45isabout9. 



20% equals 2 • 10%. 



EXAMPLE [4J Consumer Math Application 



Eric and Selena spent S25.85 for their meals at a restaurant. About 
how much money should they leave for a 15% tip? 

Since $25.85 is about $26, find 15% of $26. 

15% = 10% + 5% Think: 15% is 10% plus 5%. 

10% of $26 = $2.60 

5% of $26 = $2.60 ^ 2 = $1.30 5% is | of 10%, so divide $2.60 by 2. 

$2.60 + $1.30 = $3.90 Add the 10% and 5% estimates. 

Eric and Selena should leave about $3.90 for a 15% tip. 



Think and Discuss 

1. Describe two ways to estimate 51% of 88. 

2. Explain why you might divide by 7 or mtiltiply by y to estimate a 
15% tip. 

3. Give an example of a situation in which an estimate of a percent 
is sufficient and a situation in which an exact percent is necessary. 



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6-3 Estimating with Percents 345 



6-3 



23j'i 



Homework Help Online go.hrw.com, 



keyword ■SSQE^B W 
Exercises 1-28, 37, 39 



GUIDED PRACTICE 



See Example 1 Use a fraction to estimate the percent of each number. 

1. 30% of 86 2. 52% of 83 3. 10% of 48 



4. 27% of 63 



See Example 2 5. Darden has $35. He finds a backpack on sale for 35% off the regular price 
' of $43.99. Does Darden have enough to buy the backpack? Explain. 



See Example 3 Use 1% or 10% to estimate the percent of each number. 

L 6. 5% of 82 7. 39% of 19 8. 21% of 68 



9. 7% of 109 



See Example 4 10. Mrs. Coronado spent $23 on a manicure. About how much money should 
L she leave for a 15% tip? 



INDEPENDENT PBACTJC E 



See Example 1 Use a fraction to estimate the percent of each number. 

i 11. 8% of 261 12. 34% of 93 13. 53% of 142 14. 23% of 98 



15. 51% of 432 



16. 18% of 42 



17. 11% of 132 



18. 54% of 39 



See Example 2 19. Consumer Math A pair of shoes at The Value Store costs $20. Fancy Feet 
has the same shoes on sale for 25% off the regular price of $23.99. Wliich 
store offers the better price on the shoes? 

See Example 3 Use 1% or 10% to estimate the percent of each number. 

20. 41% of 16 21. 8% of 310 22. 83% of 70 23. 2% of 634 



24. 58% of 81 



25. 24% of 49 



26. 11% of 99 



27. 63% of 39 



See Example 4 28. Marc's lunch cost $8.92. He wants to leave a 15% tip for the service. About 
L how much should his tip be? 



Extra Practice 



D PROBLEM SOLVING 



See page EP16. 



Estimate. 

29. 31% of 180 

33. 78% of 90 



30. 18% of 150 
34. 52% of 234 



31. 3% of 96 
35. 19% of 75 



32. 2% of 198 
36. 4% of 311 



37. The new package of Marti's Snacks contains 20% more snack mix than the 
old package. There were 22 ounces of snack mix in the old package. About 
how many ounces are in the new package? 

38. Frameworks charges $60.85 for framing. Including the 7% sales tax, about 
how much will it cost to have a painting framed? 

39. Multi-Step Camden's lunch cost $11.67, and he left a $2.00 tip. About 
how much more than 15% of the bill did Camden leave for the tip? 



346 Chapter 6 Percents 



40. Sports Last season, Ali had a hit 19.3% of the times he came to bat. IfAli 
batted 82 times last season, about how many hits did he have? 

41. Business The graph shows the 
results of a survey about the 
Internet. The number of people 
interviewed was 39 1 . 

a. Estimate the number of people 
willing to give out their e-mail 
address. 

b. Estimate the number of people 
not willing to give out their 
credit card number. 

42. Estimation Sandi earns $43,000 
per year. This year, she plans to 
spend about 27% of her income 
on rent. 

a. About how much does Sandi plan to spend on rent this year? 

b. About how much does she plan to spend on rent each month? 

43. Write a Problem Use information from the graph in Exercise 41 to write 
a problem tliat can be solved by using estimation of a percent. 

w0 44. Write About It Explain why it might be important to know whether your 
estimate of a percent is too high or too low. Give an example. 

^^ 45. Challenge Use the graph from Exercise 41 to estimate how many more 
people will give out their work phone number than their Social Security 
number. Show your work using two different methods. 



Information People Are Willing 


to Give Out on the Internet 


E-mail address 




Work 
phone number 


^■■■■1 I 


c 
.— Street address 




mM 


E Home 
O phone number 

c 


J i \ 


Credit card 
number 


^^^HH i 


Social 
Security number 


:. J i M 1 1 


( 


10 20 30 40 50 60 70 80 


Percent of People 




Test Prep and Spiral Review 



46. 



Multiple Choice About 65%i of the people answering a survey said that 
they have read a "blog," or Web log, online. Sixt}'-sLx people were surveyed. 
Which is the best estimate of the number of people surveyed who have 
read a blog? 



CS) 30 



Ci:- 35 



CD 45 



CS) 50 



47. Short Response Ryan's dinner bill is $35.00. He wants to leave a 15% tip. 
Explain how to use mental math to determine how much he should leave as 
a tip. 



Find each product. (Lesson 3-3) 

48. 0.8 • 96 49. 30 • 0.04 



50. 1.6-900 



51. 0.005- 75 



52. Brandi's room was painted in a color that is a blend of 3 parts red paint and 
2 parts white paint. How many quarts of white paint does Brandi need to 
mix with 6 quarts of red paint to match the paint in her room? (Lesson 4-4) 



6-3 Estimating witli Percents 347 








Percent of a 
Number 







7.1.9 Solve problems involving ratios and proportions. Express one quantity as 
a fraction of another, given their ratio, and vice versa. Find how many... 

The human body is made 
up mosdy of water. In fact, 
about 67% of a person's total 
(100%) body weight is water. 
If Cameron weighs 90 pounds, 
about how mucli of his weight 
is water? 



. times one quantity is 
as large as another, given 
their ratio, and vice versa. 
Express one quantity as a 
fraction of another given 
the two quantities. Find the 
whole, or one part, when a 
whole IS divided into parts 
in a given ration. Solve 
problems involving two 
pairs of equivalent ratios. 



Interactivities Online ^ 



Recall that a percent is a part 
of 100. Since you want to know 
the part of Cameron's body 
that is water, you can set up 
and solve a proportion to find 
the answer. 



((5*' 



Part 
Whole 



67 
100 



11 
90 



Part 
Whole 




EXAMPLE [l 1 Using Proportions to Find Percents of Numbers 

Find the percent of each number. 



I'M^lMi 



When solving a 
problem with a 
percent greater than 
100%, the part will 
be greater than the 
whole. 



A 


67% of 90 




67 _ n 
100 90 




67 • 90 = 100 • n 




6,030 = 100» 




6,030 _ ioOh 




100 100 




60.3 = u 




67%of90is60.3. 


B 


145% of 210 




145 _ 11 




100 210 




145 • 210 = 100 • 11 


1 


30,450 = 100« 


■ 


30,450 _ 100;; 



100 100 

304.5 = n 

145% of210 is 304.5. 



Write a proportion. 

Set the cross products equal. 
Multiply. 

Divide each side by 100 to isolate the variable. 



Write a proportion. 

Set the cross products equal. 
Multiply. 

Divide each side by WO to isolate the variable. 



348 Chapter 6 Percents 



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In addition to using proportions, you can find thie percent of a 
number by using decimal equivalents. 



EXAMPLE [2I Using Decimal Equivalents to Find Percents of Numbers 



Find the percent of each number. Check whether your answer 
is reasonable. 



A 8% of 50 

8% of 50 = 0.08 • 50 
= 4 



Write the percent as a decimal. 
Multiply. 

Model 

Since 10% of 50 is 5, 

a reasonable answer % 10% 

for 8% of 50 is 4. 



8% 



50% 



100% 



5 

4 



25 



50 



B 0.5% of 36 

0.5% of 36 = 0.005 • 36 Write the percent as a decimal. 

= 0.18 Multiply. 

Estimate 

1% of 40 = 0.4, so 0.5% of 40 is half of 0.4, or 0.2. Thus 0.18 is a 
reasonable answer. 



EXAMPLE [Vj Geography Application 



Earth's total land area is 
about 57,308,738 mi^ 
The land area of Asia is 
about 30% of this total. 
What is the approximate 
land area of Asia to the 
nearest square mile? 



Find 30% of 57,308, 738 
0.30 • 57,308,738 
= 17,192,621.4 




Write the percent as a decimal. 
Multiply. 
The land area of Asia is about 17,192,621 mi"^. 



Think and Discuss 

1. Explain how to set up a proportion to find 150% of a number. 

2. Describe a situation in which you might need to find a percent of 
a number. 



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6-4 Percent of a Number 



349 



6-4 




i»irinTn 

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keyword MBibiniigw ® 

Exercises 1-26, 31, 33, 37, 39, 
41,43,45 



GUIDED PRACTICE 



See Example 1 Find the percent of each number. 

L 1. 30% of 80 2. 38% of 400 



3. 200% of 10 



4. 180% of 90 



See Example 2 Find the percent of each number. Check whether your answer is reasonable. 

L. 5. 16% of 50 6. 7% of 200 7. 47% of 900 8. 40% of 75 

See Example 3 9. Of the 450 students at Miller Middle School, 38% ride the bus to school. 
i_ How many students ride the bus to school? 



INDEPENDENT PRACTICE 



See Example 1 Find the percent of each number. 

10. 80% of 35 11. 16% of 70 



14. 5% of 58 



15. l%of4 



12. 150% of 80 
16. 103% of 50 



13. 118% of 3,000 
17. 225% of 8 



See Example 2 



See Example 3 



Find the percent of each number. Check whether your answer is reasonable. 

18. 9% of 40 19. 20% of 65 20. 36% of 50 21. 2.9% of 60 



22. 5% of 12 



23. 220% of 18 



24. 0.2% of 160 



25. 155% of 8 



26. In 2004, there were 19,396 bulldogs registered by the American Kennel 
Club. Approximately 86% of this number were registered in 2003. About 
how many bulldogs were registered in 2003? 



Extra Practice 



See page EP16, 



PRACTICE AND PROBLEM SOLVING 



Solve. 

27. 60% of 10 is what number? 

29. Wliat number is 15% of 30? 
31. 25% of 47 is what number? 
33. What number is 125% of 4,100? 



28. What number is 25% of 160? 
30. 10% of 84 is what number? 
32. What number is 59% of 20? 
34. 150% of 150 is what number? 



Find the percent of each number. If necessary, round to the nearest tenth. 

35. 160% of 50 36. 350%of20 37. 480%of25 38. 115%of200 



39. 18% of 3.4 



40. 0.9% of 43 



41. 98% of 4.3 



42. 1.22% of 56 



43. Consumer Math Fun Tees is offering a 30% discount on all merchandise. 
Find the amount of discount on a T-shirt that was originally priced at $15.99. 

44. Muiti-Step Shoe Style is discounting everything in the store by 25%. 
What is the sale price of a pair of flip-flops that was originally priced 
at $10? 



350 Chapter 6 Percents 



Qllr. 45. 




Pure gold is a soft 
metal that scratches 
easily. To make the 
gold in jewelry 
more durable, it 
is often combined 
with other metals, 
such as copper 
and nickel. 



49. 



Nutrition The United States Department of Agriculture recommends 
that women should eat 25 g of fiber each day. A granola bar provides 9% 
of that amount. How many grams of fiber does it contain? 

Physical Science The percent of pure gold in 14-karat gold is about 
58.3%. A 14-karat gold ring weighs 5.6 grams. About how many grams of 
pure gold are in the ring? 

Earth Science The apparent magnitude of the star Mimosa is 1.25. 
Spica, another star, has an apparent magnitude that is 78.4% of Mimosa's. 
What is Spica's apparent ruagnitude? 

Multi-Step Trahn purchased a pair of slacks for $39.95 and a jacket for 
$64.00. The sales tax rate on his purchases was 5.5%. Find the total cost of 
Trahn's purchases, including sales tax. 

The graph shows the results of a 
student survey about computers. 
Use the graph to predict how 
many students in your class have a 
computer at home. 



L>LH..^i-,..-.-..-..-,ii^. 





Have 


a computer 
at home 


H 


ave Internet access 
at home 




Use 


a computer 
at school 



69 



t 



45 



20 40 60 80 100 

Percent of students 



^ 50. What's the Error? A student 
used the proportion j^ = ^ to 
find 5% of 26. What did the 
student do wrong? 

51. Write About It Describe two ways to find 18% of 40. 

^52. Challenge Francjois's starting pay was $6.25 per hour. During his annual 
review, he received a 5% raise. Find Franc^ois's pay raise to the nearest cent 
and the amount he will earn with his raise. Then find 105% of $6.25. What 
can vou conclude? 



r 



Test Prep and Spiral Review 



53. Multiple Choice Of the 875 students enrolled at Sycamore Valley Middle 
School, 48% are boys. How many of the students are boys? 

CE) 250 CX> 310 CD 420 CE' 440 

54. Gridded Response A children's multivitamin has 80% of the recommended 
daily allowance of zinc. The recommended daily allowance is 15 mg. How many 
milligrams of zinc does the vitamin provide? 

Find each unit rate. (Lessori 4-2) 

55. Monica buys 3 pounds of peaches for $5.25. What is the cost per pound? 

56. Kevin types 295 words in 5 minutes. At what rate does Kevin type? 

Write each decimal as a percent. (Lesson 5 2) 

57. 0.0125 58. 0.26 59. 0.389 60. 0.099 61. 0.407 



6-4 Percent of a Number 351 




Solving Percent 
Problems 



■*^ 







7.1.9 Solve problems involving ratios and proportions. Express one quantity as a fraction of another, 
given their ratio, and vice versa Find hovu many times one quantity is as large as another,... 

Sloths may seem lazy, but their extremely slow 
movement helps them seem almost invisible 
to predators. Sloths sleep an average of 16.5 hours 
per day. To find out what percent of a 24-hour day 
16.5 hours is, you can use a proportion or an equation. 






t'A 



Proportion method 
Part -^ jt^^ 16^-^ Part 
-100 



Whole 



24 



Whole 



... given their ratio, and 
vice versa. Express one 
quantity as a fraction of 
another given the two 
quantities Find the whole, 
or one part, when a whole 
IS divided into parts m 
a given ration Solve 
problems involving two 
pairs of equivalent ratios. 



» • 24 = 100 • 16.5 
24«= 1,650 
n = 68.75 



Equation method 
What percent of 24 is 16.5? 

n • 24 = 16.5 

" 24 
n = 0.6875 



Sloths spend about 69% of the day sleeping! 




EXAMPLE 



[ij Using Proportions to Solve Problems with Percents 

! Solve. 



A What percent of 90 is 45? 

Write a proportion. 



n _ 45 
100 90 



n • 90 = 100 • 45 

90« = 4,500 

90/1 _ 4,500 
90 90 

» = 50 
50%of90is45. 



B 12 is 8% of what number? 



Set the cross products equal. 
Multiply. 

Divide each side by 90 to isolate the variable. 



8 _ 12 
100 " 




Write a proportion. 


8 • « = 100 • 


12 


Set the cross products equal. 


8/1= 1,200 




Multiply 


8« _ 1.200 
8 8 




Divide each side by 8 to isolate the variable 


n= 150 






12 is 8% of 1 


50. 





352 Chapter 6 Percents 



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EXAMPLE r2 J Using Equations to Solve Problems with Percents 



Solve. 

A What percent of 75 is 1 05? 

n • 75 = 105 Write an equation. 



n -75 _ 105 
75 75 

n= 1.4 

n = 140% 
140% of 75 is 105. 



Divide each side by 75 to isolate tlie variable. 
Write the decimal as a percent. 



B 48 is 20% of what number? 

48 = 20%) • n Write an equation. 



48 = 0.2 • n 

48 _ 0.2 • » 
0.2 0.2 

240 = n 

48 is 20% of 240. 



Write 20% as a decimal. 

Divide each side by 0.2 to isolate the variable. 



EXAMPLE fsj Finding Sales Tax 



iJJJ^llJJJjjJi 



The sales tax rate is 
the percent used to 
calculate sales tax. 



Ravi bought a T-shirt with a retail sales price of $12 and paid SO. 99 
sales tax. What is the sales tax rate where Ravi bought the T-shirt? 



Restate the question: What percent of $12 is $0.99? 
Write a proportion. 



n _ OJ39 
100 12 



» • 12 = 100 • 0.99 
\2n = 99 

]2n _ 99 
12 12 

I) = 8.25 



Set the cross products equal. 
Multiply 

Divide each side by 12. 



8.25% of $12 is $0.99. The sales tax rate where Ravi bought the T-shirt 
is 8.25%. 



Think and Discuss 

1. Describe two methods for solving percent problems. 

2. Explain whether you prefer to use the proportion method or the 
equation method when solving percent problems. 

3. Tell what the first step is in solving a sales tax problem. 



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6-5 Solving Percent Problems 353 



:.'^t^%^^^i'i:il-SitM^'j:i.i^3,i.U.'ti-L.A^.-Jjl..v\<i/-iii-i 





•Jllllll 

Homework Help Online go.hrw.com, 



keyword ■mwwii»M ® 
Exercises 1-22, 23, 25, 27, 29, 
31,35,39 



See Example 1 



See Example 2 

See Example 3 

L 



Solve. 
1. Wliat percent of 100 is 25? 

3. 6 is 10% of what number? 

5. Wliat percent of50is9? 

7. 7 is 14% of what number? 



2. Wliat percent of 5 is 4? 

4. 8 is 20% of what number? 

6. Wliat percent of 30 is 27? 

8. 30 is 15% of what number? 



9. The sales tax on a $120 skateboard at Surf 'n' Skate is $9.60. What is the 
sales tax rate? 



INDEPENDENT PRACTICE 



Solve. 

See Example 1 10. What percent of 60 is 40? 

12. What percent of 45 is 9? 
14. 56 is 140% of what number? 
See Example 2 16. Wliat percent of 80 is 10? 
18. 18 is 15% of what number? 
20. 210% of what number is 147? 



11. What percent of 48 is 16? 
13. What percent of 6 is 18? 
15. 45 is 20% of what number? 
17. What percent of 12.4 is 12.4? 
19. 9 is 30% of what number? 
21. 8.8 is 40% of what number? 



See Example 3 

L 



Extra Practice 



See page EP17. 



22. A 12-pack of cinnamon-scented pencils sells for $3.00 at a school booster 
club sale. What is the sales tax rate if the total cost of the pencils is $3.21? 



PRACTICE AND PROBLEM SOLVING 



Solve. Round to the nearest tenth, if necessary. 

23. 5 is what percent of 9? 24. Wliat is 45% of 39? 

25. 55 is 80% of what number? 26. 12 is what percent of 19? 

27. What is 155% of 50? 28. 5.8 is 0.9% of what number? 

29. 36% of what number is 57? 30. What percent of 64 is 40? 

31. Multi-Step The advertised cost of admission to a water park in a nearby 
citA>' is $25 per student. A student paid $30 for admission and received 
$3.75 in change. What is the sales tax rate in that city? 

32. Consumer Math The table shows the cost 
of sunscreen purchased in Beach City and 
Desert City with and without sales tax. 
Wliich city has a greater sales tax rate? 
Give the sales tax rate for each city. 





Cost 


Cost + Tax 


Beach City 


$10 


$10.83 


Desert City 


$5 


$5.42 



354 Chapter 6 Percents 




The viola family 
is made up of tfie 
cello, violin, and 
viola. Of the three 
instruments, the 
cello is the largest. 



33. Critical Thinking What number is always used when you set up a 
proportion to solve a percent problem? Explain. 

34. Health The circle graph shows the 
approximate distribution of blood 
types among people in the United 
States. 

a. hi a survey, 126 people had t\'pe O 
blood. Predict how many people 
were surveyed. 

b. How many of the people surveyed 
had type AB blood? 

Music Beethoven wrote 9 trios for the piano, viohn, and cello. These trios 
make up 20% of the chamber music pieces Beethoven wrote. How many 
pieces of chamber music did he write? 



U.S. 


Slood Type Distribution 




AB 





/ / /\ 11% 


45% 
L 


/ kL-^ 


A 


\ / 1 


40% 

1 


\/ / 


^ ^ 



< 



® 



36. History The length of Abraham Lincoln's first inaugural speech was 3,635 
words. The length of his second inaugural speech was about 19.3% of the 
length of his first speech. About how long was Lincoln's second speech? 

37. What's the Question? The first lap of an auto race is 2,500 m. This is 
10% of the total race distance. The answer is 10. What is the question? 



\Aj 38. Write About It If 35 is 1 10% of a number, is the number greater than or 
less than 35? Explain. 

39. Challenge Kayleen has been offered two jobs. The first job offers an 

annual salary' of $32,000. The second job offers an annual salary of $10,000 
plus 8% commission on all of her sales. How much money per month 
would Kayleen need to make in sales to earn enough commission to make 
more money at the second job? 



i 



Test Prep and Spiral Review 



40. Multiple Choice Thirty children from an after-school club went to the 
matinee. This is 20% of the children in the club. How many children are in 
the club? 



CA) 6 



® 67 



CD 150 



CE) 600 



41. Gridded Response lason saves 30% of his monthly paycheck for college. 
He earned $250 last month. How many dollars did he save for college? 



Divide. (Lesson 3-4) 
42. -3.92 ^ 7 



43. 10.68 H- 3 



44. 23.2 -4- 0.2 



45. 19.52 H- 6.1 



Find the percent of each number. If necessary, round to the nearest hundredth. 

(Lesson 6-4) 

46. 45% of 26 47. 22% of 30 48. 15% of 17 49. 68% of 98 



6-5 Solving Percent Problems 355 



'■N, 



CHAPTER 



6 



Ready To Go On? 



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SECTION 6A /| 



Quiz for Lessons 6-1 Through 6-5 

Q) 6-1 j Percents 

Write each percent as a fraction in simplest form. 
1. 9% 2. 43% 3. 5% 

Write each percent as a decimal. 
5. 22% 6. 90% 7. 29% 



4. 18% 



8. 5% 



& 



6-2 ] Fractions, Decimals, and Percents 

Write each decimal as a percent. 



9. 0.85 



10. 0.026 



11. 0.1111 



12. 0.56 



Write each fraction as a percent. Round to the nearest tenth of a percent, 
if necessary. 



13. 



14. 



25 



15. 



55 



16. 



13 
32 







6-3 j Estimating with Percents 

Estimate. 

17. 49% of 46 18. 9% of 25 
21. 18% of 80 22. 26% of 115 



19. 36% of 150 
23. 91% of 300 



20. 5% of 60 
24. 42% of 197 



25. Carlton spent $21.85 on lunch for himself and a friend. About how much 
should he leave for a 15% tip? 

(^ 6-4 ] Percent of a Number 

Find the percent of each number. 

26. 25% of 84 27. 52% of 300 
30. 41% of 122 31. 178%of35 



28. 0.5% of 40 
32. 29% of 88 



29. 160% of 450 
33. 80% of 176 



34. Students get a 15% discount off the original prices at the Iiverything 
Fluorescent store during its back-to-school sale. Find the amount of 
discount on fluorescent notebooks originally priced at $7.99. 

(^ 6-5 j Solving Percent Problems 

Solve. Round to the nearest tenth, if necessary. 

35. 14 is 44% of what number? 36. 22 is what percent of 900? 
37. 99 is what percent of 396? 38. 75 is 24% of what number? 

39. The sales tax on a $105 digital camera is $7.15. What is die sales tax rate? 



356 Chapter 6 Percents 




Focus on Problem Soliring 




r 




• Estimate or find an exact answer 

Sometimes an estimate is sufficient wlien you are solving a 
problem. Other times you need to find an exact answer. Before you 
try to solve a problem, you should decide whether an estimate will 
be sufficient. Usually if a problem includes the word about, then 
you can estimate the answer. 



Read each problem. Decide whether you need an exact answer or 
whether you can solve the problem with an estimate. Explain how 
you know. 



1 Barry has $21.50 left from his allowance. 
He wants to buy a book for $5.85 and a 
CD for $14.99. Assuming these prices 
include tax, does Barn,' have enough money 
left to buy both the book and the CDV 

2 Last weekend Valerie practiced playing 
the drums for 3 hours. This is 40% of the 
total time she spent practicing last week. 
How much time did Valerie spend 
practicing last week? 

3 Amber is shopping for a winter coat. She 
finds one that costs $157. The coat is on 
sale and is discounted 25% today only. 
About how much money will Amber save 
if she buys the coat today? 

4 Marcus is planning a budget. He plans to 
spend less than 35% of his allowance 
each week on entertainment. Last week 
Marcus spent $7.42 on entertainment. If 
Marcus gets $20.00 each week, did he 
stay within his budget? 

5 An upright piano is on sale for 20% off 
the original price. The original price is 
$9,840. What is the sale price? 



6 The Mapleton Middle School band has 
41 students. Six of the students in the 
band play percussion instruments. Do 
more than 15% of the students play 
percussion instruments? 




Focus on Problem Solving 357 



6-1 



B 



':j:/im^\ 



7.1.8 Solue pioblenis involving percents. Find the whole given a part and the 
percentage Find percentage increase or decrease. 

According to the U.S. Consumer Product 
Safety Commission, emergency rooms 
treated more than 50,000 skateboarding 
injuries in 2000. This was a 67% decrease 
from the peak of 150,000 skateboarding 
injuries in 1977. 



\m 



Vocabulary 

percent of change 
percent of increase 
percent of decrease 



EXAMPLE 



jJaipjjjjEjJj 



When a number is 
decreased, subtract 
the new amount 
from the original 
amount to find the 
amount of change. 
When a number is 
increased, subtract 
the original amount 
from the new 
amount. 



A percent can be used to describe an 
amount of change. The percent of change 
is the amount, stated as a percent, that a 
number increases or decreases. If the 
amount goes up, it is a percent of increase. 
If the amount goes down, it is a 
percent of decrease. 



You can find the percent of change by 
using the following formula. 

percent of change = 

Finding Percent of Change 



fli^,v 



■*«B(iEL 



amount of change 
original amount 



[T] Findi 



Find each percent of change. Round answers to the nearest tenth 
of a percent, if necessary. 



Find the amount of change. 
Substitute values into formula. 



A 27 is decreased to 20. 

27 - 20 = 7 

percent of change = -^ 

« 0.259259 Divide. 

«= 25.9% Write as a percent. Round. 

The percent of decrease is about 25.9%. 



B 32 is increased to 67. 

67 - 32 = 35 



percent of change 



_ 35 
32 

= 1.09375 

« 109.4% 



Find the amount of change. 

Substitute values into formula. 

Divide. 

Write as a percent. Round. 



The percent of increase is about 109.4%. 



358 Chapter 6 Percents 



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EXAMPLE [2J Using Percent of Change 



The regular price of an MPS player at TechSource is S79.99. 
This week the MPS player is on sale for 25% off. What is the 
sale price? 

Step 1 Find the amount of the discount. 
25 _ d 



100 $79.99 

25 • $79.99 = lOOrf 

1999.75 _ lOOrf 
100 100 

$20.00 « d 



Write a proportion. 

Set the cross products equal. 

IVIultiply. Then divide each side by 100. 



The amount of the discount d is $20.00. 
Step 2 Find the sale price. 

regular price - amount of discount = sale price 



$79.99 - $20.00 

The sale price is $59.99. 



= $59.99 



EXAMPLE 



(5 






The amount of 
increase is also called 
the markup. 



Business Application 

Winter Wonders buys snow globes from a manufacturer 

for S9.20 each and sells them at a 95% increase in price. 

What is the retail price of the snow globes? 

Step I Find tlie amount ii of increase. 

95% • 9.20 = n Thinl<: 95% of $9.20 is what number? 

0.95 • 9.20 = n Write the percent as a decimal. 

8.74 = ;; 
Step 2 Find the retail price. 

wholesale price + amount of increase = retail price 

$9.20 + $8.74 = $17.94 

The retail price of the snow globes is $17.94 each. 



Think and Discuss 

1. Explain what is meant by a 100% decrease. 

2. Give an example in which the amount of increase or markup is 
greater than the original amount. What do you know about the 
percent of increase? 



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6-6 Percent of Change 359 




i3JAMB£^ 



GUIDED PRACTICE 




jH^-r-lf 1111 

Homework Help Online go.hrw.com, 



keyword ■maiifjBM ® 
Exercises 1-12, 13, IS, 17, 19, 
21,23,25 



See Example 1 Find each percent of change. Round answers to the nearest tenth of a 
percent, if necessary. 

! 1. 25 is decreased to 18. 2. 36 is increased to 84. 



See Example 2 

L 

See Example 3 



3. 62 is decreased to 52. 



4. 28 is increased to 96. 



5. The regular price of a sweater is $42.99. It is on sale for 20% off. Find the 
sale price. 

6. Business The retail price of a pair of shoes is a 98% increase from its 
wholesale price. The wholesale price of the shoes is $12.50. What is the 
retail price? 



INDEPENDENT PRACTICE 



See Example 1 Find each percent of change. Round answers to the nearest tenth of a 
I percent, if necessary. 

7. 72 is decreased to 45. 8. 55 is increased to 90. 



9. 180 is decreased to 140. 



10. 230 is increased to 250. 



See Example 2 11. A skateboard that sells for $65 is on sale for 15% off. Find the sale price. 

See Example 3 12. Business A jeweler buys a ring from an artisan for $85. ITe sells the ring 
L in his store at a 135% increase in price. What is the retail price of the ring? 



Extra Practice 



See page EP17, 



PRACTICE AND PROBLEM SOLVING 



Find each percent of change, amount of increase, or amount of decrease. 
Round answers to the nearest tenth, if necessary. 

13. $8.80 is increased to $17.60. 14. 6.2 is decreased to 5.9. 



15. 39.2 is increased to 56.3. 
17. 75 is decreased by 40%. 



16. $325 is decreased to $100. 
18. 28 is increased by 150%. 



19. A water tank holds 45 gallons of water. A new water tank can hold 25% 
more water. What is the capacity of the new water tank? 

20. Business Maria makes stretchy beaded purses and sells them to Bangles 
'n' Beads for $7 each. Bangles 'n' Beads makes a profit of 28% on each 
purse. Find the retail price of the purses. 

21 . Multi-Step A store is discounting all of its stock. The original price of a 
pair of sunglasses was $44.95. The sale price is $26.97. At this discount, 
what was the original price of a bathing suit that has a sale price of $28.95? 

22. Critical Thinking Explain why a change in price from $20 to $10 is a 
50% decrease, but a change in price from $10 to $20 is a 100% markup. 



360 Chapter 6 Percents 



Economics 



other: 19% 



23. The information at right shows the expenses 
for the Kramer family for one year. 

a. The Kramers spent $2,905 on auto 
expenses. What was their income for 
the year? 

b. How much money was spent on 
household expenses? 

c. The Kramers pay $14,400 per year on 
their mortgage. What percent of their 
household expenses is this? Round 
your answer to the nearest tenth. 

24. United States health expenses were 
$428.7 billion in 1985 and $991.4 billion 
in 1995. What was the percent of increase 
in health expenses during this ten-year 
period? Round your answer to the 
nearest tenth of a percent. 

25. In 1990, the total amount of energ\' consumed for transportation in the 
United States was 22,540 trillion British thermal units (Btu). From 1950 to 
1990, there was a 165% increase in energy consumed for transportation. 
About how many Btu of energy were consumed in 1950? 

26. ^Challenge In 1960, 21.5% of U.S. households did not have a 
telephone. This statistic decreased by 75.8% between 1960 and 1990. 
In 1990, what percent of U.S. households had a telephone? 



I Med ical: 17% 




i 



Test Prep and Spiral Review 



27. Multiple Choice Find the percent of change if the price of a 20-ounce 
bottle of water increases from $0.85 to $1.25. Round to the nearest tenth. 



CS) 47.1% 



ci:> 40.0% 



CD 32.0% 



CE) 1.7% 



28. Extended Response A store buys jeans from the manufacturer for $30 

each and sells them at a 50% markup in price. At the end of the season, the store 
puts the jeans on sale for 50% off. Is the sale price $30? Explain your reasoning. 



Write each mixed number as an improper fraction. fLesson 2 9) 



29. 3-: 



30. 6| 



31. 



^1 
'4 



32. 3t 





33. 24i 



Convert each measure. (Lesson 4-5) 



34. 34 mi to feet 



35. 52 oz to pounds 



36. 164 1b to tons 



6-6 Percent of Change 361 







7.1.9 Solve problems i 

fraction of anoth 



Vocabulary 



interest 



simple interest 
principal 








. one quantity is as 
large as another, given 
their ratio, and vice versa 
Express one quantity as a 
fraction of another given 
the two quantities. Find the 
whole, or one part, when a 
whole IS divided into parts 
in a given ration Solve 
problems involving two 
pairs of equivalent ratios. 



nvolving ratios and proportions. Express one quantity as a 
er, given their ratio, and vice versa. Find how many times.. 

When you keep money in a savings 
account, your money earns interest. 
Interest is an amount of money 
that is charged for borrowing or 
using money, or an amount of 
money that is earned by saving 
money. For example, the banlc pays 
you interest to use your money to 
conduct its business. Likewise, when 
you borrow money from the bank, tlie 
bank collects interest that is paid annually 
on its loan to you. 

One type of interest, called simple interest, is money paid only on 
the principal. The principal is the amount of money deposited or 
borrowed. To solve problems involving simple interest that is paid 
annually, you can use the following formula. 



EXAMPLE 1 



Interest 



Principal 




Rate of interest per year 
(as a decimal) 



Time in years that the 
money earns interest 



Using the Simple interest Formula 

Find each missing value. 

A / = ,p= $225, r = 3%, t = 2 years 

1 = P- r- t 

/ = 225 • 0.03 • 2 Substitute. Use 0.03 for 3%. 

/ = 13.5 Multiply. 

The simple interest is $13.50. 



B /=$300,P= $1,000, ;- = 
l-P-r-t 
300 = 1,000- /-• 5 

300 = 5,000r 

300 _ 5,000r 
5,000 5,000 

0.06 = r 
The interest rate is 6%. 



, t= 5 years 

Substitute. 
IVIultiply. 

Divide eacti side by 5,000. 



362 Chapter 6 Percents 



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EXAMPLE 



C3 



PROBLEM 



SOLVING 



PROBLEM SOLVING APPLICATION 

Olivia deposits $7,000 in an account that earns 7% simple interest 
paid annually. About how long will it take for her account balance 
to reach S8,000? 

n> Understand the Problem 

Rewrite the question as a statement: 

• Find the number of years it will take for the balance to reach $8,000. 
List the important information: 

• The principal is $7,000. 

• The interest rate is 7%. 

• Her account balance will be $8,000. 



Make a Plan 

Olivia's account balance i4 includes the principal plus the interest: 

A — P + I. Once you solve for /, you can use I = P • r • r to find the time. 

*e] Solve 



A^ P+ [ 
8.000 = 7,000 + / 

-7,000 -7,000 
1,000= / 

/= P- r- t 

1,000 = 7,000 • 0.07 • t 

1,000 = 490r 

1,000 _ 4901 
490 490 

2.04 « t 

It will take just over 2 years. 



Substitute. 

Subtract 7,000 from each side. 



Substitute. Use 0.07 for 7%. 
Multiply. 

Divide each side by 490. 



Q Look Back 

The account earns 7% of $7,000, which is $490, per year. So after 
2 years, the interest will be $980, giving a total balance of $7,980. 
An answer of just over 2 years to reach $8,000 makes sense. 



^^^^^^^^^^^^■^^^^^^^B 


Think and Discuss 














1. Write the value of t 


in th 


B annual 


simple interest formula for a | 


time period of 6 months. 












2. Show how to find r 


if/ = 


$10 


P = 


$100, 


and t = 


2 years. 






^^^ 








. 



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6-7 Simple Interest 363 



6-7 



keyword ■39EB9 W 
Exercises 1-13, 15, 17, 19, 21, 23 



GUIDED PRACTICE 



See Example 1 Find each missing value. 

^. 1= ,p = $300, /- = 4%, r = 2 years 

2. / = , P = $500, r = 2%, r = 1 year 

3. / = $120, P= , ;• = 6%, r = 5 years 

4. / = $240, P = $4,000, r = , r = 2 years 



See Example 2 



5. Scott deposits $8,000 in an account that earns 6% simple interest paid 
annually. How long will it be before the total amount is $10,000? 



INDEPENDENT PRACTICE 



See Example 1 Find each missing value. 

6. / = , P = $600, ;• = 7%, r = 2 years 

i 

j 7.1= ,P = $12,000, ;-= 3%, r = 9years 

I 

i 8. /= $364, P = $1,300, /•= ,f=7years 

! 9. / = $440, P = ,r = 5%, t = 4 years 

10. / = $455, P= , ;■ = 7%, f = 5 years 

' 11. /= $231,P = $700, r= ,r = 3years 

See Example 2 12. Broderick deposits $6,000 in an account that earns 5.5% simple interest 
paid annually. How long will it be before the total amount is $9,000? 

13. Teresa deposits $4,000 in an account that earns 7% simple interest paid 
annually. How long will it be before the total amount is $6,500? 



Extra Practice 



See page EP17. 



PRACTICE AND PROBLEM SOLVING 



Complete the table. 





Principal 


Interest Rate 


Time 


Annual Simple Interest 


14. 


$2,455 


3% 




$441.90 


15. 


s 


4.25% 


3 years 


$663 


16. 


$18,500 




42 months 


$1,942.50 


17. 


$425.50 


5% 


10 years 




18. 




6% 


3 years 


$2,952 



19. Finance How many years will it take for $4,000 to double at an annual 
simple interest rate of 5%? 

20. Banking After 2 years, an account earning annual simple interest held 
$585.75. The original deposit was $550. What was the interest rate? 



364 Chapter 6 Percents 



Use the graph for Exercises 21-23. 




The 1907 paint- 
ing Portrait of 
Adele Bloch- 
Bauer I by the 
Austrian artist 
Gustav Klimt 
recently sold for 
$135 million, 
making it among 
the most expen- 
sive paintings 
ever sold. 



21. How much more interest was 
earned on $8,000 deposited for 
6 months in a statement savings 
account than in a passbook 
savings account? 

22. How much money was lost on 
$5,000 invested in S&P 500 
stocics for one year? 

23. Compare the returns on $12,000 
invested in the high-yield 1-year 
CD and the Dow lones 
industrials for one year. 



Investment Returns for 1 Year 


High-yield 1-year CD 

Statement savings 

Passbool< savings 

Dow Jones industrials 


!■ 


5.05 
1.58 
1.48 

■ 


: 

-5.7 

-10.5 




S&P 500 


-12 -9 -6 -3 3 ( 

Percent returns 





< 



Art Alexandra can buy an artist's-work-and-storage furniture set from 
her art instructor. She would buy it on credit for $5,000 at an annual 
simple interest rate of 4% for 3 years. She can purchase a similar furniture 
set online for $5,500 plus a $295 shipping and handling fee. Including 
interest, which set costs less? How much would Alexandra pay for the set? 

25. Write a Problem Use the graph in Exercises 21-23 to write a problem 
that can be solved by using the simple interest formula. 

26. Write About It Explain whether you would pay more annual simple 
interest on a loan if you used plan A or plan B. 

Plan A: $ 1 ,500 for 8 years at 6% Plan B: $ 1 ,500 for 6 years at 8% 

g§> 27. Challenge The lacksons are opening a savings account for their child's 
college education. In 18 years, they will need about $134,000. If the 
account earns 6% simple interest annually, how much money must the 
Jacksons invest now to cover the cost of the college education? 



i 



Test Prep and Spiral Review 



28. Multiple Choice lulian deposits $4,500 in a bank account that pays 3% simple 
interest annually. How much interest will he earn in 5 years? 



(S) $135 



CE) $485 



CCJ $675 



CD $5,175 



29. Short Response Susan deposits $3,000 in the bank at 6.5% annual simple 
interest. How long will it be before she has $3,500 in the bank? 

30. Small book covers are l| ft long. How many book covers can be made out 
of 40 ft of book cover material? (Lesson 3- 1 0) 

Find each percent of change. Round answers to the nearest tenth of a percent, 
if necessary. (Lesson 6-6) 

31. 154 is increased to 200. 32. 95 is decreased to 75. 33. 88 is increased to 170. 



6-7 Simple Interest 365 



CHAPTER 




To Go On? 



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^** ResourcesOnlinego.hrw.com, 
IW!Bff!]|M'; 1 n RTr.nftBlal 



<3 



2. 121 is increased to 321. 
4. 45 is increased to 60. 
6. 86 is increased to 95. 



On-the-Go Cellular Phones 


Regular 
Price 


Price with 2-year 
Contract 


$49 


Free 


$99 


$39.60 


$149 


$47.68 


$189 


$52.92 


$229 


$57,25 



GT 



Quiz for Lessons 6-6 Through 6-7 

6-6 I Percent of Change 

Find each percent of change. Round answers to the nearest tenth of a 
percent, if necessary. 

1. 37 is decreased to 17. 

3. 89 is decreased to 84. 

5. 61 is decreased to 33. 

When customers purchase a contract for 
cell phone service, providers often 
include the phone at a discounted price. 
Prices for cell phones from On-the-Go 
Cellular are listed in the table. Use the 
table for problems 7-9. 

7. Find the percent discount on tlie $99 
phone with a 2-year contract. 

8. Find the percent discount on the $149 
phone with a 2-year contract. 

9. What happens to the percent discount that On-the-Go Cellular gives on its 
phones as the price of the plione increases? 

10. Since Franic is increasing the distance of his daily runs, he needs to carry 
more water. His current water bottle holds 16 ounces. Frank's new bottle 
holds 25% more water than his current bottle. Wliat is the capacity of 
Franic's new water bottle? 

6-7 ] Simple Interest 
Find each missing value. 
11./= ,p= $750, r = 4%, r = 3 years 

12. / = $120, P = , /• = 3%, f = 5 years 

13. /= $180, P = $1500, /■ = ■, t= 2 years 

14. / = $220, P = $680, ;• = 8%. r = ^ 

15. Leslie wants to deposit $10,000 in an account that earns 5% simple interest 
paid annually so that she will have $12,000 when she starts college. How long 
will it take her account to reach $12,000? 

16. Harrison deposits $345 in a savings account that earns 4.2% simple interest 
paid annually. How long v«ll it take for the total amount in the account 

to reach $410? 



366 Chapter 6 Percents 




CONNECTIONS 



Corh Nebraska's nickname is the Cornhusker State, which 
seems appropriate because corn is Nebraska's top crop in 
terms of acres and dollar value. In 2007, nearly 1.5 billion 
bushels of corn were harvested in the state. 

For 1-2, use the table. 

1 




NEBRASKA 



The recommended daily allowance (RDA) of 
carbohydrates for a teenage girl is 130 grams. 

a. Wliat percent of the RDA of 
carbohydrates does a 
teenage girl consume 
by eating an ear of 
corn? Round to the 
nearest percent. 

b. Write the percent 
as a decimal and as 
a fraction. 




IV"; 



'' '1-Hr-: 



Nutrition Facts 

Serving Size: 

One medium ear of corn 



Amount per serving 



Calories 



78 



Carbohydrates 



i7g 



Protein 



U^ 



I A 



Fat 



^ I Dietary Fiber 




2. A student's dinner included a medium ear of corn. The 
corn provided 12% of the Calories in the meal. How many 
Calories did the student consume at dinner? 

3. In 2007, 9.4 million acres of corn were planted in Nebraska. 
In the United States, 93.6 million acres of corn were planted. 
Estimate the percent of all corn in the United States that was 
planted in Nebraska. Explain how you made the estimate. 

4. The 9.4 million acres of corn planted in Nebraska in 
2007 was an 11% increase from the amount of corn 
planted in the state in 2006. 

a. How many acres of corn were planted in 
Nebraska in 2006? 

b. Suppose 10 million acres of corn were planted 
in Nebraska in 2008. Find the percent —— '• 
increase from 2007 to 2008. Round 
to the nearest percent. 








Lighten Up 



On a digital clock, up to seven light bulbs 
make up each digit on the display. You can 
label each light bulb as shown below. 




MJiMe 




If each number were lit up for the same amount of 
time, you could find out which light bulb is lit the greatest 
percent of the time. You could also find out which light 
is lit the least percent of the time. 




bulb 



For each number 0-9, list the letters of the light bulbs that are 
used when that number is showing. The first few numbers have 
been done for you. 



n 
u 






g g 

Once you have determined which bulbs are lit for each number, 
count how many times each bulb is lit. What percent of the time is 
each bulb lit? What does this tell you about which bulb will burn 
out first? 



Percent Bingo 



Use the bingo cards with numbers and 
percents provided online. The caller has a 
collection of percent problems. The caller 
reads a problem. Then the players solve the 
problem, and the solution is a number or 
a percent. If players have the solution on 
their card, they mark it off. Normal bingo 
rules apply. You can win with a horizontal, 
vertical, or diagonal row. 

A complete copy of the rules and game pieces is available online 




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«* Game Time Extra go.hrw.com, fc 

IBajMSIOGamesEy 



368 Chapter 6 Percents 




Materials 

• 2 pieces of 
card stock 
(B^by 12 in.) 

• 21 strips of 
colored paper 

(4 by 5^ in.) 
■ glue 

• markers 



m 



-^ 



PROJECT 



Percent Strips 



This colorful booklet holds questions and answers 
about percents. 

Directions 

O Fold one piece of card stock in half. Cut along 
the crease to make two rectangles that are each 
5^ inches by 6 inches. You will use these later as 
covers for your booklet. 

On the other piece of card stock, make accordion 
folds about ^-inch wide. When you are done, there 
should be 16 panels. These panels will be the pages 
of your booklet. Figure A 

Fold up the accordion strip. Glue the covers to the 
top and bottom panels of the strip. Figure B 

O Open the front cover. Glue a strip of colored paper 
to the top and bottom of the first page. Figure C 

Turn the page. Glue a strip of colored paper to the 
back of the first page between the other two strips. 
Figure D 

Glue strips to the other pages in the same way. 




O MT- ^ 




ibL... 


J 



Putting the Math into Action 

Write a question about percents on 
the front of each strip. Write the answer 
on the back. Trade books with another 
student and put your knowledge of 
percents to the test. 



CvIApTeR 
6, 



Delmt this Word 



iLmbc/.^l 



It's in the Bag! 369 



Vocabulary 

interest 362 

percent 336 

percent of change 358 

percent of decrease 358 



percent of increase 
principal 



simple interest 



,358 
,362 
,362 



Complete the sentences below with vocabulary words from the list above. 

1 . ? is an amount that is collected or paid for the use of money. The 
equation I - P • r- tis used for calculating ? paid annually. The 
letter P represents the ? and the letter r represents the annual rate. 

2. The ratio of an amount of increase to the original amount is the ? . 



3. The ratio of an amount of decrease to the original amount is the ? . 

4. A(n) ? is a ratio whose denominator is 100. 



EXAMPLES 



EXERCISES 



6-1 ] Percents (pp. 336-338) 



Write 12% as a fraction in simplest form 
and as a decimal. 



12% = -^ 





12-^ 


4 




100 


■^ 4 




3 






2.5 





12% = li 

= 0.12 



Write each percent as a fraction in simplest 
form and as a decimal. 



5. 78% 


6. 40% 


7. 5% 


8. 16% 


9. 65% 


10. 89% 



6-2] Fractions, Decimals, and Percents (pp 340-343) 



Write as a percent. 

■i 



7 



= 7-^8 

= 0.875 
= 87.5% 



0.82 



«-«2 = m 



= 82% 



Write as a percent. Round to the nearest 
tenth of a percent, if necessary. 



11. I 
13. 0.09 



"•§ 



12. 



1 



6 
14. 0.8 

16. 0.0056 



17. Order 0.33, -2.6, 2|, and 30% from 
least to greatest. 



370 Chapter 6 Percents 



EXAMPLES 



EXERCISES 



6-3 ] Estimating with Percents (pp. 344-347) 

■ Estimate 26% of 77. 

26% of 77 == - • 77 ^^°'^° '^ about 25% and 
25% is equivalent to \. 



80 



Change 77 to 80. 
« 20 IVIultiply. 

26% of 77 is about 20. 



Estimate. 

18. 22% of 44 19. 74% of 120 

20. 43% of 64 21. 31% of 97 

22. 49% of 82 23. 6% of 53 

24. Byron and Kate's dinner cost $18.23. 
About how much money should they 
leave for a 15% tip? 

25. Salvador's lunch cost $9.85, and he left 
a $2.00 tip. About how much more than 
15% of the bill did Salvador leave for 
the tip? 



6-4 ) Percent of a Number (pp. 348-351) 

■ Find the percent of the number. 
I 125% of 610 

I 125 _ ); 

100 610 Write a proportion. 

125 • 610 = 100 • n cross products 
76,250 = lOOii IVIultiply. 

76,250 _ ioo» Divide each side by 100. 
100 100 

762.5 = n 
125% ofeiO is 762.5. 



Find the percent of each number. 

26. 16% of 425 27. 48% of 50 

28. 7% of 63 29. 96% of 125 

30. 130% of 21 31. 72% of 75 

32. Canyon Middle School has 1,247 

students. About 38% of the students are 
in the seventh grade. About how many 
seventh-graders currently attend 
Canyon Middle School? 



6-5] Solving Percent Problems (pp 352-355) 

■ Solve. 

80 is 32% of what number? 



80 = 32% • n 

80 = 0.32 • n 

80 _ 0.32 • n 
0.32 0.32 

250 = n 

80 is 32% of 250. 



Write an equation. 
Write 32% as a decimal. 
Divide each side by 0.32. 



Solve. 

33. 20% of what number is 25? 

34. 4 is what percent of 50? 

35. 30 is 250% of what number? 

36. What percent of 96 is 36? 

37. 6 is 75% of what number? 

38. 200 is what percent of 720? 

39. The sales tax on a $25 shirt purchased 
at a store in Oak Park is $1.99. What is 
the sales tax rate in Oak Park? 

40. Jaclyn paid a sales tax of $10.03 on a 
camera. The tax rate in her state is 8%. 
About how much did the camera cost? 



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Study Guide: Review 371 



EXAMPLES 



EXERCISES 



6-6j Percent of Change (pp. 358-361) 

' Find each percent of change. Round 
answers to the nearest tenth of a percent, if 
necessary. 

■ 25 is decreased to 16. 

25 — 16 = 9 Find the amount of change. 

percent of change = A 

= 0.36 

= 36% 

The percent of decrease is 36%. 

■ 13.5 is increased to 27. 

27 — 13.5 = 13.5 Find the amount of change. 



percent of change = 



13.5 



13.5 

= 1 

= 100% 



The percent of increase is 100%. 



Find each percent of change. Round 
answers to the nearest tenth of a percent, 
if necessary. 

41. 54 is increased to 81. 

42. 14 is decreased to 12. 

43. 110 is increased to 143. 

44. 90 is decreased to 15.2. 

45. 26 is increased to 32. 

46. 84 is decreased to 21. 

47. The regular price of a new pair of skis 
is $245. This week the skis are on sale 
for 15% off. Find the sale price. 

48. In 2006 the mean annual earnings for 
a person with a liigh school diploma 
was $31,071. A person with a bachelor's 
degree earned an average of $56,788 
per year. What is the percent of 
increase to tlie nearest tenth? 



6-7 ] Simple Interest (pp. 362-365) 

Find each missing value. 
;l B / = ,p= $545, ;■ = 1 .5%, t = 2 years 



I^ P- r- t 

/= 545- 0.015-2 

/= 16.35 

The simple interest is $16.35. 



Substitute. 
IVIultiply. 



I = $825, P = ,r=6%,t= II years 
l = P- V t 



825 = P- 0.06- 11 

825 = P • 0.66 
825 _ P • 0.66 
0.66 0.66 



1,250 = P 

The principal is $1,250. 



Substitute. 
IVIultiply. 

Divide each side 
by 0.66. 



Find each missing value. 

49. / = , P = $ 1 ,000, r = 3%, /- = 6 months 

50. / = $452.16, P = $1,256, r = 12%, t = i 

51 . / = , p = $675, ;• = 4.5%, t = 8 years 

52. / = $555.75, P = $950, ;■ = , 
r = 15 years 

53. /= $172.50, P= , ;■= 5%, 
f = 18 months 

54. Craig deposits $1,000 in a savings 
account that earns 5% simple interest 
paid annually. How long will it take 
for the total amount in his account to 
reach $1,350? 

55. Zach deposits $755 in an account 
that earns 4.2% simple interest paid 
annually. How long will it take for the 
total amount in the account to reach 

$1,050? 



372 Chapter 6 Percents 



Chapter Test 



Write each percent as a fraction in simplest form and as a decimal. 

1. 95% 2. 37.5% 3. 4% 




4. 0.01% 



Write as a percent. Round to the nearest tenth of a percent, if necessary. 
5. 0.75 6. 0.12 7. 0.8 8. 0.0039 



9 ^ 

^- 10 


10. 1 


11. 


5 
16 


12. 


2T 


Estimate. 

13. 48% of 8 


14. 3% of 119 


15. 


26% of 32 


16. 


76% of 280 



17. The Pattersons spent $47.89 for a meal at a restaurant. About how much 
should they leave for a 15% tip? 



Find the percent of each number. 

18. 90% of 200 19. 35% of 210 

21. 250% of 30 22. 38% of 11 



20. 16% of 85 
23. 5% of 145 



Solve. 

24. 36 is what percent of 150? 
26. 51 is what percent of 340? 
28. 70 is 14% of what number? 



25. What percent of 145 is 29? 
27. 36 is 40% of what number? 
29. 25 is 20% of what number? 



30. Hampton Middle School is expecting 376 seventh-graders next year. 
This is 40% of the expected school enrollment. How many students are 
expected to enroll in the school next year? 

Find each percent of change. Round answers to the nearest tenth, if necessary. 

31. 30 is increased to 45. 32. 115 is decreased to 46. 
33. 116 is increased to 145. 34. 129 is decreased to 32. 

35. A communit}' theater sold 8,500 tickets to performances during its first 
year. By its tenth year, ticket sales had increased by 34%. How many 
tickets did the theater sell during its tenth year? 

Find each missing value. 

36. / = , P = $500, ;■ = 5%, t = 1 year 37. / = $702, P = $1 ,200, r = 3.9%, T = 

38. / = $468, P = $900, r= ,r = 8 years 39. / = $37.50, P = , r = 10%, r = 6 months 

40. Kate invested $3,500 at a 5% simple interest rate. How many years will 
it take for the original amount to double? 



Chapter 6 Test 373 



CHAPTER 



6 



R ISTEP+ 
^ Test Prep 



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Applied Skills Assessment 

Constructed Response 

1. The graph shows the number of 
boys and the number of girls who 
participated in a talent show. 



Talent Show Participation 




a. What is the approximate percent of 
increase of girls participating in the 
talent show from 2002 to 2005? 

b. What percent of students 
participating in the talent show in 
2006 were boys? Explain how you 
found your answer. 

2. A homemaker association has 134 
members. If 31 of these members 
are experts in canning vegetables, 
are more or less than 25% of the 
members canning experts? Explain 
how you know. 

Extended Response 

3. Riley and Louie each have $5,000 to 
invest. They both invest at a 2.5% 
simple interest rate. 

a. Riley keeps her money invested 
for 7 years. How much interest 
will she earn? How much will her 
investment be worth? 



b. What is the value of Louie's 

investment if he invests for 3 years, 
then removes and spends $1,000, 
and then invests what is remaining 
for 4 more years at a rate of 4%? 

Multiple-Choice Assessment 

4. Which ratio corresponds to the similar 
figures shown? 



2.8cm/\^-8cm 7 <- 



A. 
B. 



12 cm 



5.6 cm 
4.2 




1 
1 



C. 



14 cm 
1 



D 1 



5. Which of the following is NOT 
equivalent to 12%? 

A. 0.012 C. 0.12 



B. 



12 
100 



25 



6. Which point is NOT on the graph of 
y = x^ - 3? 

A. (0, -3) C. (-2, -7) 

B. (2, 1) D. (-1, -2) 

7. Which equation is an example of the 
Identity Property? 

A. 100 + 10 = 2(50 + 5) 

B. 50 + 10 = 10 + 50 

C. 25 + (50 + 10) = (25 + 50) + 10 

D. 50 + = 50 



374 Chapter 6 Percents 



8. Which situation corresponds to the 
graph? 




Time 



A. Ty rides his bil<e up a hill, 
immediately heads back down, 
stops and rests for a while, continues 
down the hill, and then rests. 

B. Paul runs up a hill, stops a while for 
a water break, and then jogs back 
down the hill. 

C. Sue rollerskates down a hill, stops 
for lunch, and then continues along 
a flat course for a while. 

D. Eric swims across a pool, rests for a 
while when he gets to the other 
side, and then swims numerous laps 
without stopping. 

9. A basketball goal that usually sells for 
$825 goes on sale for $650. What is the 
percent of decrease, to the nearest 
whole percent? 

A. 12% C. 27% 

B. 21% D. 79% 

10. In Oregon, about 40 of the state's 
nearly 1,000 public water systems add 
fluoride to their water. What percent 
best represents this situation? 

A. 0.4% C. 40% 

B. 4% D. 400% 

11. The number of whooping cranes 
wintering in Texas reached an all time 
high in 2004 at 213. The lowest number 
ever recorded was 15 whooping cranes 
in 1941 . What is the percent of increase 
of whooping cranes wintering in Texas 
from 1941 to 2004? 



A. 7% 

B. 91% 



C. 198% 

D. 1,320% 



12. What is the value of 8^ - 2|? 



A. 5 

B. 5 



20 

13 



C. 6 



D. 6. 



20 "■ "20 

13. Which point lies outside of the circle? 
Ay 




A. (-3, 0) C. (3, 3) 

B. (1, 2) D. (-2, 1) 

Gridded Response 

14. Jarvis deposits $1,200 in an account 
that earns 3% simple interest. How 
many years will it take him to earn 
$432 in interest? 

15. Sylas finished a 100-meter freestyle 
swim in 80.35 seconds. The winner of 
the race finished in 79.22 seconds. How 
many seconds faster was the winning 
time than Sylas's time? 

16. A baseball coach has a rule that for 
every time a player strikes out, that 
player has to do 12 push ups. If Cal 
strikes out 27 times, how many push 
ups will he be required to do? 

17. Write a decimal equivalent to 65%. 

18. What is the denominator of the value 
of I + I when written in simplest form? 



Cumulative Assessment, Chapters 1-6 375 




and 







7A 


Organizing and 
Displaying Data 




7-1 


Frequency Tables, 
Stem-and-Leaf Plots, 
and Line Plots 


7.4.4 


7-2 


Mean, Median, Mode, 
and Range 


7.4.3 


7-3 


Bar Graphs and 
Histograms 


7.4.4 


7-4 


Reading and Interpreting 
Circle Graphs 


7.4.4 


7-5 


Box-and-Whisker Plots 


7.4.4 


LAB 


Explore Box-and-Whisker 
Plots 




7B 


Representing and 
Analyzing Data 




7-6 


Line Graphs 


7.4.4 


LAB 


Use Venn Diagrams to 
Display Collected Data 




7-7 


Choosing an Appropriate 
Display 


7.4.1 


LAB 


Use Technology to 
Display Data 




7-8 


Populations and Samples 


7.4.4 


7-9 


Scatter Plots 


7.4.4 


LAB 


Samples and Lines of 
Best Fit 




7-10 


Misleading Graphs 


7.4.4 



Why Learn T. 



Biologists can take random samples of a 
wildlife population, such as sea lions, to 
make estimates about population growth 
or infectious diseases that might affect 
the group. 



£?. 



Learn It Online 

Chapter Project Online go.hrw.com, 



keyword IBMIlMJM ® 



J Di 

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Tfii^. 



apter 



Make and interpret graphs, 
such as histograms and 
circle graphs. 

Make estimates relating to a 
population based on a sample. 




-^ 



i f 



376 Chapter 7 




Are You Ready? 



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IBM Msioch7 TGo. 



ST Vocabulary 

jf Choose the best term from the Hst to complete each sentence. 

1. A part of a line consisting of two endpoints and all points 
between those endpoints is called a(n) ? . 

2. A(n) ? is the amount of space between the marked 
values on the ? of a graph. 

3. The number of times an item occurs is called its ? . 



circle 
frequency 
interval 
line segment 
scale 



Complete these exercises to review skills you will need for this chapter. 

Order Whole Numbers 

Order the numbers from least to greatest. 

4. 45, 23, 65, 15, 42, 18 5. 103, 105, 102, 118, 87, 104 

6. 56, 65, 24, 19, 76, 33, 82 7. 8, 3, 6, 2, 5, 9, 3, 4, 2 



(v) whole Number Operations 



Add or subtract. 

8. 18 + 26 
12. 133 - 35 



9. 23 + 17 
13. 54 -29 



10. 75 + 37 
14. 200 - 88 



11. 98 + 64 
15. 1,055 - 899 



Locate Points on a Number Line 

Copy the number line. Then graph each number. 



8 10 12 14 16 18 20 




19. 7 



Top Speeds of Some Animals 


Animal 


Speed (mi/h) 


Elephant 


25 


Lion 


50 


Rabbit 


35 


Zebra 


40 



-I M K I 1 

7 



studm,idemmm. 



Where You've Been 

Previously, you 

• used an appropriate 
representation for displaying 
data. 

• identified mean, median, mode, 
and range of a set of data. 

• solved problems by collecting, 
organizing, and displaying data. 



In This Chapter 



You will study 

• selecting an appropriate 
representation for displaying 
relationships among data. 

• choosing among mean, 
median, mode, or range to 
describe a set of data. 

• making inferences and 
convincing arguments based 
on analysis of data. 



Where You're Going 

You can use the skills 
learned in this chapter 

• to analyze trends and make 
business and marketing 



Key 

Vocabulary /Vocabulario 



decisions. 

to strengthen a persuasive 
argument by presenting data 
and trends in visual displays. 



bar graph 




grafica de barras 


circle graph 




grafica circular 


frequency table 


tabia de frecuencia 


line graph 




grafica lineal 


line plot 




diagrama de 
acumulacion 


mean 




media 


median 




mediana 


mode 




moda 


scatter plot 




diagrama de 
dispersion 


stem-and-leaf 


plot 


diagrama de 
tallo y hojas 



Vocabulary Connections 

To become familiar with some of the 
vocabulary terms in the chapter, consider 
the following. You may refer to the chapter, 
the glossary, or a dictionary if you like. 

1. The word median comes from the Latin 
word medius, meaning "middle." What is 
the median value in a set of data? What 
other words come from this Latin root? 

2. Scatter can mean "to spread out" or "to 
occur at random." What might the data 
points on a scatter plot look like? 

3. Frequency is a measure of how often an 
event occurs or the number of like objects 
that are in a group. What do you think a 
frequency table might show? 



378 Chapter 7 Collecting, Displaying, and Analyzing Data 




...Reading / 

^ and WrLtuia 
Math X -^ 




Reading Strategy: Read a Lesson for Understanding 

Before you begin reading a lesson, find out what its main focus, or 
objective, is. Each lesson is centered on a specific objective, which is 
located at the top of the first page of the lesson. Reading with the objective 
in mind wall help guide you through the lesson material. You can use the 
following tips to help you follow the math as you read. 



Identify the objective of the lesson. Then 
skim through the lesson to get a sense of 
where the objective is covered. 



"How do I find the percent 
of a miniber?" 



As you read through the lesson, write 
down any questions, problems, or trouble 
spots you may have. 



Find the percent of each number 
8% of 50 



8% of 50 = 0.08 • 50 Write tlie percent 
as a decimal. 

= 4 Multiply. 




Work through each example, 
as the examples help 
demonstrate the objectives. 



Tftmk and Discuss 

1. Explain how to set up a proportion 
to find 150% of a number. 



Check your understanding of 
the lesson by answering the 
Think and Discuss questions. 



Try This 



Use Lesson 6-1 in your textbook to answer each question. 

1. What is the objective of the lesson? 

2. What new terms are defined in the lesson? 

3. What skills are being taught in Example 3 of the lesson? 

4. Which parts of the lesson can you use to answer Think and Discuss question 1? 



Collecting, Displaying, and Analyzing Data 379 



Frequency Tables, Stem-ai 
Leaf Plots, and Line Plots 



B 



7.4.4 Analyze data displays, including ways that they can be misleading Analyze 
ways in which the wording of questions can influence survey results, 

IMAX ' theaters, witli their huge 
screens and powerful sound 
systems, make viewers feel as 
if they are in the middle of the 
action. 

To see how common it is for 
an IMAX movie to attract such 
a large number of viewers, you 
could use a frequency table. 
A frequency table is a way to 
organize data values into 
categories or groups. By including 
a cumulative frequency column 
in your table, you can keep a 
running total of the number 
of data items. 



Vocabulary 

frequency table 
cumulative frequency 
stem-and-leaf plot 
line plot 



iiiJJiiJJJi/ijJ 



The frequency of 
a data value is the 
number of times it 
occurs. 




EXAMPLE 



Q 



Organizing and Interpreting Data in a Frequency Table 

The list shows box office receipts in millions of dollars for 

20 IMAX films. Make a cumulative frequency table of the data. 

How many films earned under $40 million? 

76, 51, 41, 38, 18, 17, 16, 15, 13, 13, 12, 12, 10, 10, 6, 5, 5, 4, 4, 2 

Step 1: Choose a scale that includes all of the data values. Then 
separate the scale into equal intervals. 

Step 2: Find the number of 
data values in each 
interval. Write these 
numbers in the 
"Frequency" column. 

Step 3: Find the cumulative 
frequency for each 
row by adding all 
the frequency values 
that are above or in that row. 

The number of films that earned under $40 million is the 
cumulative frequency of the first two rows: 17. 



IMAX Films 


Receipts 
(S million) 


Frequency 


Cumulative 
Frequency 


0-19 


16 


16 


20-39 


1 


17 


40-59 


2 


19 


60-79 


1 


20 



380 Chapter 7 Collecting, Displaying, and Analyzing Data \ Viilaij] Lesson Tutorials Online 



A stem-and-leaf plot uses the digits of each 
number to organize and display a set of 
data. Each leaf on the plot represents the 
right-hand digit in a data value, and each 
stem represents the remaining left-hand 
digits. The key shows the values of the data 
on the plot. 



Stems Leaves 



4 7 9 
6 



Key: 2J7 means 27 



EXAMPLE [?) organizing and interpreting Da,a in a S.en,.a„d.Uaf PI.. 



To represent 5 
minutes in the 
stem-and-leaf plot 
in Example 2, you 
would use as the 
stem and 5 as 
the leaf. 



The table shows the number of minutes students spent doing their 
Spanish homework. Make a stem-and-leaf plot of the data. Then 
find the number of students who studied longer than 45 minutes. 



Minutes 


Spent 


Doing 


Homework 


38 48 


45 


32 


29 48 


32 45 


36 


22 


21 64 


35 45 


47 


26 


43 29 



Step 1: Order the data from least to greatest. Since the data values 
range from 21 to 64, use tens digits for the stems and ones 
digits for the leaves. 

Step 2: List the stems from least to greatest on the plot. 

Step 3: List the leaves for each stem from least to greatest. 

Step 4: Add a key and title the graph. 



Minutes Spent Doing Homework 



The stems are 
the tens digits. 



The stem 5 has 
no leaves, so 
there are no 
data values in 
the 50's. 



Stems 


Leaves 


2 


12 6 9 9 


3 


2 2 5 6 8 


4 


3 5 5 5 7 8 8 


5 




6 


4 



Key: 3\2 means 32 



The leaves are 
the ones digits. 

The entries in 
the second row 
represent the data 
values 32, 32, 35, 
36, and 38. 



One student studied for 47 minutes, 2 students studied for 
48 minutes, and 1 student studied for 64 minutes. 
A total of 4 students studied longer than 45 minutes. 



Similar to a stem-and-leaf plot, a line plot can be used to show how 
many times each data value occurs. Line plots use a number line and 
X's to show frequency. By looking at a line plot, you can quickly see 
the distribution, or spread, of the data. 



7-7 Frequency Tables, Stenn-and-Leaf Plots, and Line Plots 381 



EXAMPLE I 3 I Organizing and Interpreting Data in a Line Plot 

Make a line plot of the data. How many miles per day did Trey run 
most often? 



Number of Miles Trey Ran Each Day During Training 


5 


6 


5 


5 


3 


5 


4 


4 


6 


8 


6 


3 


4 


3 


2 


16 


12 


12 



Step 1: The data values range from 2 to 16. Draw a number line that 
includes this range. 

Step 2: Put an X above the number on the number line that 

corresponds to the number of miles Trey ran each day. 



X X X X 
X X X X 
X X X X X X 

-\ — \ — \ — \ — \ — I — \ — \ — \ — \ — \ — h 






H — \ — \ — h* 
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 

Number of miles 

The greatest number of X's appear above the number 5. This means 
that Trey ran 5 miles most often. 



Think and Discuss 

1. Tell which you would use to determine the number of data values in 
a set: a cumulative frequency table or a stem-and-leaf plot. Explain. 




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Exercises 1-6, 7, 9, 11 



GUIDED PRACTICE 



See Example 1 

L 

See Example 2 
See Example 3 



CA 




Number of Electoral Votes for Select States (2004) 




55 


GA 15 


IN 11 


Ml 17 


NY 31 


PA 


21 


NJ 


15 


IL 21 


KY 8 


NC 15 


OH 20 


TX 


34 



1 . Make a cumulative frequency table of the data. How many of the states 
had fewer than 20 electoral votes in 2004? 

2. Make a stem-and-leaf plot of the data. How many of the states had more 
than 30 electoral votes in 2004? 

3. Make a line plot of the data. For the states shown, what was the most 
common number of electoral votes in 2004? 



382 Chapter 7 Collecting, Displaying, and Analyzing Data 



INDEPENDENT PRACTICE 



The table shows the ages of the first 18 U.S. presidents when they took office. 



-^ President Age President Age President Age 






Madison 



Monroe 



Adams 



57 



58 



57 



Harrison 



Tyler 



Polk 



Taylor 



54 



68 



51 



49 



64 



Fillmore 



Pierce 



Buchanan 



Lincoln 



Johnson 



Grant 



65 



52 



56 



46 




See Example 1 

L 

See Example 2 

L 

See Example 3 



Extra Practice 



See page EP18, 



4. Make a cumulative frequency table of the data. How many of the presidents 
were under the age of 65 when they took office? 

5. Make a stem-and-leaf plot of the data. How many of the presidents were in 
their 40s when they took office? 

6. Make a line plot of the data. What was the most common age at which the 
presidents took office? 



PRACTICE AND PROBLEM SOLVING 



Use the stem-and-leaf plot for Exercises 7-9. 

7. What is the least data value? 
What is the greatest data value? 

8. Which data value occurs most often? 

9. Critical Thinking Miich of the following is most 
likely the source of the data in the stem-and-leaf plot? 



Stems 


Leaves 





4 6 6 9 


1 


2 5 8 8 8 


2 


3 


3 


1 



Key: l\2 means 12 



CS) Shoe sizes of 1 2 middle school students 

CE) Number of hours 12 adults exercised in one month 

CO Number of boxes of cereal per household at one time 

CE) Monthly temperatures in degrees Fahrenheit in Chicago, Illinois 

10. Earth Science The table shows the masses of the largest meteorites 
found on Earth. 



Largest Meteorites 


Meteorite 


Mass (kg) 


Meteorite 


Mass (kg) 


Armanty 


23.5 


Chupaderos 


14 


Bacubirito 


22 


Hoba 


60 


Campo del Cielo 


15 


Mbosi 


16 


Cape York (Agpalilik) 


20 


Mundrabilla 


12 


Cape York (Ahnighito) 


31 


Willamette 


15 



a. Use the data in the table to make a line plot. 

b. How many of the meteorites have a mass of 15 kilograms or greater? 



7-1 Frequency Tables, Stem-and-Leaf Plots, and Line Plots 383 




The map shows the number of critically 
endangered animal species in each country' 
in South America. A species is critically 
endangered when it faces a very high risk 
of extinction in the wild in the near future. 



Which countiy has the fewest 
critically endangered species? 
Which has the most? 



12. 



13. 



14. 



15. 



Make a cumulative frequency 
table of the data. How many 
countries have fewer than 20 
critically endangered species? 

Make a stem-and-leaf plot of 
the data. 

\l^ Write About It Explain how 
changing the size of the intervals 
you used in Exercise 12 affects your 
cumulative frequency table. 

\^ Challenge In a recent year, the 
number of endangered animal 
species in the United States was 190. 
Show how to represent this number 
on a stem-and-leaf plot. 



Venezuela 30 



Guyana 7 



Colombia 74 
Ecuador 74 

Peru 35 
Bolivia 9 



Suriname 7 



- French 
Guiana 8 




Brazil 60 
Paraguay 5 



Chile 15 



Source: International Union for Conservation 
of Nature and Natural Resources 



Test Prep and Spiral Review 



20 


30 


9 


25 


28 


8 


11 


12 


7 


18 


33 


26 


10 


9 


2 



Use the data for Exercises 16 and 17. 

16. Multiple Choice How many stems would a 
stem-and-leaf plot of the data in the table have? 

CA) 1 CT) 3 

CD 2 CS) 4 

17. Extended Response Make a stem-and-leaf plot and a line plot of the data 
in the table. Wliich data display best shows the distribution of data? Explain. 

1 8. Maria has 1 8 yards of fabric. A pillowcase takes l| yards. How many pillowcases 
can Maria make with the fabric? (Lesson 3-10) 

Find each unit rate. Round to the nearest hundredth if necessary. (Lesson 4-2) 

19. 12 hr for $102 20. $2,289 in 7 mo 21. 48 points in 3 games 



384 Chapter 7 Collecting, Displaying, and Analyzing Data 



f^>iiu<L^ 



Bi" '^^^M^J^ ' 



*.* 



'■'' 



Mean, Median, Mode, 
and Range 



7.4.3 Describe how additional data, particularly outliers, added to a data set may 
affect the mean, median, and mode. 

To crack secret messages in code, you can 
list the number of times each symbol of the 
code appears in the message. The symbol 
that appears the most often represents the 
mode, which likely corresponds to the letter e. 



Vocabulary 

mean 

median 

mode 

range 

outlier 



The mode, along with the mean and the 
median, is a measure oi central tendency 
used to represent the "middle" of a data set. 

• The mean is the sum of the data values 
divided by the ntmiber of data items. 



AGf Mf ai La^JG6,1^^A^J« Coca kj^s?. ' wj: vi" 
^-Maiz-alloaV' 




• The median is the middle value of an odd number of data items 
arranged in order. For an even number of data items, the median is 
the mean of the two middle values. 

• The mode is the value or values that occur most often. Wlien all 
the data values occur the same number of times, there is no mode. 

The range of a set of data is the difference between the greatest and 
least values. 



EXAMPLE TlJ Finding the Mean, Median, Mode, and Range of a Data Set 

Find the mean, median, mode, and range of the data set. 

2, 1,8,0,2,4,3,4 



Interactivities Online ► 



The mean is 
sometimes called 
the average. 



mean: 

2+1+8 + + 2 + 4 + 3 + 4 = 24 
24 + 8 = 3 
The mean is 3. 

median: 

0, 1,2, 2,3,4,4,8 

^ = 2.5 

The median is 2.5. 

mode: 

0, 1,2, 2,3,4,4,8 
The modes are 2 and 4. 

range: 8-0 = 8 
The range is 8. 



Add the values. 
Divide the sum by the 
number of items. 



Arrange the values in order. 

There are two middle values, 
so find the mean of these 
values. 

The values 2 and 4 occur twice. 

Subtract the least value from 
the greatest value. 



fi'ldi) Lesson Tutorials OnlinE mv.hrw.com 



7-2 IVIean, Median, Mode, and Range 385 



Often one measure of central tendency is more appropriate for describing 
a set of data than another measure is. Thinic about what each measure 
tells you about the data. Then choose the measure that best answers the 
question being asked. 

EXAMPLE [2] Choosing the Best Measure to Describe a Set of Data 

The line plot shows the number of hours 15 people exercised in 
one week. Which measure of central tendency best describes these 
data? Justify your answer. 



X 

X X 
XXX 
X X X X 

H — \ — \ — h 



X 
X X 

H — \ — \ — \ — F- 



X 
X 

H — \ — h 



1 2 3 4 5 6 7 8 9 10 11 12 13 14 
Number of hours 
mean: 

0+1 + 1 + 1 + 1+2 + 2 + 2 + 3 + 3 + 5 + 7 + 7+14+14 



15 



= ^=4.2 
15 



The mean is 4.2. 



Most of the people exercised fewer than 4 hours, so the mean does 
not describe the data set best. 

median: 

0, 1, 1, 1, 1,2.2,2,3,3,5,7,7, 14, 14 
The median is 2. 

The median best describes the data set because a majority of the 
data is clustered around the data value 2. 

mode: 

The greatest number of X's occur above the number 1 on the 
line plot. 
The mode is 1. 

The mode represents only 4 of the 15 people. The mode does not 
describe the entire data set. 



In the data set in Example 2, the value 14 is much greater than the 
other values in the set. An extreme value such as this is called an 
outlier. Outliers can greatly affect the mean of a data set. 



Measure 


Most Useful When 


mean 


the data are spread fairly evenly 


median 


the data set has an outlier 


mode 


the data involve a subject in which many data points of 




one value are important, such as election results 



386 Chapter 7 Collecting, Displaying, and Analyzing Data \ 'J'hjb'j] Lessor Tutorials Online 



EXAMPLE [3] Exploring the Effects of Outliers on Measures 
of Central Tendency 



The table shows the number of art 
pieces created by students in a 
glass-blowing workshop. Identify 
the outlier in the data set, and 
determine how the outlier affects 
the mean, median, and mode of the 
data. Then tell which measure of 
central tendency best describes the 
data with and without the outlier. 
The outlier is 14. 



Name 


Nl 

i 

5 


imber of y 
Pieces 1 


Suzanne 




Glen 


1 




Charissa 


3 




Eileen 


4 




Hermann 


14 




Tom 


2 


Qhi 



Without the Outlier 



mean: 

5+1+3+4+2 



= 3 



With the Outlier 

mean: 

5+1+3 + 4+14 + 



«4.8 



Caution! 



V////f 



Since all the data 
values occur the same 
number of times, the 
set has no mode. 



The mean is 3. The mean is about 4.8. 

The outlier increases the mean of the data by about 1.8. 



median: 

1,2,3,4,5 



median: 

1,2,3,4,5, 14 

H^ = 3.5 



The median is 3. The median is 3.5. 

The outlier increases the median of the data by 0.5. 

mode: mode: 

There is no mode. There is no mode. 

The outlier does not change the mode of the data. 

The median best describes the data with the outlier. The mean and 
median best describe the data without the oudier. 



Think and Discuss 

1. Describe a situation in which the mean would best describe a 
data set. 

2. Tell which measure of central tendency must be a data value. 

3. Explain how an outlier affects the mean, median, and mode of a 
data set. 



VjilsD Lesson Tutorials Online my.hrw.com 



7-2 Mean, Median, Mode, and Range 387 



7-2 



..'■i.-ti^--W»«KS«* 



nyjdti^ 



liJ 



<iii(*riiiii[ 

Homework Help Online go.hrw.com, 



keyword MJMhWAM ® 
Exercisesl-n,13,15 



GUIDED PRACTICE 



See Example 1 Find the mean, median, mode, and range of each data set. 



See Example 2 



See Example 3 



2. 44,68,48,61,59,48,63,49 



H — \ — \ — \ — h 



+ 



4- 



150 F 200 F 250 F 300 F 350 F 400 F 450 F 



1. 5,30,35,20,5,25,20 

3. The line plot shows cooking 
temperatures required by 
different recipes. Which 
measure of central tendency 
best describes the data? 
Justify your answer. 

4. The table shows the number of glasses of water consumed in one day. Identify 
the outlier in the data set. Then determine how the outlier affects the mean, 
median, and mode of the data. Then tell which measure of central tendency 
best describes the data with and without the outlier. 



Water Consumption 


Name 


Randy 


Lori 


Anita Jana Sonya 


Victor 


Mark 


Jorge 


Glasses 


4 


12 


3 i 1 1 4 


7 


5 


4 



INDEPENDENT PRACTICE 



See Example 1 Find the mean, median, mode, and range of each data set. 

5. 92, 88, 65, 68, 76, 90, 84, 88, 93, 89 6. 23, 43, 5, 3, 4, 14, 24, 15, 15, 13 



7. 2.0,4.4,6.2,3.2,4.4,6.2,3.7 



8. 13.1, 7.5, 3.9, 4.8, 17.1, 14.6, 8.3, 3.9 



See Example 2 



9. The line plot shows the number 
of letters in the spellings of the 
12 months. Which measure of 
central tendency best describes 
the data set? Justify your answer. 



H \ — h 



H — \ — h 



01 23456789 10 11 12 



See Example 3 Identify the outlier in eacli data set. Then determine how the outlier affects 
the mean, median, and mode of the data. Then tell which measure of central 
tendency best describes the data with and without the outlier. 

10. 13, 18, 20, 5, 15, 20, 13, 20 11. 45, 48, 63, 85, 151, 47, 88, 44, 68 



Extra Practice 



See page EP18. 



PRACTICE AND PROBLEM SOLVING 



12. Health Based on the data from three annual checkups, Jon's mean height is 
62 in. At the first two checkups Jon's height was 58 in. and 61 in. What was 
his height at the third checkup? 



388 Chapter 7 Collecting, Displaying, and Analyzing Data 



®*r?. 13. 




The Leadville Trail 
100 Mountain 
Bicycle Race is a 
100-mile mountain- 
bike race held in 
Leadville, Colorado. 
Bikers climb over 
12,000 ft through- 
out the Sawatch 
Range. In 2007, 
David Wiens won 
his fifth straight 
race. 



Find the mean, median, and mode 
of tlie data displayed in tlie line plot. 
Then determine how the outlier 
affects the mean. 



X X 

XX XX 

H — \ — h 



X 

X 

XXX 

4 



^ — \ — \ — h 



X 

H — h 



2 4 6 8 10 12 14 16 18 20 22 



16. 



17. 



Critical Thinking The values in a data set are 95, 93, 91, 95, 100, 99, and 
92. Wliat value can be added to the set so that the mean, median, and 
mode remain the same? 

Sports The ages of the participants in a moimtain bike race are 14, 23, 
20, 24, 26, 17, 21,31, 27, 25, 14, and 28. Make a stem-and-leaf plot of the data 
and find the mean, median, and mode. Which measure of central tendency 
best represents the ages of the participants? Explain. 

Estimation The table shows the monthly 
rainfall in inches for six months. Estimate the 
mean, median, and range of the data. 



What's the Question? The values in a data 
set are 10, 7, 9, 5, 13, 10, 7, 14, 8, and 11. What 
is the question about central tendency that 
gives the answer 9.5 for the data set? 

18. Write About It Which measure of central 
tendency is most often affected by including 
an outlier? Explain. 

l^ 19. Challenge Pick a measure of central tendency that describes each 
situation. Explain your choice. 

a. the number of siblings in a family b. the number of days in a month 



Month 


Rainfall (in.) 


Jan 


4.33 


Feb 


1.62 


Mar 


2.17 


Apr 


0.55 


May 


3.35 


Jun 


1.14 



Test Prep and Spiral Review 



20. Multiple Choice What is the 
mean of the winning scores 
show^n in the table? 



CE) 276 
CE) 276.8 



CD 282. 
CE) 285 



Masters Tournament Winning Scores 


Year 


2001 2002 2003 2004 


2005 


Score 


272 276 281 279 


276 



21. Multiple Choice In which data set are the mean, median, and mode all the 
same number? 



CD 6,2,5,4,3,4, 1 
C5) 4,2,2, 1,3,2,3 



CE) 2,3,7,3,8,3,2 

CD 4,3,4,3,4,6,4 



22. Brett deposits $4,000 in an account that earns 4.5% simple interest. How 
long will it be before the total amount is $4,800? (Lesson 6-7) 

23. Make a stem-and-leaf plot of the following data: 48, 60, 57, 62, 43, 62, 45, 
and 51. (Lesson 7-1) 



7-2 Mean, Median, Mode, and Range 389 




Bar 



& 



7.4.4 Analyze data displays, including ways that they can be misleading. Analyze 
ways in which the wording of questions can influence survey results. 

Hundreds of different languages 
are spoken around the world. The 
graph shows the numbers of native 
speakers of four languages. 



Vocabulary 

bar graph 
double-bar graph 
histogram 

( EXAMPLE 



A bar graph can be used to display 
and compare data. The scale of a 
bar graph should include all the 
data values and be easily divided 
into equal intei-vals. 



Most Widely Spoken Languages 



English 
Hindi 


™ 








Mandarin 




Spanish 


_i 1 



200 400 600 800 1,000 

Number of speakers (millions) 



Interpreting a Bar Graph 

Use the bar graph to answer each question. 

A Which language has the most native speakers? 

The bar for Mandarin is the longest, so Mandarin has the most 
native speakers. 



EXAMPLE 




.4 Honduras 
■^^ i... ..jiNicaragua 



El "'-.i^^'"^ 
Salvador * \ 



B About how many more people speak Mandarin than speak Hindi? 

About 500 million more people speak Mandarin than speak Hindi. 



You can use a double-bar graph to compare two related sets of data. 







Making a Double-Bar Graph 

The table shows the life 
expectancies of people in three 
Central American countries. Make 
a double-bar graph of the data. 
Step 1: Choose a scale and 

interval for the vertical 

axis. 

Step 2: Draw a pair of bars for 
each country's data. Use 
different colors to show 
males and females. 

Step 3: Label the axes and give 
the graph a title. 

Step 4: Make a key to show what 
each bar represents. 



Country 


Male 


Female 


El Salvador 

Honduras 

Nicaragua 


67 
63 
65 


74 
66 
70 



Life Expectancies in 
Central America 





El Salvador 
\ai Male 19 Female 



Honduras Nicaragua 



390 Chapter 7 Collecting, Displaying, and Analyzing Data 



yjilai; Lesson Tutorials Online 



A histogram is a bar graph that shows the frequency of data within 
equal intervals. There is no space between the bars in a histogram. 



EXAMPLE 



S) 



Making a Histogram 

The table below shows survey results about the number of CDs 
students own. Make a histogram of the data. 



Number of CDs 


1 


III 


5 


Ml 


9 


Ml 


13 


M nil 


17 
18 


Mini 
jmn "" 


2 


II 


6 


III 


10 


MM 


14 


MM 1 


3 


M 


7 


Mill 


11 


MMl 


15 


MM 1 


19 


II 


4 


Ml 


8 


Mil 


12 


MM 


16 


MMl 


20 


Ml 



Step 1: Make a frequency table 

of the data. Be sure to 

use a scale that includes 

all of the data values 

and separate the scale 

into equal intervals. 

Use these intervals on 

the horizontal axis of 

your histogram. 
Step 2: Choose an appropriate scale and 

interval for the vertical axis. The 

greatest value on the scale should 

be at least as great as the greatest 

frequency. 

Step 3: Draw a bar for each interval. 
The height of the bar is the 
frequency for that interval. Bars 
must touch but not overlap. 

Step 4: Label the axes and give the 
graph a title. 



Number of CDs 


Frequency 


1-5 


22 


6-10 


34 


11-15 


52 


16-20 


35 



E^^^iaa 




fo' VV .^V3 

Number of CDs 



Think and Discuss 

1. Explain how to use the frequency table in Example 3 to find the 
number of students surveyed. 

2. Explain why you might use a double-bar graph instead of two 
separate bar graphs to display data. 

3. Describe the similarities and differences between a bar graph and 
a histogram. 



'MbD Lesson Tutorials OnlinE mv.hrw.com 



7-3 Bar Graphs and Histograms 391 




keyword HiaEBOl W 
Exercises 1-10, 11, 17 



See Example 1 



GUIDED PRACTICE 



See Example 2 



See Example 3 



The bar graph shows the average 
amount of fresh fruit consumed per 
person m the United States in 1997. Use 
the graph for Exercises 1-3. 

1 . Which fruit was eaten the least? 

2. About how many pounds of apples 
were eaten per person? 

3. About how many more pounds of 
bananas than pounds of oranges 
were eaten per person? 

4. The table shows national average SAT scores 
for three years. Make a double-bar graph of 
the data. 

5. The list below shows the ages of musicians in a 
local orchestra. Make a histogram of the data. 

14, 35, 22, 18, 49, 38, 30, 27, 45, 19, 35, 46, 27, 21, 32, 30 



Fresh Fruit Consumption 




Year 


Verbal 


Math 


1980 


502 


492 


1990 


500 


501 


2000 


505 


514 



INDEPENDENT PRACTICE 



See Example 1 



The bar graph shows the maximum 
precipitation in 24 hours for several 
states. Use the graph for Exercises 6-8. 

6. Which state received the most 
precipitation in 24 hours? 

7. About how many inches of 
precipitation did Virginia receive? 

8. About how many more inches 
of precipitation did Oklahoma 
receive than Indiana? 



See Example 2 




9. 



See Example 3 10. 



Extra Practice 



See page EP18, 



The table shows the average annual 
income per capita for three U.S. states. 
Make a double-bar graph of the data. 

The list below shows the results of a 
typing test in words per minute. Make 
a histogram of the data. 

62, 55, 68, 47, 50, 41, 62, 39, 54, 70, 56, 47, 71, 55, 60, 42 



State 


2000 


2005 


Alabama 


$23,521 


$29,136 


Indiana 


$26,933 


$31,276 


Ohio 


$27,977 


$32,478 



392 Chapter 7 Collecting, Displaying, and Analyzing Data 



In 1896 and 1900, William McKinley, a Republican, and 
William Jennings Bryan, a Democrat, ran for president of 
the United States. The table shows the number of electoral 
votes each man received in these elections. 

1 1 . Use the data in the table to 
make a double-bar graph. 
Label the horizontal axis 
with the years. 

12. Estimation In 1896, about how many more 
electoral votes did McKinley get than Br\?an? 



13. 



14. 



Candidate 


1896 


1900 


McKinley 


271 


292 


Bryan 


176 


155 



The frequency table shows 
the number of years the first 
42 presidents spent in office. 
Find the median and mode 
of the data. 

Use the frequency table to 
make a histogram. What 
percent of the presidents 
spent 12-14 years in office? 



Years in Office 


Frequency 


0-2 


7 


3-5 


22 


6-8 


12 


9-11 





12-14 


1 




William McKinley 



15. vvp Write About It What does your histogram show you about 
the number of years the presidents spent in office? 



i 



Test Prep and Spiral Review 



Use the graph for Exercises 16 and 17. 

16. Multiple Choice In which year did the Democrats 
get the fewest number of electoral votes? 



CSj 1988 
CD 1996 



C^. 2000 
CS:' 2004 



Electoral Votes Cast 



S 500 



200 

100 






17. Gridded Response In which year was the 
difference between the number of electoral votes 
for the Republicans and Democrats the least? 



Determine whether the ratios are proportional. (Lesson 4 3) 
10 15 ,„ 5 10 ^„ 2 3 



I Democrats 



i 1992 1996 
I Republicans Year 



2000 2004 



18 



24' 36 



19. 



22' 27 



20. 



ZO' 30 



21. 



72 9_ 
96' 12 



Find the mean, median, mode, and range of each data set. (Lesson 7-2) 

22. 42, 29, 49, 32, 19 23. 15, 34, 26, 15, 21, 30 24. 4, 3, 3, 3, 3, 4, 1 



7-3 Bar Graphs and Histograms 393 






Reading and Interpreting 
Circle Graphs 







7.4.4 Analyze data displays, including ways that they can be misleading Analyze 
ways in which the wording of questions can influence survey results, 

A circle graph, also called a pie chart, shows how 
a set of data is divided into parts. The entire circle 
contains 100% of the data. Each sector, or slice, 
of the circle represents one part of the entire 
data set. 



Vocabulary 

circle graph 
sector 



EXAMPLE 




The circle graph 
compares the number 
of species in each group 
of echinoderms. 
Echinoderms are marine 
animals that live on the 
ocean floor. The name 
ecliiuoderm means 
"spiny-skinned." 



Life Science Application 

Use the circle graph to answer each question. 

A Which group of echinoderms includes the greatest number of 
species? 

The sector for brittle stars and basket stars is the largest, so this 
group includes the greatest number of species. 

B Approximately what percent of echinoderm species are sea stars? 

The sector for sea stars makes up about one-fourth of the circle. 
Since the circle shows 100% of the data, about one-fourth of 
100%, or 25%, of echinoderm species are sea stars. 

C Which group is made up of fewer species — sea cucumbers or 
sea urchins and sand dollars? 

The sector for sea urchins and sand dollars is smaller than the 
sector for sea cucumbers. This means there are fewer species of 
sea urchins and sand dollars than species of sea cucumbers. 



394 Chapter 7 Collecting, Displaying, and Analyzing Data VlJaLiI Lesson Tutorials OnlinE 



EXAMPLE 



f3 



Interpreting Circle Graphs 

Leon surveyed 30 people about pet ownership. The circle graph 
shows his results. Use the graph to answer each question. 



How many people do not own pets? 

The circle grapli sliows that 50% 
of the 30 people do not own pets. 
50% of 30 = 0.5-30 

= 15 
Fifteen people do not own pets. 

How many people own cats only? 

The circle graph shows that 20% 
of the 30 people own cats only. 
20% of 30 = 0.2 -30 

= 6 
Six people own cats only. 



Pet Survey Results 



Dogs only 

20% 



No pets 

50% 




EXAMPLE |3J Choosing an Appropriate Graph 




Shenandoah National 
Park, located near 
Waynesboro, Virginia, 
covers 199,017 acres. 
The highest moun- 
tain in the park is 
Hawksbill Mountain, 
standing at 4,050 ft. 



Decide whether a bar graph or a circle graph would best display 
the information. Explain your answer. 

A the percent of a nation's electricity supply generated by each of 
several fuel sources 

A circle graph is the better choice because it makes it easy to see 
what part of the nation's electricity comes from each fuel source. 

B the number of visitors to Shenandoah National Park in each of 
the last five years 

A bar graph is the better choice because it makes it easy to see 
how the number of visitors has changed over the years. 

C the comparison between the time spent in math class and the 
total time spent in school each day 

A circle graph is the better choice because the sector that 
represents the time spent in math class could be compared to 
the entire circle, which represents the total time spent in school. 





Think and Discuss 






1. Describe two ways a circle 


graph 


can be used to compare data. 


2. Compare the use of circle 
display data. 


graphs 


with the use of bar graphs to 



'Mi'j\ Lesson Tutorials OnlinE inv.hrw.com 7-4 Reading and Interpreting Circle Graphs 395 



7-4 



J^i,i^:^^!i^-fS'di>ii-iJ:*jifi.Zi^'lA''M>^.^^ 



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keyword ■BiaiifcBM ® 
Exercises 1-10, 11, 13 



See Example 1 



See Example 2 



Girriifiisif^SiiOitE 



The circle graph shows the estimated 
spending on advertising in 2000. 
Use the graph for Exercises 1-3. 

1 . On which Vi,rpe of advertising was the 
least amount of money spent? 

2. Approximately what percent of spending 
was on radio and magazine advertising? 

3. Television and magazine advertising 
made up about 50% of all advertising 
spending in 2000. If the total amount 
spent was $100,000, about how much 
was spent on television and magazine 
advertising? 



Money Spent on Advertising 



Television 



Outdoor 




Newspaper 



Radio 



Magazine 



Source USA Today 



See Example 3 Decide whether a bar graph or a circle graph would best display the 
information. Explain your answer. 

4. the lengths of the five longest rivers in the world 
L 5. the percent of citizens who voted for each candidate in an election 



See Example 1 



See Example 3 



INDEPENDENT PRACTICE 



The circle graph shows the results 
of a survey of 100 teens who were 
asked about their favorite sports. 
Use the graph for Exercises 6-8. 

6. Did more teens pick basketball or 
tennis as their favorite sport? 

7. Approximately what percent of teens 
picked soccer as their favorite sport? 



See Example 2 8. 



According to the sui'vey, 5% of teens 
chose golf. What is the number of 
teens who chose golf? 



Sports Survey Results 



Tennis 



Soccer 



Golf 




Basketball 



Baseball 



Decide whether a bar graph or a circle graph would best display the 
information. Explain your answer. 

9. the number of calories eaten at breakfast compared with the total number 
of calories eaten in one day 

10. the number of inches of rain that fell each month in Honolulu, Hawaii, 
during one year 



396 Chapter 7 Collecting, Displaying, and Analyzing Data 



PRACTICE AND PROBLEM SOLVING 



Extra Practice 



Seepage EP18. 



Geography The circle graph shows the 
percent of Earth's land area covered by each 
continent. Use the graph for Exercises 11-13. 

1 1 . List the continents in order of size, from 
largest to smallest. 

12. Approximately what percent of Earth's 
total land area is Asia? 

13. Approximately what percent of Earth's 
total land area is North America and 
South America combined? 



Area of Continents 



North 
America 



South 
America 

Europe 




Asia 



Africa 



14. Critical Thinking A group of 200 students were asked how tliey like to spend 
their free time. Of the students surveyed, 47% said they like to play on the 
computer, 59% said they like to go to the mall, 38% said they like to go to the 
movies, and 41% said they like to play sports. Can you make a circle graph to 
display this data? Explain. 

15. What's the Error? The table shows the t>'pes 
of pets owned by a group of students. A circle 
graph of the data shows that 25% of the students 
surveyed orai a dog. Why is the graph incorrect? 

@ 16. Write About It What math skills do you use 

when interpreting information in a circle graph? 

^ 17. Challenge Earth's total land area is approximately 57,900,000 square miles. 
Antarctica is almost 10%) of the total area. What is the approximate land area 
of Antarctica in square miles? 



Pet 


Number of Students 


Cat 


MMM 


Dog 


MMi 


Fish 


M 


Other 


M 



L 



Test Prep and Spiral Review 



Use the graph for Exercises 18 and 19. 
18. Multiple Choice Approximately what percent of 
the medals won by the United States were gold? 



U.S. Distribution of Medals 

Summer Olympics, 

1988-2004 



CK) 25% ^:S) 40% CD 50% (E) 75% 

19. Short Response The United States won a total of 
502 medals in the Summer OlyTnpics from 1988 to 
2004. About how many of these were bronze medals? 
Show yotrr work. 

20. lose has an American flag that measures 10 inches by 
19 inches. He paints a picture of a flag that is 60 inches 

by 1 14 inches. Will his painted flag be similar to the American flag? (Lesson 4-8) 




Compare. Write <, >, or =. (Lesson 6-2) 



21. 0.1 



0.09 



22. 1.71 



24 



23. 1,25 



125% 



24. 32.5 



69% 



7-4 Reading and Interpreting Circle Graphs 397 




7.4.4 Analyze data displays, including ways thatthey can be misleading Analyze 
ways in which the wording of guestions can influence survey results 

Carson is planning a deep-sea fishing trip. 
He chooses a fishing charter based on the 
number offish caught on difl'erent charters. 

A box-and-whisker plot uses a number 
line to show the distribution of a set 
of data. 



Vocabulary 

box-and-whisker plot 
lower quartile 
upper quartile 
interquartile range 



i EXAMPLE 



Caution! 



I////// 

To find the median of 
a data set with an 
even number of 
values, find the mean 
of the two middle 
values. 



To make a box-and-whisker plot, first 
divide the data into four parts using 
quartik's. The median, or middle quartile, 
divides the data into a lower half and an 
upper half. 'Hie median of the lower half 
is the lower quartile , and the median 
of the upper half is the upper quartile . 




9 



Making a Box-and-Whisker Plot 

Use the data to make a box-and-whisker plot. 

26, 17, 21, 23, 19, 28, 17, 20, 29 

Step 1: Order the data from least to greatest. Then find the least 
and greatest values, the median, and the lower and upper 
quartiles. 



Least 
value 



17 




20 



Lower quartile 




Greatest 
value 



17+ 19 



Median 



Upper quartile 



26 + 28 



27 



Step 2: Draw a number line. Above the number line, plot a point for 
each value in Step 1. 

• • • • • 



H — I — h 
10 



H \ h 



H — \ — I — y- 



H \ \ 1 \ 1 h 



15 



20 



25 



30 



Step 3: Draw a box from the lower to the upper quartile. Inside the 
box, draw a vertical line through the median. Then draw the 
"whiskers" from the box to the least and greatest values. 



-^IZI 



H h 



H 1 1 1 1 1 1 1 1 1 \ 1 1 1 1 h 



10 



15 



20 



25 



30 



398 Chapter 7 Collecting, Displaying, and Analyzing Data \ 'yjilap] Lesson Tutorials OnlinE 



The interquartile range of a data set is the difference between tlie 
lower and upper quartiles. It tells how large the spread of data around 
the median is. 

You can use a box-and-whisker plot to analyze how data in a set are 
distributed. You can also use box-and-whisker plots to help you compare 
two sets of data. 



EXAMPLE r2j Comparing Box-and-whisker Plots 



The box-and-whisker plots below show the distribution of the 
number of fish caught per trip by two fishing charters. 



H — \ — \ — I — \ — h 



H — I — \ — \ — \ — h 



l = Reel-to-Reel 

Charters 
I = Mud Puppy 

Charters 



20 30 40 50 60 

Number of fish 



70 80 



A Which fishing charter has a greater median? 

The median number offish caught on Reel-to-Reel Charters, 
about 54, is greater than the median number of fish caught 
on Mud Puppy Charters, about 51. 

B Which fishing charter has a greater interquartile range? 

The length of the box in a box-and-whisker plot indicates the 
interquartile range. Reel-to-Reel Charters has a longer box, so 
it has a greater interquartile range. 

C Which fishing charter appears to be more predictable in the 
number of fish that might be caught on a fishing trip? 

The range and interquartile range are smaller for Mud Puppy 
Charters, which means that there is less variation in the data. 
So the number offish caught on Mud Puppy Charters is more 
predictable. 



Tiiink and Discuss 

1. Describe what you can tell about a data set from a 
box-and-whisker plot. 

2. Explain how the range and the interquartile range of 

a set of data are different. Wliich measure tells you more 
about central tendency? 



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7-5 Box-and-whisker Plots 399 








Homework Help Online go.hrw.com 



keyword ■BHIilifl.-M ® 

Exercises 1-8,9, 11, 19 



GUIDED PRACTICE 



See Example 1 Use the data to make a box-and-whisker plot. 

1. 46 35 46 38 37 33 49 42 



35 40 37 



See Example 2 



Use the box-and-whisker plots of inches flown by two different paper 
airplanes for Exercises 2-4. 



2. Which paper airplane has a greater 
median flight length? 

3. Which paper airplane has a greater 
interquartile range of flight lengths? 

4. Which paper airplane appears to have 
a more predictable flight length? 



Airplane 
Airplane B 




i 


r ' 
















1 1 






, 






■ 



H — \ — I — \ — I — \ — \ — \ — \ — h 



160 170 180 190 200 
Length of flight (in.) 



210 



INDEPENDENT PRACTICE 



See Example 1 Use the data to make a box-and-whisker plot. 

5. 81 73 88 85 81 72 86 72 79 



75 76 



See Example 2 Use the box-and-whisker plots of apartment rental costs in two different 
cities for Exercises 6-8. 



6. Which city has a greater median 
apartment rental cost? 

7. Which city has a greater interquartile 
range of apartment rental costs? 

8. Which city appears to have a more 
predictable apartment rental cost? 



City A •-V 



City B < 



-\ \ 1 \ 1 \ 1 \ \ 1 h 



375 425 475 525 575 
Rental cost (S) 



625 



Extra Practice 



See page EP9. 



PRACTICE AND PROBLEM SOLVING 



The points scored per game by a basketball player are shown below. Use the 
data for Exercises 9-11. 

12 7 15 23 10 18 39 15 20 8 13 

9. Make two box-and-whisker plots of the data on the same number line: 
one plot with the outlier and one plot without the outlier. 

10. How does the outlier affect the interquartile range of the data? 

1 1 . Which is affected more by the outlier: the range or the interquartile range? 



12. Make a box-and- 
whisker plot of the 
data shown in the 
line plot. 



H — h 



+ 



X 

X 

X X 

-f 



1 — h 



X 

X X 

X X X X X 

X X X X X X 

H — \ — \ — \ — \ — h 



H — I — h 



400 



40 44 

Chapter 7 Collecting, Displaying, and Analyzing Data 



48 



52 



X 

-+- 
56 



13. Sports The table shows the countries that were the top 15 medal winners 
in the 2004 Olympics. 




Country 


Medals 


Country 


Medals 


Country 


Medals 


USA 


103 


Russia 


92 


China 


63 


Australia 


49 


Germany 


48 


Japan 


37 


France 


33 


Italy 


32 


Britain 


30 


Korea 


30 


Cuba 


27 


Ukraine 


23 


Netherlands 


22 


Romania 


19 


Spain 


19 



a. Make a box-and-whisker plot of the data. 

b. Describe the distribution of the number of medals won. 



14. Measurement The stem-and-leaf plot shows the 
heights in inches of a class of seventh graders. 

a. Make a box-and-whisker plot of the data. 

b. Three-fourths of the students are taller 
than what height? 

c. Three-fourths of the students are shorter 
than what height? 



Student Heights 
Stems Leaves 



356688899 
111112 2 2 4 
Key: 5\3 means 53 



15. What's the Error? Using the data 2, 9, 5, 14, 8, 13, 7, 5, and 8, a student 
found the upper quartile to be 9. What did the student do wrong? 

^p 16. Write About It Two box-and-whisker plots have the same median and 
equally long whiskers. If the box of one plot is longer, what can you say 
about the difference between the two data sets? 

^ 17. Challenge An outlier is defined to be at least 1.5 times the interquartile 
range. Name the value that would be considered an outlier in the data 
set 1,2,4,2, 1,0, 6,8, 1,6, and 2. 



m 



Test Prep and Spiral Review 



Use the graph for Exercises 18 and 19. 

18. Multiple Choice Wliat is the difference between 
the interquartile ranges for the two data sets? 

CE) 21 CD 9 

CD 18 CD> 



H \ \ — \ — I — \ — \ — \ h 



12 



18 



24 



30 



19. Gridded Response What is the lower quartile of the box-and-whisker 
plot with the greater range? 

20. A tree casts a 21.25 ft shadow, while a 6 ft tall man casts a 10.5 ft shadow. 
Estimate the height of the tree. (Lesson 4-9) 

21. Mari spent $24.69 on lunch with her mom. About how much should she 
leave for a 15% tip? (Lesson 6-3) 



7-5 Box-and-Whisker Plots 401 



A LAB 



Explore 
Box-and-Whisker Plots 



Use with Lesson 7-5 



S^., 



Learn It Online 

Lab Resources Online go.hrw.com, 

MSlOLab? ■Go, 



You can use a graphing calculator to analyze data 
in box-and-whisker plots. 



Activity 1 



Ms. Garza's math class took a statewide math test. The data below are 
the scores of her 19 students. 

79, 80, 61, 66, 74. 92, 88, 75, 93, 61, 77, 94, 25, 79, 86, 85, 48, 99, 80 

Use a graphing calculator to make a box-and-whisker plot of the data. 

To make a list of the scores, press il=iBl| and choose Edit. Enter each 
value under List 1 (LI). 

Use the STAT PLOT editor to set up the box-and-whisker plot. 

STAT PLOT ^^^^ 

f'''^^^ 'ffPpj . Press m§3i to select Plotl. Turn the plot On and 

use the arrow to select the plot type. The box-and-whisker plot is the 
fifth type shown. 

The plot's values will come from the values listed in LI, so Xlist: LI 
should be visible. The Freq should also be set at 1. 

Press W{^^ and select 9: ZoomStat to display the plot. 

Press iSjSBSli and use the arrows to see the values of the least value 



LI 



79 
8fi 
K 



L1(19) =80 



— -_ 


■ -|| 


aos Plots pi«ti 
iSroff 

Type: Li: Ui; Jh, 
1 Kh- jji; (^ 

1 Xlist: Li 
Fre^: 1 


1 





■a> 



(minX), greatest value (maxX), median (Med), and lower (Q1) 
and upper (Q3) quartiles. 



Thinic and Discuss 



1. Wliat five values do you need to construct a box-and-whisker plot? 

Wliat values must you find before you can identify the upper and lower quartiles? 

2. Wliat does the box-and-whisker plot tell you about the data? 



Try This 



1 . Survey your classmates to find the number of U.S. states that each student has visited. Use 
your calculator to make a box-and-whisker plot of the data. 

2. Identify the least value, greatest value, range, median, lower quartile and upper quartile. Wliat 
is the range between the upper and lower quartile? 



402 Chapter 7 Collecting, Displaying, and Analyzing Data 



Activity 2 



Ray surveys 15 seventh-grade students and 15 teachers at his 
school to find the number of hours they sleep at night. 
The table shows the results. 





Average Number of Hours of Sleep Per Night 


Students 


9,7, 10, 6, 11, 7, 9, 10, 10, 7,9, 10,8, 9, 11 


Teachers 


7, 6, 8, 9, 8, 7, 10, 6, 7, 9, 6, 7, 5, 7, 8 



Use a graphing calculator to make a box-and-whisker plot for each 
set of data. 

Enter the first set of student data in LI. 

Press ipj^ to move right into the L2 column. 
Enter the teacher data. 

Set up Ploti as shown in Activity' 1. Repeat the steps to set up Plot2. 
Set the Xlist to L2 by pressing VSh 



Press jjIgiiTI I and select 9: ZoomStat to display both 
box-and-whisker plots. Press IHSl to display the statistics and 



use the left and right arrows to move along the plots. Use the up 
and down arrows to move between plots. The display in the left 
corner tells which plot (PI or P2) and which list (LI or L2) 
the statistics are for. 



Thinic and Discuss 



1 . How can you use the box-and-whisker plots to compare the 
ranges of the data sets? 

2. Make a Conjecture What do the graphs tell you about the 
sleeping habits of students and teachers? 





1 






L1 


L£ 


L3 2 






9 

? 

10 

6 

11 

? 

9 












1 





fVAl JHH F-lci:^ 

SB Off 
Type:L:. k± .Dn. 

SL «!!• L^ 

Klisf.Lz 



t*WBW'-A-!{V-.;^-:;;i.;^=-:iM;VW"S^.'^?^.vS:W45S 



1 






P1:L1 1 

f— - 


+ - 
















""•I-" , r 




'U 









Try Tliis 



1 . Survey the boys and the girls in your class to find how many minutes they each 
talk on the phone. Use your calculator to make separate box-and-whisker plots 
for each set of data. 

2. What are the least and greatest values and the median and lower and upper 
quartile for each box and-whisker-plot? 

3. Are there any differences in the plots? What do these differences tell you about 
boys talking on the phone as compared to girls? 



7-5 Technology Lab 403 



CHAPTER 



7 



SECTION 7A 



Ready To Go On? 



^£*9Learn It Online 



ResourcesOnlinego.hrw.com, 



(2r 



















Quiz for Lessons 7-1 Through 7-5 

7-1 ] Frequency Tables, Stem-and-Leaf Plots, and Line Plots 

The list shows the top speeds of various land animals. 

42 55 62 48 65 51 47 59 67 61 49 54 55 52 44 

1 . Make a cumulative frequency table of the data. 

2. Make a stem-and-leaf plot of the data. 

3. Make a line plot of the data. 

7-2 ] Mean, Median, Mode, and Range 

The list shows the life spans in years of vampire bats in captivity. 
18 22 5 21 19 21 17 3 19 20 29 18 17 

4. Find the mean, median, mode, and range of the data. Round your 
answers to the nearest tenth of a year. 

5. Which measure of central tendency best represents the data? Explain. 

7-3 ] Bar Graphs and Histograms 

6. The table shows the numbers of students in the 
sixth and seventh grades who participated in school 
fairs. Make a double-bar graph of the data. 

7. The list below shows the numbers of tracks on a 
group of CDs. Make a histogram of the data. 

13, 7, 10, 8, 15, 17, 22, 9, 11, 10, 16, 12, 9, 20 

7-4 j Reading and Interpreting Circle Graphs 

Use the circle graph for problems 8 and 9. 

8. Approximately what percent of students picked 
cheese as their favorite topping? 

9. Out of 200 students, 25% picked pepperoni as their 
favorite pizza topping. How many students picked 
pepperoni? 

7-5 ) Box-and-Whisker Plots 

10. Make a box-and-whisker plot of the data 14, 8, 13, 20, 
15, 17, 1, 12, 18, and 10. 

1 1 . On the same number line, make a box-and-whisker plot 
ofthedataS, 8, 5, 12,6, 18, 14,8, 15, and 11. 

12. Which box-and-whisker plot has a greater interquartile range? 



School Fair Participation 


Fair 


Sixth 
Grade 


Seventh 
Grade 


Book 


55 


76 


Health 


69 


58 


Science 


74 


98 



Favorite Pizza Toppings 



Pepperoni 



Cheese 




Green 



P PP Sausage 



iVIushrooms 



404 Chapter 7 Collecting, Displaying, and Analyzing Data 



Focus cm Problem Soliiing 






Solve 

• Choose an operation: addition or subtraction 

In order to decide whether to add or subtract to solve a problem, 
you need to determine what action is taking place in the problem. 
If you are combining or putting together numbers, you need to 
add. If you are taking away or finding how far apart two numbers 
are, you need to subtract. 




Determine the action in each problem. Then determine which 
operation could be used to solve the problem. Use the table for 
problems 5 and 6. 



O Betty, Raymond, and Helen ran a 

three-person relay race. Their individual 
times were 48 seconds, 55 seconds, and 
51 seconds. What was their total time? 

The Scots pine and the sessile oak are 
trees native to Northern Ireland. The 
height of a mature Scots pine is 111 feet, 
and the height of a mature sessile oak is 
90 feet. How much taller is the Scots pine 
than the sessile oak? 

Mr. Hutchins has $35.00 to buy supplies 
for his social studies class. He wants to 
buy items that cost $19.75, $8.49, and 
$7.10. Does Mr. Hutchins have enough 
money to buy all of the supplies? 

O The running time for the 1998 
movie yi;7f~ is 83 minutes. Jordan 
has watched 25 minutes 
of the movie. How many 
minutes does he have 
left to watch? 



Sizes of Marine Mammals 


Mammal 


Weight (kg) 


Killer whale 


3,600 


Manatee 


400 


Sea lion 


200 


Walrus 


750 



The table gives the approximate weights 
of four marine mammals. How much 
more does the killer whale weigh than 
the sea lion? 

O Find the total weight of the manatee, the 
sea lion, and the walrus. Do these three 
mammals together weigh more or less 
than the killer whale? 




7-6 



Line Graphs 







7.4.4 Analyze data displays, including ways thatthey can be 

misleading. Analyze ways in which the wording of questions... 

You can use a line grapli 
to show how data changes 
over a period of time. In a 
line graph, line segments 
are used to connect data 
points on a coordinate grid. 
The result is a visual record 
of cliange. 



Vocabulary 

line graph 
double-line graph 







..can influence survey 
results. 



Line graphs can be used for a 
variety of reasons, including 
showing the growth of a dog over time 



EXAMPLE [1| Making a Line Graph 

Make a line graph of the data 
in the table. Use the graph to 
determine during which 2-month 
period the puppy's weight 
increased the most. 




To plot each point, 
start at zero. Move 
right for the time 
and up for the 
weight. 



Age (mo) 


Weight (lb) ' 





0.2 


2 


1.7 


4 


3.8 


6 


5.1 


8 


6.0 


10 


6.7 


12 


7.2 



Step 1: Determine the scale and 
intei"val for each axis. 
I^lace units of time on 
the horizontal axis. 



Step 2: Plot a point for each pair of values. Connect 
the points using line segments. 

Step 3: Label the axes and give the graph a title. 



Growth Rate of a Puppy 




The graph shows the steepest line segment between 2 and 4 months. 
This means the puppy's weight increased most between 2 and 4 months. 



406 Chapter 7 Collecting, Displaying, and Analyzing Data [VJil^^J Lesson Tutorials Online 



You can use a line graph to estimate values between data points. 



EXAMPLE 



Using a Line Graph to 
Estimate Data 

Use the graph to estimate the 
population of Florida in 1990. 

To estimate the population in 
1990, find the point on the line 
between 1980 and 2000 that 
corresponds to 1990. 
The graph shows about 
12.5 million. In fact, the 
population was 12.9 million 
in 1990. 



Florida Population 




A double-line graph shows change over time for two sets of data. 



EXAMPLE 



9 



Russia 



^^' Nome 



Canada 



Alaska 



Anchorage* 



-U 



i^ 



Gulf of Alaska 



Making a Double-Line Graph 

The table shows the normal daily 
temperatures in degrees Fahrenheit 
in two Alaskan cities. Make a 
double-line graph of the data. 



Average Temperatures 




Month 


Nome 


Anchorage 


Jan 


7 


15 


Feb 


4 


19 


Mar 


9 


26 


Apr 


18 


36 


May 


36 


47 


Jun 


46 


54 



Plot a point for each 
temperature in Nome and 
connect the points. Then, 
using a different color, 
plot a point for each 
temperature in Anchorage 
and connect the points. 
Make a key to show what 
each line represents. 



Think and Discuss 

1. Describe how a line graph would look for a set of data that 
increases and then decreases over time. 

2. Give an example of a situation that can be described by a double- 
line graph in which the two sets of data intersect at least once. 



7-6 Line Graphs 407 



7-6 



''. Homework Help Online go.hrw.com, 



keyword ■mBiiiBMiM ® 

Exercises 1-7, 9, 15 



See Example 1 

See Example 2 
See Example 3 



GUIDED PRACTICE 



The table at right shows average movie theater ticket prices 
in the United States. Use the table for Exercises 1 and 2. 

1 . Make a line graph of the data. Use the graph to determine 
during which 5-year period the average ticket price 
increased the least. 

2. Use the graph to estimate the average ticket price in 1997. 

3. The table below shows the amount of apple juice and 
raw apples in pounds consumed per person in the 
United States. Make a double-line graph of the data. 





2001 


2002 


2003 


2004 


2005 


Apple Juice 


21.4 


21.3 


21.4 


23.1 


24.0 


Raw Apples 


17.5 


15.6 


16.0 


16.9 


19.1 



Year 


Price ($) 


1965 


1.01 


1970 


1.55 


1975 


2.05 


1980 


2.69 


1985 


3.55 


1990 


4.23 


1995 


4.35 


2000 


5.39 


2005 


6.41 



INDEPENDENT PRACTICE 



The table at right shows the number of teams 
in the National Basketball Association (NBA). 
Use the table for Exercises 4-6. 

See Example 1 4. Make a line graph of the data. Use the 
graph to determine during which 5-year 
period the number of NBA teams 
increased the most. 



See Example 2 



5. During which 5-year period did the 
number of teams increase the least? 

6. Estimation Use the graph to estimate 
the number of NBA teams in 1988. 



Year 


Teams 


1965 


9 


1970 


14 


1975 


18 


1980 


22 


1985 


23 


1990 


27 j 


1995 


27 M 


2000 


29 ■ 


2005 


30 ^ 




See Example 3 7. The table below shows the normal daily temperatures in degrees 

Fahrenheit in Peoria, Illinois, and Portland, Oregon. Make a double-line 
graph of the data. 





Jul 


Aug 


Sept 


Oct 


Nov 


Dec 


Peoria 


76 


73 


66 


54 


41 


27 


Portland 


68 


69 


63 


55 


46 


40 



Extra Practice 



See page EP19. 



PRACTICE AND PROBLEM SOLVING 



8. Critical Thinking Explain how the intervals on the vertical axis of a line 
graph affect the look of the graph. 



408 Chapter 7 Collecting, Displaying, and Analyzing Data 



•• ' ^'* * • . 9. Life Science The table shows the numbers of endangered species of 
\ n ni.^'x vertebrates for selected years between 1998 and 2004. 






1998 


2000 


2002 


2003 


2004 


Number of Species (thousands) 


3.31 


3.51 


3.52 


3.52 


5.19 



Wildfires can also 
be started natu- 
rally by lightning 
or lava. Fires can 
start when the 
lava flow ignites 
the vegetation. 
This is common 
in Hawaii. 



a. Make a line graph of the data in the table. 

b. Estimate the number of endangered species of vertebrates in 1999. 

Earth Science The graph shows the 
number of acres burned by wildfires 
in the United States from 2001 to 2006. 

a. During which years did wildfires 
burn more than 8 million acres? 



® 




b. Explain whether the graph would 
be useful in predicting future data. 

11. What's the Error? Denise makes a 
line plot to display how her town's 
population has changed over 1 years. '°'""' "■'*'°"'" '"•^'■'s'^"'^ '"' ^emer 
Explain which type of graph would be more appropriate. 

12. Write About It Explain the benefit of drawing a double-line graph rather 
than two single-line graphs for related sets of data. 

13. Challenge A line graph shows that a town's population was 4,500 in 1980, 
5,300 in 1990, and 6,100 in 2000. Assuming the population continues to 
grow at the same rate, what population will the line graph show in 2010? 



m 



Test Prep and Spiral Review 



Use the graph for Exercises 14 and 15. 

14. Multiple Choice During which period did 
the average cost of a major league baseball 
ticket increase the most? 



CX) 1991-1993 
CE) 1993-1997 



CD 1997-2001 
CE) 2001-2005 




1 5. Short Response Use the line graph to 
estimate the average cost of a major league baseball ticket in 2003. Explain. 

Write as a percent. Round to the nearest tenth of a percent, if necessary. Lesson 6-2) 

16. 0.15 17. 1.36 18. I 19. ^ 

20. Decide whether a bar graph or a circle graph would best display the average 
temperature for each day of one week. Explain your answer. (Lesson 7-4) 



7-6 Line Graphs 409 



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Use Venn Diagrams 

to Display Collected Data 



Wse w/f/i Lesson 7-6 



£?. 



Learn It Online 

Lab Resources Online go.hrw.com, 
■BBB£]MS10Lab7 BG7- 



You can use a Venn diagram to display relationships in data. 

Use ovals, circles, or other shapes to represent individual data sets. 



Activity 1 



At Landry Middle School, 127 students play a team sport, 145 play a 
musical instrument, and 31 do both. Make a Venn diagram to display 
the relationship in the data. 



O Draw and label two overlapping circles to represent 
the sets of students who play a team sport and a 
musical instrument. Label one "Team sport" and 
the other "Musical instrument." 



Team 
sport 






Musical 
instrument 



Write "31" in the area where the circles overlap. 
This is the number of students who play a musical 
instrument and a team sport. 

To find the number of students who play a team sport only, begin 
with the number of students who play a team sport, 127, and 
subtract the number of students who do both, 31. 

team sport - both = team sport only 

127 - 31 = 96 

Use the same process to find the number of students who play a 
musical instrument only. 

musical instrument - both = musical instrument only 



145 



- 31 = 



114 



O Complete the Venn diagram by adding the 

number of students who play o)ily a team sport 
and the number of students who play only a 
musical instrument to the diagram. 



Team 4«fe Musical 
Sport 31 instrument 
96 114 



410 Chapter 7 Collecting, Displaying, and Analyzing Data 



Think and Discuss 



1. Explain why some of the numbers that were given in Activity 1, 
such as 127 and 145, do not appear in the Venn diagram. 

2. Describe a Venn diagram that has three individual data sets. How 
many overlapping areas does it have? 



Try This 



Responding to a survey about favorite foods, 60 people said they like 
pasta, 45 said they like chicken, and 70 said they like hot dogs. Also, 
15 people said they like both chicken and pasta, 22 said they like both 
hot dogs and chicken, and 17 said they like both hot dogs and pasta. 
Only 8 people said they like all 3 foods. 

1 . How many people like only pasta? 

2. How many people like only chicken? 

3. How many people like only hot dogs? 

4. Make a Venn diagram to show the relationships in the data. 



Activity 2 



Q Interview your classmates to find out what kinds of movies they 
like (for example, action, comedy, drama, and horror). 

Q Make a Venn diagram to show the relationships in the collected 
data. 



Think and Discuss 



1. Tell how many individual sets and how many overlapping areas a 
Venn diagram of the movie data will have. 

2. Describe what a Venn diagram of student ages might look like. 
Would there be any overlapping sets? Explain. 



Try This 



1. Interview your classmates to find out what kinds of sports they like 
to play. Make a Venn diagram to show the relationships in the data. 

2. The Venn diagram shows the types of exercise that 
some students do. 



a. How many students were surveyed? 

b. How many students jog? 

c. How many students like to both bike and walk? 



biking 



12 



walking 



^'h 



jogging 



7-6 Hands-On Lab 411 



\ 



& 



7-7 



Choosing an Appropriate 
Display r^ 



7.4,1 Choose the appropriate display for a set of data from bar graphs, line graphs, 
circle graphs, line plots and histograms. Justify your choice. 

On a field trip to a butterfly park, 
students recorded the number of 
species of each butterfly family 
they saw. Wliich type of graph 
would best display the data 
they collected? 

There are several ways to display 
data. Some types of displays are 
more appropriate than others, 
depending on how the data is 
to be analyzed. 




ill 



X 
XXX 



Use a bar graph to 
display and compare 
data. 



Use a line plot to 

show the frequency 
of values. 



® 



Use a circle graph 

to show how a set 
of data is divided 
into parts. 

Use a line graph 

to show how data 
change over a 
period of time. 



CD 







1 

3 


79 
6 



Use a Venn diagram 

to show relationships 
between two or more 
data sets. 

Use a stem-and-leaf plot 

to show how often data 
values occur and how 
they are distributed. 



EXAMPLE [lj Choosing an Appropriate Display 



The students want to create a display to show the number of 
species of each butterfly family they saw. Choose the type of 
graph that would best represent the data in the table. Explain. 



There are distinct 
categories sliowing 
tine number of 
species seen in eacli 
butterfly family. 



A bar graph can be used to display data in categories. 

The students want to create a display to show the population of 
butterflies in the park for the past few years. Choose the type of 
graph that would best represent this data. Explain. 

A line graph would best represent data that gives population 
over time. 



Butterfly Family 


Number of Species 


Gossamer-wing 


7 


Skippers 


10 


Swallowtails 


5 


Whites and sulphurs 


4 



412 Chapter 7 Collecting, Displaying, and Analyzing Data {y'fld'j] Lesson Tutorials Online 



EXAMPLE {2J Identifying the Most Appropriate Display 

The table shows the amount of 
time the students spent at the 
different exhibits at the butterfly 
park. Explain why each display 
does or does not appropriately 
represent the data. 



Exhibit 


Time (min) 


Butterflies 


60 


Insects 


45 


Invertebrates 


30 


Birds 


15 



Stems 


Leaves 


1 


5 


2 




3 





4 


5 


5 




6 


/ 



A stem-and-leaf plot shows how often data 
values occur and how they are distributed. 

Key: 2\0 means 20 

There are only four data values, and how often they occur and 
how they are distributed are not important. 



Butterflies 


Insects 




60 


45 


A Venn diagram shows the 
relationship between two 


Invertebrates 


Birds 


or more data sets. 


30 


15 





There is no relationship among the times spent at each exhibit. 

C Birds —|__^^^ Butterflies 

Invertebrates /\ \ ^ ^i^^,^ ^^^^^ ^^^^^ ^^^ 

a set of data is divided 
into parts. 
Insects 

This circle graph appropriately shows the proportionate 
amount of time spent at each exhibit. 




H — \ \ h 



H — I — I — I — h 



10 15 20 25 30 35 40 45 50 55 60 65 
How often the data values occur is not important 



A line plot shows 
frequency of values. 



^^^B^^^n^^^^^^^n^^^^i^^^^Hiiim 


ThiHk and Discuss 




1. Explain how data displayed in a stem 


-and-leaf plot and data 


displayed in a line plot are similar. 




2. Describe a set of data that could best be displayed in a line graph. 1 



^fiiibii Lesson Tutorials OnlinE mv.hrw.com 



7-7 Choosing an Appropriate Display 413 



7-7 



<iit<*Tiiiiii 

Homework Help Online qo.hrw.com, 



keyword ■mmwiBiM ® 
Exercises 1-8, 15 



GUIDED PRACTICE 



See Example 1 Choose the type of graph that would best represent each type of data. 

1. the prices of the five top-selhng 42-inch plasma televisions 

2. the height of a person from birth to age 21 

See Example 2 The table shows Keiffer's earnings for a month. 
I Explain why each display does or does not 
appropriately represent the data. 



Week 


1 1 2 


3 


4 


Earnings (S) 


20 30 


15 


25 



.^^i..L:^^-J^^f.. 



-^ 30 
I 10 






Week 1 Week 2 Week 3 Week 4 



X X X X 

-\ — I — I — \ — h- 

10 20 30 



INDEPENDENT PRACTICE 



See Example 1 Choose the type of graph that would best represent each type of data. 

5. the number of tracks on each of the 50 CDs in a CD collection 

6. the number of runners in a marathon for the last five years 

See Example 2 The table shows the number of people who participate in various activities. 
Explain why each display does or does not appropriately represent the data. 



Activity 


Biking Hiking Skating Jogging 


Number of People 


35 


20 


25 15 



Activity Participation 




Stems 


Leaves 


1 


5 


2 


5 


3 


5 



Key: l\5 means 15 



See page EP19. 



PRACTICE AND PROBLEM SOLVING 



9. The data gives the number of boolcs 25 students read last summer. 
7, 10, 8, 6, 0, 5, 3, 8, 12, 7, 2, 5, 9, 10, 15, 8, 3, 1, 0, 4, 7, 10, 8, 2, 11 
Make the type of graph that would best represent the data. 



414 Chapter 7 Collecting, Displaying, and Analyzing Data 



J 




Food 


Protein (g) 


Egg 


6 


Milk 


8 


Cheese 


24 


Roast beef 


28 



Komodo Dragons 
have a poor sense 
of hearing and 
sight. To make up 
for this lack of 
senses, they use 
their tongue to 
taste and smell. 
With a favorable 
wind they can 
smell their meal 
from 6 miles away. 



13. 



©14. 



^15. 



Nutrition The table shows the amount of protein 
per serving in various foods. Draw two different 
displays to represent the data. Explain your choices. 

Yoko wants to use a stem-and-leaf plot to show 
the growth of the sweet peas that she planted last 
year. She measured how much the vines grew each 
month. Explain why Yoko's display choice may or 
may not best represent the data. 

Life Science Komodo Dragons are the world's 
largest lizard species. The table shows the weights of 
some adult male Komodo Dragons. Make the type of 
graph that would best represent the information. 

Choose a Strategy Five friends worked together 

on a project. Matti, Jerad, and Stu all worked the 

same length of time. Tisha worked a total of 3 hours, 

which was equal to the total amount of time that 

Matti, Jerad, and Stu worked. Pablo and Matti together worked -, of the 

total amount of time that the five friends worked. Make the type of graph 

that would best represent the information. 

Write About It Is a circle graph always appropriate to represent data 
stated in percents? Explain your answer. 

Challenge The table shows the results of a survey of 50 people about 
their favorite color. What type of display would you choose to represent 
the data of those who chose blue, green, or red? Explain. 



Weight (lb) 


Frequency 


161-170 


4 


171-180 


8 


181-190 


12 


191-200 


11 


201-210 


7 



Color 


Blue 


Yellow 


Green 


Red 


Other 


Number 


14 


4 


6 


14 


12 



Test Prep and Spiral Review 



16. Multiple Choice Which t\T3e of display would be most appropriate to 
compare the monthly rainfall for five cities? 



CSj Line graph d) Bar graph 



(X) Circle graph CS) Stem-and-leaf plot 



17. Extended Response Nathan's family budgets SI, 000 a month for 

expenses. They budget $250 for food, $500 for rent, $150 for transportation, 
and $100 for utilities. Tell which type of graph would best represent the 
data, justify your response, and draw the display. 



Write each decimal as a percent. (Lesson 6-2) 
18. 0.27 19. 0.9 



20. 0.02 



21. 0.406 



22. Of the 75 campers at Happy Trails Summer Camp, 36% are scheduled to go 
horseback riding on Tuesdays. How many campers are scheduled to go 
horseback riding on Tuesdays? {Lesson 6-4) 



7-7 Choosing an Appropriate Display 415 




There are several ways to display data, including bar graplis, line 
graphs, and circle graphs. A spreadsheet provides a quick way to 
create these graphs. 



Activity 



Use a spreadsheet to display the Kennedy Middle School 
Student Council budget shown in the table at right. 

O Open the spreadsheet program, and enter the data as 
shown below. Enter the activities in column A and the 
amount budgeted in column B. Include the column 
titles in row 1. 





A 


i B 




C :\ 1 


1 


Activity 


Amoun 


($) 




2 


Assemblies 




275 




3 


Dances 




587 




4 


Spring Festival 




412 




5 


Awards Banquet 




384 




6 


Other 




250 




7 











Student Council Budget 


Activity 


Amount (S) 


Assemblies 


275 


Dances 


587 


Spring Festival 


412 


Awards Banquet 


384 


Other 


250 



Highlight the data by 
clicking on cell Al and 
dragging the cursor to 
cell B6. Clickjhe Chart 
Wizard icon B . Then 
click FINISH to choose 
the first type of column 
graph. 



Activity 
Assemblies 
Dances 
Spring Festn. 
Awards Bang 
Other 



*g Chart Wizard -Step I or4- Chart Type 



/standard Typea \/ Custom Type? \ 

Chart lype Chart suti- type 



Iti Column 
E Bar 
llLi l-ine 
<> Pie 

I XY (Scatter) 

Hk Area 
& Doughnut 
^ Radar 
fS Surface 
?i Bubble 



m 



inii 


_ I I 


m 




Im 


\M 



Clustered Column. Compares values across 
^ate^ories 



Presa and Hold to Viev Sample 



J '""^> L "««> I Lii 



416 Chapter 7 Collecting, Displaying, and Analyzing Data 



The bar graph of the data appears as shown. Resize or reposition the 
graph, if necessary. 





A 


B 


1 


Activity 


Amount ($) 


2 


Assemblies 


275 


3 


Dances 

Spring Festival 
Avi^ards Banquet 
Other 


587 


4 
5 
6 


412 
384 
250, 


7 






8 




9 




10 
11 
12 











13 


1 
1 


14 


i 


IS 




16 







Student Council Budget 



700 
600 
500 
400 
300 
200 4- 
100-1- 




I g Amount ($)| 



>;?' cy <^ 



..-^ 






c,^- 



To see a circle graph of the data, select the bar graph (as shown above). 
Click the Chart Wizard icon and choose "Pie," which is the circle graph. 
Then click FINISH to choose the first type of circle graph. 





A ; B C 


D E F 


G 




1 


Activity 


Amount ($L 


■ 


■ 


I 




2 
3 
4 
5 
6 


Assemblies 
Dances 

Spring Festival 
Awards Banquet 


275 
587 
412 
384 


Student Council Budget 




Other 


250, 


\ ) 


H Assemblies 
H Dances 
D Spring Festival 
■ Awards Banquet 
□ Other 




7 








8 
9 
10 










11 
12 












\ y 






13 
14 
15 




--'- — 


-^ 












16 




■ 







Think and Discuss 



1. Which graph best displays the Student Council budget? Wliy? 

2. Would a line graph be an appropriate display of the Student Council 
budget data? Explain. 



Try This 



1. The table shows the number of 
points scored by members of a girls' 
basketball team in one season. Use 
a spreadsheet to create a bar graph 
and a circle graph of the data. 

2. Which type of graph is a better display of the data? Wliy? 

3. Formulate a question and survey your classmates. Use the 
Chart Wizard to make the graph that best displays your 
data. Which type of graph did you use? Why? 



Player 


Ana 


Angel 


Mary 


Nia 


Tina 


Zoe 


Points Scored 


201 


145 


89 


40 


21 


8 



7-7 Technology Lab 417 







7.4.4 Analyze data displays, including ways that they can be misleading. Analyze 
ways in which the wording of questions can influence survey results. 

In 2002, there were claims that 
Chronic Wasting Disease (CWD), 
or Mad Elk Disease, was spreading 
westward across North America. 
In order to verify claims such as 
these, the elk population had to 
be tested. 



Vocabulary 

population 
sample 

random sample 
convenience sample 
biased sample 



Helpful«i| 



A random sample is 
more likely to be 
representative of a 
population than a 
convenience sample is. 



When information is gathered 
about a group, such as all the 
elk in North America, the entire 
group is called the population. 
Because testing each member of 
a large group can be difficult or 
impossible, researchers often 
study a part of the population, 
called a sample . 

For a random sample , members of the population are chosen at 
random. This gives eveiy member of the population an equal chance 
of being chosen. A convenience sample is based on members of the 
population that are readily available, such as 30 elk in a wildlife 
preservation area. 




EXAMPLE |lj Analyzing Sampling Methods 

Determine which sampling method will better represent the entire 
population. Justify your answer. 



Football Game: Student Attendance 


Sampling Method 


Results of Survey 


Arnie surveys 80 students by randomly 
choosing names from the school directory. 


62% attend football 
games 


Vic surveys 28 students that were sitting 
near him during lunch. 


81% attend football 
games 



Arnie's method produces results that better represent the entire 
student population because he uses a random sample. 
Vic's method produces results that are not as representative of the 
entire student population because he uses a convenience sample. 



418 Chapter 7 Collecting, Displaying, and Analyzing Data [ViJa:;] Lesson Tutorials Online 



A biased sample does not fairly represent the population. A study of 
50 elk belonging to a breeder could be biased because the breeder's elk 
might be less likely to have Mad Elk Disease than elk in the wild. 

EXAMPLE [T] Identifying Potentially Biased Samples 

Determine whether each sample may be biased. Explain. 

A The first 50 people exiting a movie are surveyed to find out wfhat 
type of movie people in the town like to see. 

The sample is biased. It is likely that not ever\'one in the town 
likes to see the same type of movie that those 50 people just saw. 



B A librarian randomly chooses 100 books from the library's 
database to calculate the average length of a library book. 

The sample is not biased. It is a random sample. 



Given data about a random sample, you can use proportional reasoning to 
make predictions or verif>' claims about the entire population. 



EXAMPLE [T] Verifying Claims Based on Statistical Data 



emembei'' 



In the proportion 
f = §, the cross 
products, a ■ d and 
b • c are equal. 



A biologist estimates that more than 700 of the 4,500 elk at a 
wildlife preserve are infected with a parasite. A random sample 
of 50 elk shows that 8 of them are infected. Determine whether 
the biologist's estimate is likely to be accurate. 

Set up a proportion to predict the total number of infected elk. 
infected elk in sample _ infected elk in population 



size of sample 



size of population 



50 4,500 

• 4,500 = 50 • .V 

36,000 = 50.V 

36,000 _ 50.V 



Let X represent the number of 
infected elk at the preserve. 
The cross products are equal. 
Multiply. 

Divide each side by 50. 



50 50 

720 = A- 
Based on the sample, you can predict that there are 720 infected elk 
at the preserve. The biologist's estimate is likely to be accurate. 



^^^^^^^^^^^^^^^^^^^^^^^^^^^^^B 


Think and Discuss 






1. Describe a situation in which you would want to 


use a sample 


rather than survey the entire population. 






2. Explain why it might be difficult to obtain 


a truly 


random sample 


of a very large population. 







'Mb'j Lesson Tutorials Online mv.hrw.com 



7-8 Populations and Samples 419 



7-8 



M i..ilt^^iH*^tiuili H a,tiia. i A»t > i/yf>Mt' 



\, :^jy:^fn 



^Dl^Lfl^ife^ 




MrFTTiifltJTImT 

Homework Help Online qo.hrw.com, 



keyword ■aWlil'AM . ® 
Exercises 1-8, 9,11 



See Example 1 



GUIDED PRACTICE 



1. Determine which sampling method will better represent the entire 
population. Justify your answer. 



Lone Star Cars: Customer Satisfaction 


Sampling Method 


Results of Survey 


Nadia surveys 200 customers on the car lot one 
Saturday morning. 


92% are satisfied 


Daria mails surveys to 100 randomly-selected customers. 


68% are satisfied 



See Example 2 



See Example 3 



See Example 1 



Determine whether each sample may be biased. Explain. 

2. A company randomly selects 500 customers from its computer database 
and then surveys those customers to find out how they like their service. 

3. A city-hall employee sui^veys 100 customers at a restaurant to learn about 
the jobs and salaries of city residents. 

4. A factory produces 150,000 light bulbs per day. The manager of the factory 
estimates that fewer than 1,000 defective bulbs are produced each day. In a 
random sample of 250 light bulbs, there are 2 defective bulbs. Determine 
whether the manager's estimate is likely to be accurate. Explain. 



INDEPENDENT PRACTICE 



5. Determine which sampling method will better represent the entire 
population. Justify your answer. 



Midville Morning News: Subscription Renewals 


Sampling Method 


Results of Survey 


Suzanne surveys 80 subscribers in her neighborhood. 


61% intend to 
renew subscription 


Vonetta telephones 150 randomly-selected subscribers. 


82% intend to 
renew subscription 



See Example 2 Determine whether each sample may be biased. Explain. 

6. A disc jockey asks the first 10 listeners who call in if they liked the last 
song that was played. 

7. Members of a polling organization survey 700 registered voters by 
randomly choosing names from a list of all registered voters. 



See Example 3 8. A university has 30,600 students. In a random sample of 240 students, 
20 speak three or more languages. Predict the number of students at the 
university who speak three or more languages. 



L 



420 Chapter 7 Collecting, Displaying, and Analyzing Data 



PRACTiLCE^AND: PROBLEM SOLVING 



Extra Practice 



See page EP19. 



.•OH-. 




North American 
fruit flies are 
l<nown to 
damage clierries, 
apples, and blue* 
berries. In the 
Mediterranean, 
fruit flies are a 
threat to citrus 
fruits. 



Explain whether you would survey the entire population or use a sample. 

9. You want to know the favorite painters of employees at a local art museum. 

10. You want to know the types of calculators used by middle school students 
across the countiy. 

You want to know how many hours per week the students in your social 
studies class spend on their homework. 

Life Science A biologist chooses a random sample of 50 out of 750 fruit 
flies. She finds that 2 of them have mutated genes causing deformed 
wings. The biologist claims that approximately 30 of the 750 fruit thes 
have deformed wings. Do you agree? Explain. 

13. A biased question is one that leads people to a certain answer. Kelly 
decides to use a random sampling to determine her classmates' favorite 
color. She asks, "Is green your favorite color?" Is this cjuestion biased? If 
so, give an example of an unbiased question. 

14. Critical Thinl<ing Explain why surveying 100 people who are listed in 
the phone book may not be a random sample. 

15. Write About It Suppose you want to know whether the seventh- 
grade students at your school spend more time watching TV or using a 
computer. How might you choose a random sample from the population? 

^16. Challenge A manager at XQJ Software 
surveyed 200 company employees to 
find out how many of the employees 
walk to work. The results are shown in the 
table. Do you think the manager chose a 
random sample? Why or why not? 



Employees at XQJ Software 




Total 
Number 


Number 
Who Walk 


Population 


9,200 


300 


Sample 


200 


40 




Test Prep and Spiral Review 



17. Multiple Choice Banneker Middle School has 580 students. Wei surveys 
a random sample of 30 students and finds that 12 of them have pet dogs. 
How many students at the school are likely to have pet dogs? 

CS) 116 CE) 232 CS;- 290 CD 360 

18. Short Response Give an example of a biased sample. Explain why it is biased. 



Write each percent as a decimal. (Lesson 6-1) 
19. 52% 20. 7% 

Find the percent of each number. (Lesson 6-4) 
23. 11% of 50 24. 48% of 600 



21. 110% 



25. 0.5% of 82 



22. 0.4% 



26. 210% of 16 



7-8 Populations and Samples 421 



7-9 



Scatter Plot 







7.4.4 Analyze data displays, inc 
ways in which the wordiii 



Vocabulary 

scatter plot 
correlation 
positive correlation 
negative correlation 
no correlation 



Q 



influence survey results 




, •O'CS^ ""^-S!?^^-?! 



The supersaurus, one of the largest known dinosaurs, could weigh as 
much as 55 tons and grow as long as 100 feet from head to tail. The 
tyrannosaurus, a large meat-eating dinosaur, was about one-third the 
length of the supersaurus. 

Two sets of data, such as the length and the weight of dinosaurs, may 
be related. To find out, you can make a scatter plot of the data values 
in each set. A scatter plot has two number lines, called axes — one for 
each set of data values. Each point on the scatter plot represents a 
pair of data values. These points may appear to be scattered or may 
cluster in the shape of a line or a curve. 



EXAMPLE ]1| Making a Scatter Plot 

Use the data to make a scatter 
plot. Describe the relationship 
between the data sets. 
Step 1: Determine the scale and 
interval for each axis. 
Place units of length on 
the horizontal axis and 
units of weight on the 
vertical axis. 



Step 2: Plot a point for each pair of values. 

Step 3: Label the axes and title the graph. 

The scatter plot shows that a dinosaur's 
weight tends to increase as its length 
increases. 



Name 


Length 
(ft) 


Weight 
(tons) 


Triceratops 


30 


6 


Tyrannosaurus 


39 


7 


Euhelopus 


50 


25 


Brachiosaurus 


82 


50 


Supersaurus 


100 


55 





Dinosaur 


Sizes 


t/1 




* I 


c 
o /in 






; i i i 1 i 




• ; 


1 1 




; ! 

• • :' r 

\ 1 1 H 


! 1 j 
-H 1 ' 



40 80 120 
Length (ft) 



422 Chapter 7 Collecting, Displaying, and Analyzing Data {Tj^bv] Lesson Tutorials Online 



A correlation is the description of the relationship between two data sets. 
There are three correlations that can describe data displayed in a scatter plot. 
Positive Correlation Negative Correlation No Correlation 



The values in both 
data sets increase at 
the same time. 



The values in one 
data set increase 
as the values in the 
other set decrease. 



The values in both 
data sets show no 
pattern. 



EXAMPLE [2] Determining Relationships Between Two Sets of Data 



Write positive correlation, negative correlation, or no correlation 
to describe each relationship. Explain. 



C 
01 



6i 


r - 





4 




• 






:•• 


2- 


/ 







— \ 1 h- 


-H 1 1 \ 1 



W 60 



E 
o 
c 
o 



40 - 
20 - 



12 3 4 
Width (cm) 

The graph shows tliat as 
width increases, length 
increases. So the graph 
shows a positive correlation. 



0) 

-5 

3 



H \ 1 1 1 1 H 

12 3 
Engine size (L) 



The graph shows that as engine 
size increases, fuel economy 
decreases. So the graph shows 
a negative correlation. 



C the ages of people and the number of pets they own 

The number of pets a person owns is not related to the person's 
age. So there seems to be no correlation between the data sets. 



Think and Discuss 

1. Describe the type of correlation you would expect between the 
number of absences in a class and the grades in the class. 

2. Give an example of a relationship between two sets of data that 
shows a negative correlation. 



'Mh'j Lesson Tutorials OnlinE mv.hrw.com 



7-9 Scatter Plots 423 





keyword ■mMllBigjM ® 

Exercises 1-8, 9, 11 



GUIDED PRACTICE 



See Example 1 1. 



The table shows the typical weights 
(in kilograms) and heart rates 
(in beats per minute) of several 
mammals. Use the data to make 
a scatter plot. Describe the 
relationship between tlie data sets. 



See Example 2 Write positive correlation, negative correlation, or no correlation to describe 
each relationship. Explain. 

2. Math Score and Shoe Size 3. Work Experience 



Mammal 


Weight 


Heart Rate 


Ferret 


0.6 


360 


Human 


70 


70 


Llama 


185 


75 


Red deer 


110 


80 


Rhesus monkey 


10 


160 



o 
1/1 



— , — , — ; « — , — t — 

• • '- :»_._.- 

. ^ a .. ■ — 

• • 



Math score 



^ - 
o 






11: 






— 






^ 


g 


, 1 t 


• 




o 




— 


'— :-—•-*- 




> - 






1 — 1— 


• 

H ! 1 i \ h 


— h- 


— 1 1 



Age 



4. the time it takes to drive 100 miles and the driving speed 



See Example 1 



INDEPENDENT PRACTICE 



5. The table shows solar energy cell 
capacity (in megawatts) over several 
years. Use the data to make a scatter 
plot. Describe the relationship between 
the data sets. 



See Example 2 Write positive correlation, negative correlation, or no correlation to describe 
each relationship. Explain. 

6. SalM 7. 



Year 


Capacity 


Year 


Capacity 


1990 
1991 
1992 


13.8 
14.9 
15.6 


1993 
1994 
1995 


21.0 
26.1 
31.1 



Extra Practice 



See page EP19. 







Sales 






: 1 ; , . r • : 




— 


-'—I- ^ .^ •■ i ~M 


13 . 


— 




01 


~-i 


. J ■ M i '^ 






^ 




• 




— 1 — ♦ — i 


1 1 1 1 i \ 1 1 





Car's 


Milec 


ige an 


d \/a 


lue 




- -• — 




1 








; 




^ 


1 


' 


<u 


— 




1 






3 
(D - 

> 


— •—;- 


• 








: 




— 1 — 1 — 








t 




1 ; 
-I 1 


1 — 1 — 1- 


1 
— 1 h- 


H 1 



Advertising cost Mileage (thousands) 

8. the number of students in a district and the number of buses in the district 



424 Chapter 7 Collecting, Displaying, and Analyzing Data 



Critical Thinking For Exercises 9-11, tell whether 
you would expect a positive correlation, a negative 
correlation, or no correlation. Explain your answers 

9. the average temperature of a location and the 
amount of rainfall it receives each year 

10. the latitude of a location and the amount 
of snow it receives each year 

1 1 . the number of hours of daylight and the 
amount of rainfall in a day 

12. The table shows the approximate latitude and average Fief Mounta 
temperature for several locations in the Southern Antarctica 
Hemisphere. Construct a scatter plot of the data. 
What can you conclude from this data? 




San Rafael Falls, Ecuador 



13. 



^ Challenge A location's elevation 
is negatively correlated to its average 
temperature and positively correlated 
to the amount of snow it receives. 
What kind of correlation would you 
expect between temperature and the 
amount of snowfall? Explain. 



Location 


Latitude 


Temperature 


Quito, Ecuador 


0°S 


55 °F 


Melbourne, Australia 


38° S 


43 °F 


Tucuman, Argentina 


27° S 


57 °F 


Tananarive, Madagascar 


19° S 


60 °F 


Halley Research Station, 


76° S 


20 °F 


Antarctica 







Test Prep and Spiral Review 



14. Multiple Choice Use the scatter plot to 
determine which statements are true. 

I The data shows a positive correlation. 

II The data shows a negative correlation. 

III The data shows no correlation. 

IV As the years increase, the prize money 
increases. 



C£) I only 



CD I and IV 



Indianapolis 500 Winner's Prize Money 



^ 2,000,000 



>- 1,500,000 

g 1,000,000 

E 500,000 

S 



■ t » t f 



• • 



1900 1920 1940 1960 1980 2000 2020 

Year 



Cc;) II and IV 



2) III only 



15. Short Response Give an example of two data sets that you would expect to 
have a positive correlation. Explain your answer 

Find the percent of each number. If necessary, round to the nearest tenth. (Lesson 6-4) 

16. 95% of 80 17. 120% of 63 18. 62% of 14 19. 7% of 50 

20. The regular price of a computer monitor at the electronics store is $499. This month 
the monitor is on sale for 15% off. Find the sale price of the monitor (Lesson 6-6) 



7-9 Scatter Plots 425 



T^LAB 



Samples and Lines of 
Best Fit 



Use after Lesson 7-9 



You can use a graphing calculator to display relationships 
between variables in a scatter plot. 



-^V Learn It Online 

~'* LabResourcesOnlinego.hrw.com, i 



keyword MblMBlSa ® 



Activity 1 



O Survey at least 30 students in your grade to find the following 

information. Record your data in a table like the one below. (Your 
table will have at least 30 rows of data.) For 15, use numbers for the 
month. For example, enter "1" for lanuary, "2" for February, etc. 



LI 
Height (in.) 


L2 
Age (yr) 


L3 
Length of 
Foot (in.) 


L4 

Length of 

Forearm (in.) 


L5 
Month 
of Birth 


66 


12 


11 


10 


3 


63 


13 


8 


9 


10 


65 


12 


10 


9.5 


7 



Q Press jmjil 



to enter all the data into a graphing calculator. 



© Create a scatter plot for height and length of foot. 

STAT PLOT ^^^^ 

a. Press WSM 93 for Plot 1. 

b. Select On, and use the arrow keys to select the scatter plot 
for Type. 

c. Use the down arrow to move the cursor to Xlist. 
Press 1 to select LI. 



d. Move the cursor to Ylist. Press 3 to select L3. 

e. Press fffl^ and then 9: ZoomStat to view your graph. 



Think and Discuss 



1. Describe the relationship between height and length of foot that is 
shown in the scatter plot from Activity 1. 

2. Wliat relationships would you expect to see between the other 
variables in the table? 



L1 


IZ 


LS 1 


bb 

es 


11 


11 
B 

10 



_ Plots Mo« 
Off 

lype; M li:^ Jhb 

HI*" *QH \y^ 

Xlist:Li 
Vlist:Ls 
Mark: Q ♦ . 



wmmmmmm 



426 Chapter 7 Collecting, Displaying, and Analyzing Data 



J 



Try This 



1. Create a scatter plot of each of the other pairs of variables in your data-collection table. 
Which variables show a positive correlation? a negative correlation? no correlation? 

A line of best fit is a straight line that comes closest to the points on a scatter plot. You can 
create a line of best fit on the calculator. 



Activity 2 



O Follow the steps from Activity 1, part 3 to display a scatter plot that shows the 
relationship between height and length of forearm. 



O Use 



to move the cursor between points on the graph. 



Use the coordinates of two points to estimate the slope of a 
line that would best fit through the data points on the graph. 



O Press 



and then use the right arrow key to select CALC 4: 
LinReg (ax + b). Then press 1 MB 4 



to find the equation of the line of best fit. 







"" = 




F-1:L1AH 


a 


K 




H-iS 


1=10 















O Press 



laail 5: Statistics Use the right arrow key to 



select EQ 1: RegEQ and press j^^BHj to put the equation for 
the line of best fit into the equation editor. 

Press fMM\\ to see the line of best fit graphed with the data 
points on the scatter plot. 



Think and Discuss 



1 . Discuss how estimating the line of best fit gets easier the more 
data points you have. 

2. Explain whether the sample from your class is representative 
of the population. 

3. What type of correlation does the line of best fit help show? 
What is the relationship between these two variables? 



V-H8iiKar^<M gr° 



Try This 



a. Press i^*^' BSli ^Pf^i^\ 3: mean 
your 30 classmates. 



to find the mean height of 



b. Calculate the mean height of three students from the original survey who 
sit closest to you. What kind of sample is this? How does the mean height 
of this sample compare to the mean of the population from part a? Explain 
why they might be different. 

c. Calculate the mean height of 15 students from the original survey. How 
does this number compare with the mean of the population? Is it closer to 
the mean than the answer you got in part b? 



r-i-jiiA'msaf 



7-9 Technology Lab 427 



7-10 



isle 







Interactivities Online ► 



7.4.4 Analyze data displays, including ways that they can be misleading. Analyze 
ways in which the wording of questions can influence survey results. 

Advertisements and news articles 
often use data to support a point. 
Sometimes tlie data is presented in 
a way that influences how the data 
is interpreted. A data display that 
distorts information in order to 
persuade can be inisleading. 

An axis in a graph can be "brolcen" 
to make the graph easier to read. 
However, a broken axis can also be 
misleading. In the graph at right, 
the cost per text message with 
Company B looks like it is twice as 
much as the cost with Company A. 
In fact, the difference is only $0.01 
per text message. 




Text Message Costs 




EXAMPLE !lj Social Studies Application 



Both bar graphs show the percent of people in California, 
Maryland, Michigan, and Washington who use seat belts. Which 
graph could be misleading? Why? 









100 




people 

O O 




O 




■£ 40 

OI 

u 

a! 20 
Q- 












Seat Beit Use 



Graph A 




CA MD Ml 

State 



Graph B 



u 

oJ 82 




CA MD Ml 

State 



WA 



Graph B could be misleading. Because the vertical axis on graph B is 
broken, it appears that the percent of people in California who wear 
seat belts is twice as great as the percent in Michigan. In fact, it is 
only 5% greater. People might conclude from graph B that the percent 
of people in California who wear seat belts is much greater than the 
percents in the other states. 



428 Chapter 7 Collecting, Displaying, and Analyzing Data | y]ii-i<j\ Lesson Tutorials OnlinE 




At the 1988 
Summer Olympics 
Jackie Joyner- 
Kersee earned 
gold medals in 
the long jump 
and heptathlon 
events. A hep- 
tathlon consists 
of seven separate 
events given over 
the course of 
two days. 



[Zj Analyzing Misleading Graphs 

Explain why each graph could be misleading. 



Women's Long Jump 




Because the vertical axis is broken, the distance 
jumped in 1988 appears to be over two times as far as 
in 1984. hi fact, the distance jumped in 1988 is less than 
0.5 meter greater than in the other years. 




The scale of the graph is wrong. Equal distances on 
the vertical axis should represent equal intervals of 
numbers, but in this graph, the first $18,000 in sales is 
larger than the next $18,000. Because of this, you can't 
tell from the bars that Pizza Perfect's sales were twice 
those of Pizza Express. 



fljB^^mm^^^^^B^^^^^^^^^^^^^^i 


Think and Discuss 




1. 


Explain how to use the scale of a graph to d 
misleading. 


ecide if the graph is 


2. 


Describe what might indicate that a graph 


s misleading. 


3. 


Give an example of a situation in which a 
might be used to persuade readers. 


misleading graph 



Yi'hp Lesson Tutorials OnlinE mv.firw.com 



7-70 Misleading Grapiis 429 



7-10 



\L'^ttei'tfi^vX\'rau:'i«i.«jft;wo?M'»*iiL.'S[*W'5aa^^^ 




,ii^j'^J33^ 




keyword IBBiliiiaiil ® 

Exercises 1-6, 7 






See Example 1 1. Which graph could be misleading? Why? 



20 



oi c 18 

2j= 16 

< c 14 

~ o' 



Graph A 



20 
<u c 15 

5^ 10 

< c 5 



Graph B 



Femur Tibia Fibula Humerus 

Bone 



IT 



Femui Tiljia Fibula Humerus 

Bone 



See Example 2 Explain why each graph could be misleading. 

2. WgffifflffTfffffflfffffTfffflM 3. 



^ 155,000 




95,000 

O O-" 

£ 2004 2005 2006 2007 

•^ Year 




INDEPENDENT PRACTICE 



See Example 1 4. Which graph could be misleading? Why? 



Average Number of Daylight Hours in Anchorage 



5 20 
o 
-= 15 

(U 

-Q 5 

E 
= 



Graph A 




20 



.a 

E 



Mar Apr May Jun 

Month 



Graph B 




Mar Apr May Jun 

Month 



See Example 2 Explain why each graph could be misleading. 



CD Sales 



3,000 

2,000 

■^ 1,000 

£ 750 

ID 500 

250 





1/1 



I 



; 



CD Music 
Palace World 



Threatened Birds 






Myanmar 

*^ 

3 Thailand 

o 
I) 

Vietnam 


■■1 


C 


44 45 46 47 48 

Number of species 



430 Chapter 7 Collecting, Displaying, and Analyzing Data 



PRACTICE AND PROBLEM SOLVING 



Extra Practice 



Seepage EP19. 



7. Business Explain why the graphs below are misleading. Then tell how you 



can redraw them so that they are not misleading. 





6,000 


i/=t 








t- 


4,000 


3 




O 




F 


2,000 


< 






Week 1 Week 2 Week 3 Week 4 



^ 5,000 

■£ 3,000 

£ 1,000 

< 




Week 1 Week 2 Week 3 Week 4 



« 



10. 

11. 



Social Studies The Appalachian Trail is a 2,160-mile 
footpath that runs from Maine to Georgia. The bar 
graph shows the number of miles of trail in three 
states. Redraw the graph so that it is not misleading. 
Then compare the two graphs. 

Choose a Strategy Tanya had $1.19 in coins. None 
of the coins were dollars or 50-cent pieces. Josie asked 
Tanya for change for a dollar, but she did not have tlie 
correct change. Wliich coins did Tanya have? 

Write About It Why is it important to closely examine graphs in ads? 

Challenge A company asked 10 people about their favorite 
brand of toothpaste. Three people chose Sparkle, one chose 
Smile, and six chose Purely White. An advertisement for 
Sparkle states, "Three times as many people prefer Sparkle 
over Smile!" Explain why this statement is misleading. 





Test Prep and Spiral Review 



Use the graph for Exercises 12 and 13. 

12. Multiple Choice Which statement is NOT a 
reason that the graph is misleading? 

Ca:' Broken interval on the vertical axis 
CD The title 

CO Vertical scale is not small enough 
CE) Intervals are not equal 

13. Short Response Redraw the graph so that it is not misleading. 



School Dance 
Admissions Soar 




Solve. Write each answer in simplest form. (Lesson 3-1 1) 
14. h = ^ 15 



X + - — - 
•*■ ^ 3 6 



16. -iv^f 



17. x-^ = 



Write positive, negative, or no correlation to describe each relationship. (Lesson 7-9) 

18. height and test scores 19. speed of a car and time required to travel a distance 



7-70 Misleading Graphs 431 



CHAPTER 



7 



SECTION 7B 



Ready To Go On? 



#^Leam it Online 

»*■ ResourcesOnlinego.hrw.com, 

lBBW!l|Msin RTri07B^ 



Quiz for Lessons 7-6 Through 7-10 

7-6 ] Line Graphs 

The table shows the value of a truck as its mileage increases. 

1. Make aline graph of the data. 

2. Use the graph to estimate the value of the truck when it 
has 12,000 miles. 



Mileage 
(thousands) 


Value of 
Truck ($) 





20,000 


20 


18,000 


40 


14,000 


60 


11,000 


80 


10,000 



& 



7-7 ] Choosing an Appropriate Display 

The table shows worldwide earthquake frequency. 

3. Choose the type of graph that would best display 
this data. 

4. Create the graph that would best display the data. 



(^ 7-8 ] Populations and Samples 



Earthquake Frequency 


Category 


Annual 
Frequency 


Great 


1 


Major 


18 


Strong 


120 


Moderate 


800 



Determine whether each sample may be biased. 
Explain. 

5. Rickie surveys people at an amusement park to find out the average 
size of people's immediate family. 

6. Theo surveys every fourth person entering a grocery store to find out 
the average number of pets in people's homes. 

7. A biologist estimates that there are 1,800 fish in a quarry. To test this estimate, 
a student caught 150 fish from the quarry, tagged them, and released them. 

A few days later, the student caught 50 fish and noted that 4 were tagged. 
Determine whether the biologist's estimate is likely to be accurate. 



er 



7-9 ] Scatter Plots 

J. Use the data to make a scatter plot. 



Write positive correlation, negative 
correlation, or no correlation to 
describe the relationship between 
the data sets. 



Cost ($) 


2 


3 


4 


5 


Number of Purchases 


12 


8 


6 


3 



(^ 7-10] Misleading Graphs 

10. Which graph is misleading? 
Explain. 



o 
o 

o 



.iLiLn 


^ 


4^ 




1 



o 
o 
o 



30 



20 



0) 10 




May July Sep Nov 



May July Sep Nov 



432 Chapter 7 Collecting, Displaying, and Analyzing Data 




CHAPTER 



CONNECT 




The Utah Jazz in 1979, the New Orleans Jazz moved to Salt 
Lake City, giving the state of Utah its first professional sports 
team. Since then, the Jazz have appeared frequently in the 
National Basketball Association's postseason playoffs. 

For 1-5, use the table. 

1 . Make a stem-and-leaf plot to display the number of wins. 

2. Find the mean, median, mode, and range of the data. 

3. Which season, if any, was an outlier? How does removing 
this season from the data set affect the mean, median, 
and mode? 

4. A sports writer wants to present a graph that shows 
how the number of wins changed 
over time. 

a. Which Vs^pe of graph should the writer use? Why? 

b. Make the graph. 

c. In general, what does the graph tell you about 
the team? 

5. Make a box-and-whisker plot of the data. 



UTAH 





K^l 



/ 



Salt Lake 
i^City 



Wins by the Utah Jazz 


Season 


Wins 


1999-2000 


55 


2000-2001 


53 


2001-2002 


46 


2002-2003 


47 


2003-2004 


42 


2004-2005 


26 


2005-2006 


41 


2006-2007 


51 



.t 



v.- 



17 




433 




Code Breaker 



A ayptogram is a message written in code. One of the most common 
types of codes is a substitution code, in which each letter of a text is 
replaced with a different letter. The table shows one way to replace the 
letters in a text to make a coded message. 



Original Letter 


A 


B 


C 


D 


E 


F 


G 


H 


I 


J 


K 


L 


M 


Code Letter 


J 


E 





H 


K 


A 


U 


B 


L 


Y 


V 


G 


P 


Original Letter 


N 





P 


Q 


R 


S 


T 


U 


V 


W 


X 


Y 


Z 


Code Letter 


X 


N 


s 


D 


Z 


Q 


M 


W 


c 


R 


F 


T 


I 



With this code, the word MATH is written PJMB. You can also use the 
table as a key to decode messages. Try decoding the following message. 

J EJZ UZJSB OJX EKWQKH MN HLQSGIT HJMJ. 

Suppose you want to crack a substitution code but are not given the key. 
You can use letter frequencies to help you. The bar graph below shows 
the number of times each letter of the English language is likely to 
appear in a text of 100 letters. 



etter Frequencies 




A B C D E F G H I J 



T U V W X Y Z 



From the graph, you can see that E is the mode. In a coded text, the 
letter that appears most frequently is likely to represent the letter E. The 
letter that appears the second most frequently is likely to represent the 
letter T. Count the number of times each letter appears in the following 
message. Then use the letter frequencies and a bit of guesswork to 
decode the message. (Hint: In this code, P represents the letter M.) 

KSQ PQUR, KSQ PQHGUR, URH KSQ PXHQ KQWW VXE DXPQKSGRT 
UCXEK U DQK XZ HUKU. 



434 Chapter 7 Collecting, Displaying, and Analyzing Data 




glue 

colored paper 

magnetic strip 

tape 

empty CD case 

graph paper 

stapler 



PROJECT 



Graph Match 



Use an empty CD case to make a magnetic matching 
game about different types of graphs. 

Directions 

O Trim the card stock to 4:^ inches by 5 inches. On tlie 
card stock, write "Match the Name and Number" and list 
the numbers 1 through 5 as shown. Cut small 
rectangles from the magnetic strip and glue these 
next to the numbers. Figure A 

Glue colored paper to the rest of the magnetic strip. 
Write the names of five different types of graphs on 
the strip. Cut these apart to form magnetic rectangles 
with the names of the graphs. Figure B 

Put a magnetic name of a graph next to each number 
on the card stock. Then tape the card stock to the 
inside back cover of an empty CD case. Figure C 

O Cut out five squares of graph paper that are each 
4^ inches by 4^ inches. Label the squares 1 through 
5. Draw a different type of graph on each square, 
making sure to match the types that are named on 
the magnetic rectangles. 



Staple the graphs together to make a 
booklet. Insert the booklet into the cover 
of the CD case. 

Putting the Math into Action 

Exchange your game with a partner. Can 
you match each graph with its name? 




BAR 6RAPH 



BOX-AND-WHISKER 
PLOT 

STEM-AND-LEAF 
PLOT 

CIRCLE GRAPH 








iS.iiv:S.;!:-zc. 



It's in the Bag! 435 




Vocabulary 

bar graph 390 

biased sample 419 

box-and-whisker plot . . 398 

circle graph 394 

convenience sample ... 418 

correlation 423 

cumulative frequency . . 380 

double-bar graph 390 

double-line graph 407 

frequency table 380 

histogram 391 



interquartile range 399 

line graph 406 

line of best fit 427 

line plot 381 

lower quartile 398 

mean 385 

median 385 

mode 385 

negative correlation 423 

no correlation 423 

outlier 386 



population 418 

positive correlation 423 

random sample 418 

range 385 

sample 418 

scatter plot 422 

sector 394 

stem-and-leaf plot 381 

upper quartile 398 



Complete the sentences below with vocabulary words from the list above. 

1. When gathering information about a en) ? researchers 
often study part of the group, called a(n) ? 



2. The sum of the data values divided by the number of data 
items is called the ? of the data. 



EXAMPLES 



EXERCISES 



7-1 ] Frequency Tables, Stem-and-Leaf Plots, and Line Plots (pp. 380-384) 



Make a line plot of the data. 

X 

XX XX 

XXX XXX 

-H — I I I I I I I I — I I I I 



14 16 18 20 22 24 26 
15, 22, 16, 24, 15, 25, 16, 22, 15, 24, 18 



Use the data set 35, 29. 14, 19. 32, 25, 27. 16, 
and 8 for Exercises 3 through 5. 

3. Make a cumulative frequency table. 

4. Make a stem-and-leaf plot of the data. 

5. Make a line plot of the data. 



7-2] Mean, Median, Mode, and Range (pp. 385-389) 



Find the mean, median, mode, and range 
of the data set 3, 7, 10, 2, and 3. 

Mean: 3 + 7+ 10 -1-2-1-3 = 25 ^=5 

Median: 2, 3, 3, 7, 10 

Mode: 3 Range: 10 - 2 = 8 



Find the mean, median, mode, and range 
of each data set. 

6. 324, 233, 324, 399, 233, 299 

7. 48,39,27,52,45,47,49,37 

8. When is the median the most useful 
measure of central tendency? 



436 Chapter 7 Collecting, Displaying, and Analyzing Data 



EXAMPLES 



EXERCISES 



7-3] Bar Graphs and Histograms (pp. 390-393) 

I Make a bar graph of the chess club's 
results: W, L, W, W, L, W. L, L, W, W, W, L, W. 



Chess Club Results 




9. Make a double-bar graph of the data. 



Favorite Pet 


Girls 


Boys 


Cat 


42 


31 


Dog 


36 


52 


Fish 


3 


10 


Other 


19 


7 



7-4 J Reading and Interpreting Circle Graphs (pp 394-397) 



About what 
percent of 
people said 
yellow was 
their favorite 
color? 
about 25% 



Favorite Colors 




7-5l Box-and-Whisker Plots (pp 398-401) 

I Use the data to make a box-and-whisker 
plot: 14, 10, 23, 16, 21, 26, 23, 17, and 25. 



I I I I I I I I I I I I I I I I I I I I I » 
10 15 20 25 30 



Use the circle graph at left for Exercises 10 
and 11. 

10. Did more people choose purple or 
yellow as their favorite color? 

11. Out of the 100 people surveyed, 35% 
chose blue as their favorite color. How 
many people chose blue? 

12. Decide whether a bar graph or a circle 
graph would best display the percent 
of U.S. citizens living in different 
countries. 



Use the following data for Exercises 13-14: 
33, 38, 43, 30, 29, 40, 51, 27, 42, 23, and 31. 

13. Make a box-and-whisker plot. 

14. What is the interquartile range? 



\^-6} Line Graphs (pp. 406-409) 

I Make a line graph of the rainfall data: 
Apr, 5 in.; May, 3 in.; Jun, 4 in.; Jul, 1 in. 




15. Make a double-line graph of the data in 
the table. 



U.S. Open Winning Scores 




1995 


1996 


1997 


1998 


1999 


Men 


280 


278 


276 


280 


279 


Women 


278 


272 


274 


290 


272 



VldaLi Lesson Tutorials Online mv.hrw.com 



Study Guide: Review 437 



EXAMPLES 



EXERCISES 



7-7 j Choosing an Appropriate Display (pp. 412-415) 



Choose the type of graph that would best 
represent the population of a town over a 
10-year period. 

Line graph 



Choose the type of graph that would best 
represent these data. 

1 6. number of dogs in a kennel each day 

1 7. number of exports from different 
countries 



7-8] Populations and Samples (pp. 418-421) 

I In a random sample of 50 pigeons at a 
park, 4 are found to have a beak 
deformation. Is it reasonable to claim 
that about 20 of the pigeon population of 
2,000 have this deformation? Explain. 



No; ^ is not closely proportional to 



20 
2,000' 



7-9] Scatter Plots (pp. 422-425) 

I ■ Write positive, negative, or no correlation 
I to describe the relationship between date 

of birth and eye color. 

|i 

I There seems to be no correlation between 
the data sets. 



18. Fourteen out of 35 people surveyed 
prefer Brand X detergent. Is it 
reasonable for the store manager to 
claim that about 2,500 of the town's 
6,000 residents will prefer Brand X 
detergent? 

Determine whether each sample may be 
biased. Explain 

1 9. A newspaper reporter randomly 
chooses 100 different people walking 
down the street to find out their 
favorite dessert. 

20. The first 25 teenagers exiting a clothing 
store are surveyed to find out what 
types of clothes teenagers like to buy. 



21. Use the data to make a scatter plot. 

Write positive, negative, or no correlation. 



Customers 47 


56 


35 


75 


25 


Sales ($) 495 


501 


490 


520 


375 



7-10] Misleading Graphs (pp. 428-431) 



Explain why the graph 
could be misleading. 

The vertical axis is 
broken, so it appears 
that A's sales are twice 
more than B's. 



■■SWlSt/rilTn 


900 
S 850 

01 

TO 800 





I 


A B 



22. Explain why 
the graph could 
be misleading. 



Temperatures 




438 Ctiapter 7 Collecting, Displaying, and Analyzing Data 



Chapter Test 



Use the data set 12, 18, 12, 22, 28, 23, 32, 10, 29, and 36 for problems 1-8. 

1. Find the mean, median, mode, and range of tlie data set. 

2. How would the outlier 57 affect the measures of central tendency? 

3. Make a cumulative frequency table of the data. 

4. Make a stem-and-leaf plot of the data. 

5. Make a line plot of the data. 6. Make a histogram of the data. 
7. Make a box-and-whisker plot of the data. 8. What is the interquartile range? 




Use the table for problems 9 and 10. 

9. The table shows the weight in poimds of several 
mammals. Make a double-bar graph of the data. 

10. Which mammal shows the greatest weight 
difference between the male and the female? 

Use the circle graph for problems 1 1 and 12. 

1 1 . Approximately what percent of the students are 
seventh-graders? 

12. If the school population is 1,200 students, are more 
than 500 students in eighth grade? Explain. 

Use the table for problems 13 and 14. 

13. The table shows passenger car fuel rates in miles 
per gallon for several years. Make a line graph of the 
data. During which 2 -year period did the fuel rate 
decrease? 

14. Estimate the fuel rate in 2005. 

1 5. What type of graph would best display student 
attendance at various sporting events? 



Mammal 


Male 


Female 


Gorilla 


450 


200 


Lion 


400 


300 


Tiger 


420 


300 



School Population 



Grade 8 



Grade 7 




Grade 6 



Year 


2000 


2002 


2004 


2006 


Rate 


21.0 


20.7 


21.2 


21.6 



For problems 16 and 17, write positive correlation, negative 
correlation, or no correlation to describe each relationship. 

16. size of hand and typing speed 

17. height from which an object is dropped and time it 
takes to hit the ground 

18. Explain why the graph at right could be misleading. 



Sports Participation 




Chapter Test 439 





Test Tackier 




Short Response: Write Short Responses 

Short-response test items are designed to test your understanding of a math 
concept, hi your response, you usually have to show your work and explain 
your answer. Scores are based on a 2-point scoring chart called a rubric. 



EXAMPLE 



Short Response The following data represents the 
number of hours Leann studied each day after school 
for her history test. 

0, 1,0, 1,5,3,4 

Find the mean, median, and mode for the data set. 
Which measure of central tendency best represents 
the data? Explain your answer. 

Here are some responses scored using the 2-point rubric. 
2-point response: 



-1- ; + 4 


1 
7 


+ 5 + 3 + H _ 


2 The mean 


isZ. 




1(1)3 


H 


5 


The 


medlar 


is 1. 






(00 3 


H 


5 


The 


modes 


are and 1. 






The measure o-f central tendency that best represer 


ts 


the data is 


ih 


e mean. 


because if shows the 


averacje 




number of hours 


thai 


Leann 


studied fcefore 


l-isr test. 





1 -point response: 



+ 


1 + + 


1 


+ 5 + 3 +H 


2 


The 


mean 


isZ. 






7 











1(7)3 


H 


5 


The 


median is 


1. 






005 


H 


5 


The 


modes 


are 


and 1. 





Scoring Rubric 

2 points: The student 
correctly answers the 
question, shows all work, 
and provides a complete 
and correct explanation. 

1 point: The student 
correctly answers the 
question but does not 
show all work or does not 
provide a complete 
explanation; or the student 
makes minor errors 
resulting in an incorrect 
solution but shows all 
work and provides a 
complete explanation. 

points: The student 
gives an incorrect answer 
and shows no work or 
explanation, or the 
student gives no response. 



Notice that there is no explanation 
given about the measure of central 
tendency that best represents the data. 



0-point response: 



The mean is 2. the median is 2, and the mode is 0. 



Notice that the answer is incorrect 
and there is no explanation. 



440 Chapter 7 Collecting, Displaying, and Analyzing Data 




Underline or highlight what you are 
being asked to do in each question. 
Be sure to explain how you get your 
answer in complete sentences. 



Read each test item and use the scoring 
rubric to answer the questions that follow. 

Item A 

; Short Response The box-and-whisker 
■ plot shows the height in inches of 

seventh-grade students. Describe the 

spread of the data. 



1 f 



—{ — \ — I — I — \ — \ — \ — \ — \ — I — h- 
50 52 54 56 58 60 62 64 66 68 70 

Student's Answer 



There are more students heivjeen 58 
and 10 inches fall than there are 
faefween 50 and 58 inches tall because 
the third cjuarfile is farther from the 
median than the first cjuartile is- 



1. What score should the student's answer 
receive? Explain your reasoning. 

2. What additional information, if any, 
should the student's answer include in 
order to receive full credit? 

Item B 

Short Response Explain the type of 
graph you would use to represent the 
number of each type of car sold at a car 
dealership in May. 

Student's Answer 



/ \^/ould use 


a 


bar 


graph 


to show/ ho^^y 




many of each 


car 


model 


v/as sold durinq 


the month. 













What score should the student's answer 
receive? Explain your reasoning. 

What additional information, if any, 
should the student's answer include in 
order to receive full credit? 



Item C 

Short Response Create a scatter plot of 
the data and describe the correlation 
between the outside temperature and the 
number of people at the public pool. 



Temperature (°F) 


70 75 80 


85 


90 


Number of People 


20 22 40 


46 


67 



Student's Answer 



There is a positive correlation bet\^een 
the temperature and the number of people 
at the public pool because as it qets hotter, 
more people w/ont to cjo sv/imminy. 



5. What score should the student's answer 
receive? Explain your reasoning. 

6. What additional information, if any, 
should the student's answer include in 
order to receive full credit? 

Item D 

Short Response A survey was conducted 
to determine which age group attended 
the most movies in November. Fifteen 
people at a movie theater were asked 
their age, and their responses are as 
follows: 6, 10, 34, 22, 46, 11, 62, 14, 14, 5, 
23, 25, 17, 18, and 55. Make a cumulative 
frequency table of the data. Then explain 
which group saw the most movies. 

Student's Answer 



Groups 


Frequency 


Cumulative 
Frequency 


0-13 


H 


H 


1H-Z6 


1 


11 


11 -HO 


1 


11 


H1-5H 


1 


13 


55-68 


1 


15 



1 . What score should the student's answer 
receive? Explain your reasoning. 

8. What additional information, if any, 
should the student's answer include in 
order to receive full credit? 



Test Tackier 441 



CHAPTER 



7 



ra ISTEP+ 
^ Test Prep 






Learn It Online 

State Test Practice go.hrw.com, ^ 

WMHIfa^MTi TestPreplGql " 




Applied Skills Assessment 

Constructed Response 

1. The graph shows the results of a survey. 
Aaron read the graph and determined 
that more than | of the students chose 
drama as their favorite type of movie. Do 
you agree with Aaron? Why or why not? 



Favorite Types of Movies 



c 10 



-5 6 



5 2 



I I I ■ I 



.# <>" 



*•* / ,/ 



2. A land developer purchases 120 acres 
of land and plans to divide one part 
into five 5-acre lots, another part into 
two 10-acre lots, and the rest into 
^-acre lots. Each lot will be sold for a 
future home site. How many total lots 
can the developer plan to sell? 

Extended Response 

3. Mr. Parker wants to identify the types of 
activities in which high school students 
participate after school, so he surveys 
the twelfth-graders in his science classes. 
The table shows the results of the survey. 



Activity Boys Girls 


Play sports 


36 


24 


Talk to friends 


6 


30 


Do homework 


15 


18 


Work 


5 


4 



a. Use the data in the table to 
construct a double-bar graph 



b. What is the mean number of girls 
per activity? Show your work. 

c. What type of sample is used? Is this 
sample representative of the 
population? Explain. 

Multiple-Choice Assessment 

4. Which expression is true for the 
data set? 15, 18, 13, 15, 16, 14 

A. Mean < mode 

B. Median > mean 

C. Median = mean 

D. Median = mode 

5. What is the first step to complete in 
simplifying this expression? 

I + [3 - 5(2)] ^ 6 

A. Multiply 5 and 2. 

B. Divide by 6. 

C. Subtract 5 from 3. 

D. Divide 2 by 5. 

6. What is the slope of the line shown? 

Ay 




C. 2 



D. —: 



442 



V.WI oil u«^ L a uv-zuuic uai yia^ii. 

Chapter 7 Collecting, Displaying, and Analyzing Data 



On Monday the temperature was 
-13 °F. On Tuesday the temperature 
rose 7 °F. What was the temperature 
on Tuesday? 

A. -20 °F C. -6 °F 

B. -8 °F D. 7 °F 

Which model best represents the 
fraction |? 



A. 



9. Ron eats ^ cup of cereal every day as 
part of his breakfast. He has had a 
total of 16 cups of cereal this year. 
How many days has he eaten cereal? 

A. 4 days C. 32 days 

B. 16 days D. 64 days 

10. A store is offering lip gloss at 25% off 
its original price. The original price of 
lip gloss is $7.59. What is the sale price? 

A. $5.69 C. $3.80 

B. $4.93 D. $1.90 

11. What is the mode of the data given 
in the stem-and-leaf plot? 



Stems 


Leaves 


6 


122 59 


7 


04678 


8 


33356 



A. 25 

B. 62 



Key: 7\0 means 70 

C. 76 

D. 83 



12. Solve 8 + 34x = -60 for x. 

A. x= -5 C. x= -2 

B. x= -0.97 D. x = 2 

13. Which statement is best supported by 
the data? 



Soccer League Participation 




A. More students played soccer in 2005 
than in 2002. 

B. From 2001-2007, soccer 
participation increased by 100%. 

C. From 2002-2006, soccer 
participation decreased by 144%. 

D. Participation increased between 
2004 and 2005. 



# 



Read a graph or diagram as closely as 
you read the actual test question. 
These visual aids contain important 
information. 



Gridded Response 

14. To the nearest hundredth, what is the 
difference between the median and 
the mean of the data set? 

14, 11, 14, 11, 13, 12, 9, 15, 16 

15. What value represents the upper 
quartile of the data in the box-and- 
whisker plot below? 



--H — \ — I — \ — \ — \ — \ — \ — I — \-^ 
8 10 12 14 16 18 20 22 24 26 

16. The key in a stem-and-leaf plot states 
that 2I5 means 2.5. What value is 
represented by l|8 ? 



Cumulative Assessment, Chapters 1-7 443 



CHAPTER 



8A Lines and Angles 

8-1 Building Blocks of 
Geometry 

LAB Explore Complementary 
and Supplementary 
Angles 

8-2 Classifying Angles 

LAB Explore Parallel Lines and 
Transversals 

8-3 Line and Angle 
Relationships 

LAB Construct Bisectors and 
Congruent Angles 

8B Circles and Polygons 

8-4 Properties of Circles 

LAB Construct Circle Graphs 

8-5 Classifying Polygons 

8-6 Classifying Triangles 

8-7 Classifying Quadrilaterals 

8-8 Angles in Polygons 







7.3.1 



r: If 



Use facts about distance 
and angles to analyze 
figures. 

• Find unknown measures of 
angles. 









mti 



5l!l||iil|^M| 



IL 1 , _ 



L. 



w 



A ^ 1 "■ 



111' L- 

'// If Liu:»a 



R' 



^,- 



.\ ^ 



I 






I ' ! /I ■ I < 



A 



./J I 



\ 



8C 


Transformations 




8-9 


Congruent Figures 


7.3.4 


8-10 


Translations, Reflections, 
and Rotations 


7.3.2 


LAB 


Explore Transformations 




EXT 


Dilations 




8-11 


Symmetry 


7.3.4 


LAB 


Create Tessellations 





\trr 



i 






y Learn This? 



j The deck of the Brooklyn Bridge is 

suspended by vertical cables. Reinforcement | 
cables intersect the suspenders and fornn 
geometric shapes such as quadrilaterals 

y^^' Learn It Online 

^^ Chapter Project Online go.hrw.com, 



ll«:X'. 






7-^. 



*■ I . I 



t 



kevword ^illitmMiKM ® 



Wf' 



tf m 



hapter 



X.. 




' ^r,M 



Are You Ready?^ ^ 



0^ Vocabulary 

Choose the best term from the list to complete each sentence. 

1. An equation showing that two ratios are equal is acn) ? 



2. The coordinates of a point on a grid are written as a(n) ? . 

3. Acn) ? is a special ratio that compares a number to 100 and 
uses the symbol %. 

4. The number —3 is acn) ? . 



Learn It Online 

Resources Online go.hrw.com, 

IBWBTmsioayrs Mf-o 



decimal 
integer 
percent 
proportion 
ordered pair 



Complete these exercises to review skills you will need for this chapter. 

Percents and Decimals 



Write each decimal as a percent. 

5. 0.77 6. 0.06 

Write each percent as a decimal. 

9. 42% 10. 80% 



7. 0.9 



8. 1.04 



12. 131% 



11. 1% 

Find the Percent of a Number 

Solve. 

13. WTiat is 10% of 40? 14. Wiat is 12% of 100? 15. WOiat is 99% of 60? 
16. Wliatis 100%of81? 17. Wliat is 45% of 360? 18. What is 55% of 1,024? 



Inverse Operations 



Use the inverse operation to write an equation. Solve. 

19. 45 + /; = 97 20. /; - 18 = 100 21. ;; - 72 = 91 



23. 5 X f = 105 24. b ^ 13 = 



25. /.■ X 18 = 90 



(2/ Graph Ordered Pairs 

Use the coordinate plane at right. Write the ordered 
pair for each point. 

27. points 28. point B 



22. ;/ + 23 = 55 
26. (1-^7 = 8 

6 
44 



29. point C 
31. pointf 



30. point D 
32. point F 




H 1 1 1 1 1- 



// ' 



Geometric Figures 445 



Where You've Been 

Previously, you 

© identified angle and line 
relationships. 

• identified similar figures. 

® graphed points on a coordinate 
plane. 



ffllii^h is ChapteKfflH 



You will study 

• classifying pairs of angles 
as complementary or 
supplementary. 

• classifying triangles and 
quadrilaterals. 

• graphing translations and 
reflections on a coordinate 
plane. 

• using congruence and 
similarity to solve problems. 

Where You're Going 

You can use the skills 
learned in this chapter 

• to solve problems related to 
architecture and engineering. 

® to use transformations to 
create patterns in art classes. 



Key 

Vocabulary /Vocabulario 



angle 


angulo 


congruent 


congruentes 


image 


imagen 


line symmetry 


simetria axial 


parallel lines 


lineas paralelas 


perpendicular lines 


Ifneas perpendiculares 


polygon 


poligono 


rotation 


rotacion 


transformation 


transformacion 



vertex 



vertice 



Vocabulary Connections 

To become familiar with some of the 
vocabulary terms in the chapter, consider 
the following. You may refer to the chapter, 
the glossary, or a dictionary if you like. 

1 . Congruent comes from the Latin word 
cougniere, meaning "to agree or 
correspond." If two figures are congruent, 
do you think they look the same or 
different? 

2. Polygon comes from the Greek words 
polus, meaning "many," and gonia, 
meaning "angle." What do you think 
a shape called a polygon includes? 

3. Rotation can mean "the act of spinning 
or turning." How do you think a figure is 
moved when you perform a rotation on it? 



446 Chapter 8 




.^Reading /" 

^ and WriM*va 
Math X -^ 



CHAPTER 




Writing Strategy: Keep a iViatii Journal 

Keeping a math journal can help you improve your writing and reasoning 
skills and help you make sense of math topics that might be confusing. 

You can use your journal to reflect on what you have learned in class or to 
summarize important concepts and vocabular^'. Most important, though, 
your math journal can help you see your progress throughout the year. 



Journal Entry: 
Read the entry 
Lydia wrote in her 
math journal about 
similar figures. 



r 






, Ocfoher 15 

of their correspondmg s Jss are prop 



—4 



fl 53- 



3 




JV 
4m 




„„ e,.«l Are ih. .or.^pond^ng s^d. I-3*' 

propori'ionall 




XY - YZ XZ 6 « '0 

Y.sl Th. ratios of the corr.spond/.g sid.s are 
X.a^er..t.ABCa.6^KYZare.r.slar. 




Tr^ Tiiis 



Begin a math journal. Make an entry every day for one week. Use the 
following ideas to begin your entries. Be sure to date each entry. 

• What 1 already know about this lesson is . . . 

• The skills 1 need to be successful in this lesson are . . . 

• What challenges did I have? How did I handle these challenges? 



Geometric Figures 447 



Building _ 
of Geometr 



Vocabulary 

point 

line 

plane 

ray 

line segment 

congruent 



liJJJJJjjji 

A number line is an 
example of a line, 
and a coordinate 
plane is an example 
of a plane. 



Points, lines, and planes 
are the most basic figures 
of geometry. Otiier geometric 
figures, such as line segments 
and rays, are defined in terms 
of these building bloci<s. 

Artists often use basic geometric 
figures when creating their 
works. For example, Auguste 
Herbin used line segments in 
his painting called EigJ}t I, 
which is shown at right. 



A point is an exact location. 
It is usually represented as a 
dot, but it has no size at all. 




•A point /4 

Use a capital letter 
to name a point. 



A line is a straight path that 
has no thickness and extends 
forever in opposite directions. 



XY, YX, or i 

Use two points on the 
line or a lowercase 
letter to name a line. 



A plane is a flat surface 
that has no thickness 
and extends forever. 



/ s 



«. 



Q» plane QRS 

Use three points in any 
order, not on the same 
line, to name a plane. 



EXAMPLE [1J Identifying Points, Lines, and Planes 

Identify the figures in the diagram. 




A three points 

A, E, and D Choose any / 

three points. / 

B two lines ~ 

BD, CE Choose any two points on a line to name a line. 

c a plane 

plane ABC Choose any three points not on the same line to 
name a plane. 



448 Chapter 8 Geometric Figures 



'A'.k'j\ Lessor Tutorials Online mv.hrw.com 



A ray is a part of a line. It Inas one 
endpoint and extends forever in 
one direction. 



Name the endpoint 
first when naming a ray. 



A line segment is a part of a line 
or a ray that extends from one 
endpoint to another. 



M 



LM or IVIL 

Use tne endpoints to 

name a line segment. 



EXAMPLE [2] Identifying Line Segments and Rays 

Identify the figures in the diagram. 



A three rays 

RQ. RT. and SQ 



Name the endpoint 
of a ray first. 




B three line segments 

RQ, QS, and ST Use the endpoints in any 

order to name a line segment. 



Figures are congruent if tliey have the same shape 
and size. Line segments are congruent if they have 
the same length. 

You can use ticlc marks to indicate congruent Hue 
segments. In the triangle at right, line segments AB 
and BCare congruent. 




20 m y V 20 m 



16m 



EXAMPLE [3J Identifying Congruent Line Segments 

Identify the line segments that are congruent in the figure. 



The symbol = means 
"is congruent to." 




QR = SR 
QS=Pf 
QP=Sf 



One tick mark 
Two tick marks 
Three tick marks 



Think and Discuss 

1. Explain why a line and a plane can be named in more than two 
ways. How many ways can a line segment be named? 

2. Explain why it is important to choose three points that are not on 
the same line when naming a plane. 



fiiibu Lesson Tutorials Online mv.hrw.com 



8-1 Building Blocks of Geometry 449 



li 



8-1 




<iii<iriiiii[ 

Homework Help Online go.hrw.com, 



keyword ■QgUOggH (J) 

Exercises 1-12, 21 



GUIDED PRACTICE 



See Example 1 Identify the figures in the diagram. 

1. three points 

2. two lines 

3. a plane 

4. three rays 

5. three line segments 



See Example 2 

i 

See Example 3 




6. Identify the line segments that are 
congruent in the figure. 




INDEPENDENT PRACTICE 



See Example 1 Identify the figures in the diagram. 

I 7. three points 

8. two lines 

9. a plane 
See Example 2 10. three rays 

[ 11. three line segments 

See Example 3 12. Identify the line segments that are 
congruent in the figure. 



Z. 





Extra Practice 



PRACTICE AND PROBLEIV! SOLVING 



See page EP20. 



13. Identify the points, lines, line segments, and rays 
that are represented in the illustration, and tell 
what plane each is in. Some figures may be in 
more than one plane. 

14. Critical Thinking How many different line 
segments can be named in the figure below? 
Name each segment. 



1/1/ 



X Y 




15. Draw a diagram in which a plane, 5 points, 4 rays, 

and 2 lines can be identified. Then identify these figures. 



450 Chapter 8 Geometric Figures 



16. The artwork at right, by Diana Ong, is called Blocs. 

a. Copy the line segments in the artwork. Add 
tick marks to show line segments that appear 
to be congruent. 

b. Label the endpoints of the segments, incltiding 
the points of intersection. Then name four pairs 
of line segments that appear to be congruent. 

17. Draw a figure that includes at least three sets of 
congruent line segments. Label the endpoints and 
use notation to tell which line segments are congruent. 

18. Critical Thinking Can two endpoints be shared by 
two different line segments? Make a drawing to 
illustrate your answer. 

19. ^ Write About It Explain the difference between 
a line, a line segment, and a ray. Is it possible to 
estimate the length of any of these figures? If so, 

tell which ones and why. 

20. 



^ Challenge The sandstone sculpture at right, by 
Georges Vantongerloo, is called Iiirerrelario?! of Vohiuies. 
Explain whether two separate faces on the front of the 
sculpture could be in the same plane. 




i 



Test Prep and Spiral Review 



21. Multiple Choice Identify' the line segments 
that are congruent in the figure. 



I AB,BC 
III BC.CD 

CA) I only 



II AB, CD 
\\l BC,AD 
cX) I and III 



Cc:) II and IV 



CE) II only 



22. Short Response Draw a plane that contains each of the following: 
points A, B, and C; line segment AB; ray BC; and line AC. 



Find each product or quotient. (Lesson 2-4) 
23. -48- (-3) 24. -2- (-6) 



25. -56 4- 



26. 5- (-13) 



Find each percent of change. Round answers to the nearest tenth of a percent, 
if necessary. (Lesson 6-6) 

27. 85 is decreased to 60. 28. 35 is increased to 120. 29. 6 is decreased to 1. 



8-1 Building Blocks of Geometry 451 



naM<,-bv\ 



Explore Complementary and 
Supplementary Angles 



Use with Lesson 8-2 



£?. 



Learn It Online 

lab Resources Online go.hrw.com, 



keyword IBHIllBg;! m 



REMEMBER 

• An angle is formed by two rays with a common endpoint, 
called the vertex. 



Activity 1 



You can use a protractor to measure angles in units called 
degrees. Find the measure of A AVB. 

Q Place the center point of the protractor on the vertex 
of the angle. 

Q Place the protractor so that AV passes through the 
0° mark. 

Q Using the scale tha t starts with 0° along AV, read the 
measure where VB crosses the scale. The measure of 

AAVB is 50°. 



Think and Discuss 



1. Explain how to find the measure of ZB\/C without moving the protractor. 




Try This 



Use the protractor in Activity 1 to find the measure of each angle. 

1. /.AVC 2. /LAVZ 3. ADVC 



Activity 2 



Copy and measure 


each 


pair of angles. 






Type of Angle Pair 


Examples 


Nonexamples 


Complementary 


1. 


a/ 

/ B 


2- \ 


\X 






'■ \ 


4. 


\ ^^ 






< > 




A ,^^H , 




* 





452 Chapter 8 Geometric Figures 



Type of Angle Pair 


Examples 


Nonexamples 


Supplementary 


5. /< 


6. \ 


1. y y 
ym ^ ^ f^X 





Think and Discuss 



1. Make a Conjecture For each type of angle pair, complementary and 

supplementary, make a conjecture about how the angle measurements are related. 



Try This 



Use a protractor to measure each of the angle pairs below. Tell whether 
the angle pairs are complementary, supplementary, or neither. 

1. 





5. Make a Conjecture The two angles in Exercise 4 form a straight angle. 
Make a conjecture about the number of degrees in a straight angle. 

6. Use a protractor to find four pairs of 
complementary angles and four pairs of 
supplementary angles in the figure at right. 




8-2 Hands-On Lab 453 



8-2 



Clas 




Vocabulary 

angle 

vertex 

right angle 

acute angle 

obtuse angle 

straight angle 

complementary 
angles 

supplementary 
angles 

Interactivities Online ► 



As an airplane takes off, the 
path of the airplane forms an 
angle with the ground. 

An angle is formed by two rays 
with a common endpoint. The 
two rays are the sides of the 
angle. The common endpoint 
is the vertex. 

Angles are measured in 
degrees (°). An angle's measure 
determines the type of angle it is. 



A right angle is an angle that measures 

exactly 90°. The symbol n indicates a right angle. 




An acute angle is an angle that measures greater 
than 0°and less than 90°. 



An obtuse angle is an angle that measures 
greater than 90° but less than 180°. 



\ 

v 



A straight angle is an angle that measures 
exactly 180°. 



EXAMPLE (T) Classifying Angles 




You can name this 
angle /.ABC, ^CBA, 
Zfi, or Z1. 



Tell whether each angle is acute, right, obtuse, or straight. 



The angle measures greater 
than 90° but less than 180°, 
so it is an obtuse angle. 



The angle measures less 
than 90°, so it is an acute 
angle. 



If the sum of the measures of two angles is 90°, then the angles are 
complementary angles . If the sum of the measures of two angles is 
180°, then the angles are supplementary angles . 



454 Chapter 8 Geometric Figures 



'J'nj-zu] Lesson Tutorials OnlinE mv.hrw.com 



EXAMPLE [Zj Identifying Complementary and Supplementary Angles 



HelpfulHint 



If the angle you are 
measuring appears 
obtuse, then its 
measure is greater 
than 90°. If the angle 
is acute, its measure 
is less than 90°. 



Use the diagram to tell whether the angles are complementary, 
supplementary, or neither. 




Reading Math 






Read mZDXf as 
"the measure of 
angle DXE." 



A /_DXEanA/J\XB 

m^DXE = 55° and mAAXB = 35° 

Since 55° + 35° = 90°, ADXE and AAXB are complementary. 

B ADXE and ABXC 

mADXE = 55°. To find mABXC, start with the measure that XC 
crosses, 75°, and subtract the measure that XB crosses, 35°. 
m^BXC = 75° - 35° = 40°. 

Since 55° + 40° = 95°, A DXE and ABXC are neither 
complementary' nor supplementar\'. 

C AAXC and ^CXE 

mAAXC = 75° and mACXE = 105° 

Since 75° + 105° = 180°, ZAXC and ACXE are supplementary. 



EXAMPLE 



S) 



Finding Angle Measures 

Angles R and V are supplementary. If mZ/? is 67°, what is mZl/? 

Since /.R and ZVare supplementary, mAR + mZl''= 180°. 

m/LR + mAV= 180° 

67° + mAV- 180° Substitute 67' for m/.R. 

-67° -67° Subtract 67° from both sides. 

mAV= 113° 
The measure of Z Vis 113°. 



Thmk and Discuss 

1. Describe three different ways to classify an angle. 

2. Explain how to find the measure of ZP if Z.P and AQ are 

complementary angles and mZQ = 25°. 



'J'iiibu Lesson Tutorials Online my.hrw.com 



8-2 Classifying Angles 455 




keyword ■gJMIlBBM ® 
Exercises 1-18, 19, 21, 23 



GUIDED PRACTICE 



See Example 1 Tell whether each angle is acute, right, obtuse, or straight 
1. P *■ 2. K. 3 



See Example 2 




See Example 3 



Use the diagram to tell whether the angles are 
complementary, supplementary, or neither. 

4. ZAXB and ABXC 5. ABXC and ADXE 
6. ADXE and AAXD 7. ACXD and AAXB 

8. Angles L and P are complementary. 
If mAP is 34°, what is mZL? 

9. Angles B and C are supplementary. 
If mZB is 1 19°, what is mziC? 




INDEPENDENT PRACTICE 



See Example 1 Tell whether each angle is acute, right, obtuse, or straight. 
10. ^ \ 11. \ ^ 12. 





See Example 2 Use the diagram to tell whether the angles are 
complementary, supplementary, or neither. 

13. ANZO and AMZN 14. AMZN and AOZP 



L 15. ZLZ/VandZA/ZP 16. ZiVZO and ZLZM /w^f^"' 






See Example 3 17. Angles Fand O are supplementary. 
If mZF is 85°, what is mZO? 

1 8. Angles / and K are complementary. 
If mAK is 22°, what is mZ/? 



f1-° 




,lll.lMl„:l,:iMU:ll.,Mi:.l,lJ»Jllii.|.lnl:i.l,l:l,:,l:l.l,.,|.|:i 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EP20. 



Classify each pair of angles as complementary or supplementary. Then find the 
unknown angle measure. 



19. 




20. 




21. 




456 Chapter 8 Geometric Figures 



22. 
23. 



Critical Thinking The hands of a clock form an acute angle at 1:00. What 
type of angle is formed at 6:00? at 3:00? at 5:00? 



Geography Imaginary curves around 
Earth show distances in degrees from 
the equator and Prime Meridian. On a 
flat map, these curves are displayed as 
horizontal lines (latitude) and vertical 
lines (longitude). 

a. What r\'pe of angle is formed where 
a line of latitude and a line of 
longitude cross? 

b. Estimate the latitude and longitude 
of Washington, D.C. 



45 N 



40" N • 



35' N 



30" N 



:i: 




VCashlrigtoi i! 



A)* 



\ 



\, 



i{-D.Cj 



w 



"3) 




ATLANTIC 
OCEAN 



90 W 



85 W 80 W 75 W 70 W 



^24. 



What's the Error? A student states that when the sum of two angles 
equals the measure of a straight angle, the two angles are complementary. 
Explain why the student is incorrect. 



Write About It Explain why two obtuse angles cannot be supplementaiy 
to one another. 



25. 

33 26. Challenge Find mABAC in the figure 




Test Prep and Spiral Review 



Use the diagram for Exercises 27 and 28. 

27. Multiple Choice Wliich statement is NOT true? 

CSj ZB/IC is acute. 

CD ADAE is a right angle. 

<X) /LFAE and /LEAD are complementary angles. 

CS) /-EAD and /LDAC are supplementaiy angles. 

28. Multiple Choice What is the measure of ZFAD? 
CT) 30° CD 120° (Sj 150° 




CD 180° 



Find the mean, median, mode, and range of each data set. (Lesson 7-2) 

29.6,3,5,6,8 30.14,18,10,20,23 31.41,35,29,41,58,24 

32. Identify and name the figure at right. (Lesson 8-1) • — 



K 



8-2 Classifying Angles 457 



'mA 



LAB 



Explore Parallel Lines 
and Transversals 



Use with Lesson 8-3 



REMEMBER 

• Two angles are supplementary if the sum of their measures is 180°. 

• Angles with measures greater than 0° but less than 90° are acute. 

• Angles with measures greater than 90° but less than 180° are obtuse. 



£?. 



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Parallel lines are lines in the same plane that never cross. When two 
parallel lines are intersected by a third line, the angles formed have 
special relationships. This third line is called a transversal. 

In San Francisco, California, many streets are parallel such as Lombard St. 
and Broadway. 

Lombard St. 



Broadway 



Columbus Ave. is a transversal that runs diagonally across them. The eight 
angles that are formed are labeled on the diagram below. 




Columbus Ave. 



Activity 



O Copy the table below. Then use a protractor to measure angles 1-8 in the diagram. 
Write these measures in your table. 



Angle Number 


Angle Measure 


1 




2 




3 




4 




5 




6 




7 




8 






458 Chapter 8 Geometric Figures 



O Use the table you completed and the corresponding diagram for the 
following problems. 

a. Angles inside the parallel lines are imerior angles. Name them. 

b. Angles outside the parallel lines are exterior angles. Name them. 

c. Angles 3 and 6 and angles 4 and 5 are alternate interior angles. 
What do you notice about the measures of angles 3 and 6? What 
do you notice about the measures of angles 4 and 5? 

d. Angles 2 and 7 and angles 1 and 8 are alternate exterior angles. How 
do the measures of each pair of alternate exterior angles compare? 

e. Angles 1 and 5 are corresponding angles because they are in the 
same position relative to the parallel lines. How do the measures 

of angles 1 and 5 compare? Name another set of corresponding angles. 

f. Make a Conjecture What conjectures can you make about the measures of 
alternate interior angles? alternate exterior angles? corresponding angles? 



Think and Discuss 



1 . FG and LO are parallel. Tell what you know about the -* 
angles that are labeled 1 through 8. 

2. Angle 2 measures 125°. What are the measures of 
angles 1, 3, 4, 5, 6, 7, and 8? 

3. A transversal intersects nvo parallel lines and one of the angles 
formed measures 90°. Compare the measures of the remaining 
angles formed by the three lines. 




Try This 



Use a protractor to measure one angle in each diagram. Then find the 
measures of all the other angles without using a protactor. Tell how to find 
each angle measure. 




3\4 



5\5 



3. 



, 1 


1 

2 , 


3 

5 


4 

6 


7 

1 


8 




8-3A Hands-On Lab 459 



8-3 




B 



Line and Angle 
Relationships 



7.3.1 Identify and use basic properties of angles formed by transversals 
intersecting pairs of parallel lines. 

Wlien lines, line segments, or rays 
intersect, they form angles. If the angles 
formed by two intersecting lines measure 
90°, the lines are perpendicular lines . 

Some lines in the same plane do 
not intersect at all. These lines are 
parallel lines . Segments and rays that are 
parts of parallel lines are also parallel. The 
blue lines in the photograph are parallel. 



Vocabulary 

perpendicular lines 
parallel lines 
skew lines 
adjacent angles 
vertical angles 
transversal 



Skew lines do not intersect, and yet 
they are also not parallel. They lie in 
different planes. The yellow lines in the 
photograph are skew. 




EXAMPLE 1 



Interactivities Online ► 



The symbol _L means 
"is perpendicular to." 
The symbol || means 
"is parallel to." 



Identifying IParailel, Perpendicular, and 
Skew Lines 

Tell whether the lines in the figure appear 
parallel, perpendicular, or skew. 



A AB and AC 

ABLAC 

B CE and BD 

CE and BD are skew. 

C AC and BD 

acWbd 









t 


jf 


1 


f/i f/ 






A 1 j/ 

1 
1 


C 




B 


^ 

^ 
• 


D 


y 






" 





The lines appear to intersect to form 
right angles. 



The lines are in different planes and do 
not intersect. 

The lines are in the same plane and do 
not intersect. 



Adjacent angles have a common vertex and a 
common side, but no common interior points. 
Angles 2 and 3 in the diagram are adjacent. 
Adjacent angles formed by two intersecting lines 
are supplementary. 




460 Chapter 8 Geometric Figures 



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Reading Math 



Angles with the 
same number of tick 
marks are congruent 



/^' •- / Vertical angles are the opposite angles formed 
^ by two intersecting lines. Angles 1 and 3 in the 
diagram are vertical angles. Vertical angles have 
the same meastire, so they are congruent. 





A transversal is a line that intersects two or more lines that lie in the 
same plane. Transversals to parallel lines form special angle pairs. 






Alternate interior 
angles 



Alternate exterior 
angles 



Corresponding 
angles 



PROPERTIES OF TRANSVERSALS TO PARALLEL LINES 



If two parallel lines are intersected by a transversal, 

• corresponding angles are congRient, 

• alternate interior angles are congaient, 

• and alternate exterior angles are congruent. 



EXAMPLE [2] Using Angle Relationships to Find Angle Measures 

Line n \\ line p. Find the measure of each angle. 

A Z6 7/6 



mZB = 55° Vertical angles are congruent. P 55/5 

^ 4/3 

B ^1 *n m 

mZl = 55° Corresponding angles are congruent 

C Z7 

mZ7 + 55° = 180° Adjacent angles formed by two 

— 55° — 55° intersecting lines are supplementary. 

mZ7 = 125° 

D Z3 

mZ3 = 55° Alternate interior angles are congruent. 




Think and Discuss 

1. Draw a pair of parallel lines intersected by a transversal. Use tick 
marks to indicate the congruent angles. 

2. Give some examples in which parallel, perpendicular, and skew 
relationships can be seen in the real world. 



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8-3 Line and Angle Relationships 461 




;i(^2B333 



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Exercises 1-12,13,15,17,19,21,23 



GUIDED PRACTICE 



See Example 1 Tell whether the lines appear parallel, perpendicular, or skew 

1. /Land WW 

2. LM and Sv 

3. LM and m 

See Example 2 Line r || line s. Find the measure of each angle. 

4. A5 

5. Z2 

6. Z6 




INDEPENDENT PRACTICE 



See Example 1 Tell whether the lines appear parallel, perpendicular, or skew. 

7. UX and YZ 

8. YZ and XY 

9. UXandVW 

See Example 2 Line k || line m. Find the measure of each angle. 

10. Zl 
I 11. ^4 
I 12. A6 



u 








■»-. 










, 


, --^ 






* xT*- ]Y 




■»-- 




J^ 






1/ 




~ — ^ 


. -7 






"'"n/'-~^ r 




^ 


i\2 


k 




3\4 






30^\^5 


m 

— »• 



Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EP20. 



For Exercises 13-16, use the figure to complete each statement. 

13. Lines .v and )/ are ? . 1 

14. Lines ;; and x are 

15. Z.3andZ4are_ 

16. /12andZ7are 



J . They are also L 

_? . They are also ?_ 




17. Critical Thinking A pair of complementary angles are congruent. What 
is the measure of each angle? 

18. Multi-Step Two lines intersect to form four angles. The measure of one 
angle is 27°. Draw a diagram to show the measures of the other three 
angles. Explain your answer. 



462 Chapter 8 Geometric Figures 



\ 



Tell whether each statement is always, sometimes, or never true. 

19. Adjacent angles are congruent. 20. hitersecting lines are skew. 

21. Vertical angles are congruent. 22. Parallel lines intersect. 

23. Construction In the diagram of the 
partial wall frame shown, the vertical 
beams are parallel. 

a. Angle Oi?r measures 90°. How are 
OR and RS related? 

b. PT crosses two vertical crossbeams. 
What word describes PT'? 

c. How are Z. I and Z.2 related? 

24. Critical Thinking Two lines intersect to form congruent adjacent anj 

Wliat can you sav about the two lines? 



^0-- 




jes. 



25. Choose a Strategy Trace the dots in the figure. 
Draw all the lines that connect three dots. How 
many pairs of perpendicular lines have you drawn? 

C£) 8 CD 9 CD 10 CD> 14 









26. Write About It Use the definition of a straight angle to explain why 
adjacent angles formed by two intersecting lines are suppiementar\'. 

jg>27. Challenge The lines in the parking lot 
appear to be parallel. How could you 
check that the lines are parallel? 




i 



Test Prep and Spiral Review 



Use the diagram for Exercises 28 and 29. Line r\\ line s. 

28. Multiple Choice What is the measure of ^3? 

CSj 125° • CD 75° CD 65° 

29. Multiple Choice Wliat is the measure of Z6? 

CD 125° CD 75° CH) 65° 



CD 55° ^ 



CD 55° 




Add or subtract. Estimate to check whether each answer is reasonable. (Lesson 3-2) 

30. 3.583-1-2.759) 31. -9.43 + 7.68 32. -1.03 + (-0.081) 

Classify each pair of angles as complementary or supplementary. Then find the 
unknown angle measure. (Lesson 8-2) 



33. 




34. 



35. 



148"^ 




8-3 Line and Angle Relationships 463 



,(\v\6<>-ov\ 



Construct Bisectors and 
Congruent Angles 



Use with Lesson 8-3 



REMEMBER 

• Congruent angles have the same measure, and 
congruent segments are the same length. 



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To bisect a segment or an angle is to divide it into two congruent parts. 
You can bisect segments and angles, and construct congruent angles 
without using a protractor or ruler. Instead, you can use a compass 
and a straightedge. 



Activity 



O Construct a perpendicular bisector of a line segment. 

a. Draw a line segment JS on a piece of paper. ^*®P i^ 

b. Place your compass on endpoint / and, using an 

opening that is greater than half the length of • 

JS, draw an arc that intersects JS. 

c. Place your compass on endpoint S and draw an arc 
using the same opening as you did in Step b. The arc 
should intersect the first arc at both ends. 

d. Draw a line to connect the intersections of the arcs. 
Label the intersection of /S and the line point K. 

Measure JS, JK, and KS. What do you notice? 

The bisector of/S is a perpendicular bisector because all of the 
angles it forms with JS measure 90°. 

Q Bisect an angle. 

a. Draw an acute angle GHE on a piece of paper. Label the vertex H. 

b. Place the point of your compass on H and draw an 
arc through both sides of the angle. Label points G 
and £■ where the arc crosses each side of the angle. 

c. Without changing your compass opening, draw 
intersecting arcs from point G and point E. Label 
the point of intersection D. 

d. Draw HD. 

Use your protractor to measure angles GHE, GHD, and DHE. 
What do vou notice? 



Y 



Step b 



K 




A 



Stepd 




464 Chapter 8 Geometric Figures 



Construct congruent angles. 



a. Draw /LABM on your paper. 




b. To construct an angle congruent to /_ABM, 
begin by drawing a ray, and label its endpoint C. 



c. With your compass point on B, 
draw an arc through Z^45M. 



d. With the same compass opening, place the 
compass point on C and draw an arc through 
the ray. Label point D where the arc crosses the ray. 



e. With your compass, measure the arc in /^BM. 



f. With the same opening, place your compass point 
on D, and draw another arc intersecting the first one. 
Label the intersection F. Draw CF . 



Use your protractor to measure /LABM and /LFCD. 
What do you find? 



Think and Discuss 



1 . How many bisectors would you use to di\dde an angle into four 
equal parts? 

2. An 88° angle is bisected, and then each of the two angles formed are 
bisected. What is the measure of each of the smaller angles formed? 




Try This 



Use a compass and a straightedge to perform each construction. 

1. Draw and bisect a line segment. 

2. Trace and then bisect /.GOB. 

3. Draw an angle congruent to AGOB. 




8-3B Hands-On Lab 465 




Ready To Go On? 



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SECTION 8A 



Quiz for Lessons 8-1 Through 8-3 

^ 8-1 I Building Blocks of Geometry 

Identify the figures in the diagram. 
1. three points 2. three lines 

3. a plane 4. three line segments 

5. three rays 

6. Identify the line segments that are congruent 
in the figure. 




A B 




Q) 8-2 1 Classifying Angles 



Tell whether each angle is acute, right, obtuse, or straight. 
7. n ^ 8. ^^ *■ 9. 




Use the diagram to tell whether the angles are 
complementary, supplementary, or neither. 



11. ^DXE and AAXD 
13. ADXE and ^AXB 



12. AAXB and ACXD 
14. ABXC and Z.DXE 



1 5. Angles R and S are complementary. If mZS 
is 17°, what is mZ./?? 

16. Angles D and Fare supplementary. If mZD 
is 45°, what is mZF? 




& 



8-3 ] Line and Angle Relationships 

Tell whether the lines appear parallel, perpendicular, or skew. 

17. KL and MN 18. /L and MTV 

19. KL and JL 20. T} and MN 

Line a \\ line b. Find the measure of each angle. 

21. Z3 22. Z4 

23. Z8 24. Z6 

25. Zl 26. Z5 







^ 




/ 


Wi 




/c - 






' 






466 Chapter 8 Geometric Figures 







Focus on Problem Solving 





Understand the Problem 

• Restate the problem in your own words 

By writing a problem in your own words, you may understand 
it better. Before writing the problem, you may need to reread it 
several times, perhaps aloud, so that you can hear yourself saying 
the words. 

Once you have WTitten the problem in your own words, check to 
make sure you included all of the necessary information to solve it. 




Write each problem in your own words. Check to make sure you 
have included all of the information needed to solve the problem. 



Q The diagram shows a ray of light being 

reflected off a mirror. The angle of reflection 
is congruent to the angle of incidence. 
Use the diagram to find the measure of the 
obtuse angle formed by the reflected light. 



Angle of | Angle of 




Mirror 



At the intersection showoi, the turn from 
northbound Main Street left onto Jefferson 
Street is dangerous because the turn is too 
sharp. City planners have decided to change 
the road to increase the angle of the turn. 
Explain how the measures of angles 1, 3, and 
4 change as the measure of angle 2 increases. 



Jefferson Street 3 




Parallel lines s and r are intersected by a 
transversal r. The obtuse angles formed 
by lines 5 and t measure 134°. Find the 
measure of the acute angles formed 
by the intersection of lines t and r. 




Many fashion designers use basic geometric 
shapes and patterns in their textile designs. 
In the textile design shown, angles 1 and 2 
are formed by two intersecting lines. Find 
the measures of Z.1 and Z.2 if the angle 
adjacent to /.2 measures 88°. 




Focus on Problem Solving 467 



8-4 



Vocabulary 

circle 

center of a circle 

arc 

radius 

diameter 

chord 

central angle 

sector 



of Circles 



^m 



Completed in 1893 for the Chicago World's 
Fair, the first Ferris wheel could carry up to 
2,160 people. George Ferris relied on the 
idea of a circle when he modeled his design 
on a bicycle wheel. 

A circle is the set of all points in a plane 
that are the same distance from a given 
point, called the center of a circle . 

A circle is named by its center. For 
example, if point A is the center of a circle, 
then the name of the circle is circle A. 
There are special names for the different 
parts of a circle. 




Navy Pier Ferris Wheel, Chicago 



Arc 

Part of a circle 
named by its 
endpoints 



Radius 

Line segment whose 
endpoints are the 
center of a circle 
and any point on 
the circle 




Diameter 

Line segment that 
passes through 
the center of a circle, 
and whose endpoints 
lie on the circle 



Chord 

Line segment 
whose endpoints 
are any two points 
on a circle 



EXAMPLE 



Reading Math 



Radii is the plural 
form of radius. 



[lj Identifying Parts of Circles 

Name the parts of circle P. 



A 


radii 






PA. PB, PC, PD 


B 


diameter 

BD 




C 


chords 






AD, DC, AB. 


BC.BD 




468 Chapter 8 Geometric Figures 



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A central angle of a circle is an 
angle formed by two radii. A 
sector of a circle is the part of the 
circle enclosed by two radii and an 
arc connecting them. 

The sum of the measures of all of 
the nonoverlapping central angles 
in a circle is 360°. We say that there 
are 360° in a circle. 

EXAMPLE i2l PROBLEM SOLVING APPLICATION 




Sector 



Central angle 



Keep the Penny? 



PROBLEM 



SOLVING 




The circle graph shows the results 
of a survey to determine how 
people feel about keeping the 
penny. Find the central angle 
measure of the sector that shows 
the percent of people who are 
against keeping the penny. 

P!f* Understand the Problem 

List the important information: source usa Today 

• The percent of people who are against keeping the penny is 32%. 

Make a Plan 

The central angle measure of the sector that represents those people 
against keeping the penny is 32% of the angle measure of the whole 
circle. The angle measure of a circle is 360°. Since the sector is 32% of 
the circle graph, the central angle measure is 32% of 360°. 
32% of 360° = 0.32 -360° 

•1] Solve 

0.32 • 360°= 115.2° Multiply. 

The central angle of the sector measures 115.2°. 

Q Look Back 

The 32% sector is about one-third of the graph, and 120° is 
one-third of 360°. Since 1 15.2° is close to 120°, the answer is 
reasonable. 



Tfiink and Discuss 

1. Explain why a diameter is a chord but a radius is not. 

2. Draw a circle with a central angle of 90°. 



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8-4 Properties of Circles 469 



8-4 



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keyword ■BaiiligB ® 
Exercises 1-8, 9, IT, 13 



GUIDED PRACTICE 



See Example 1 Name the parts of circle O. 

1. radii 

2. diameter 



3. chords 



See Example 2 




The circle graph shows the results 
of a survey in which the following 
question was asked: "If you had to 
describe your office environment 
as a type of television show, which 
would it be?" Find the central angli 
measure of the sector that shows 
the percent of people who 
described their workplace as a 
courtroom drama. 



Describe Your Workplace 



Soap opera 

— 27% 



Science 
fiction 

7% - 




Medical 
emergency 

— 18% 



Courtroom 
drama 

— 10% 



Source USA Today 



INDEPENDENT PRACTICE 



See Example 1 Name the parts of circle C. 

5. radii 

6. diameters 



7. chords 



See Example 2 8. 




The circle graph shows 
the areas from which the 
United States imports 
bananas. Find the central 
angle measure of the sector 
that shows the percent of 
banana imports from South 
America. 



y// 



U.S. Banana Imports 



Central 

America 

51.0% 




^ 



Rest of 
the world 

5.5% 

South 
America 

43.5% 



Source US Bureau of the Census Trade Data 



Extra Practice 



See page EP21. 



PRACTICE AND PROBLEM SOLVING 



9. What is the distance between the centers of 
the circles at right? 

10. A circle is divided into five equal sectors. 
Find the measure of the central angle of 
each sector. 



4 cm 




470 Chapter 8 Geometric Figures 



Surveys The results of a survey asking 
"What word(s) do you use to address a 
group of two or more people?" are shown in the 
graph. Use the graph for Exercises 11 and 12. 

1 1 . Find the central angle measure of 
the sector that shows the percent 
of people who say "you guys" to 
address two or more people. 

12. Find the central angle measure of the sector 
that shows the percent of people who say 
"y'all" to address two or more people. 

13. If AB II CD in the circle at right, what is the measure of /LI? 
Explain your answer. 

,^ 14. Write a Problem Find a circle graph in your science or 
social studies textbook. Use the graph to write a problem 
that can be solved by finding the central angle measure of 
one of the sectors of the circle. 



15. Write About It Compare central angles of a circle with 
sectors of a circle. 

16. Challenge Find the angle measure between the minute 
and hour hands on the clock at right. 



What People Say 
in the United States 






r 



Test Prep and Spiral Review 



Use the figure for Exercises 17 and 18. 

17. Multiple Choice Which statement is NOT true about the figure? 

CA^i GI is a diameter of the circle. 

CS) GI is a chord of the circle. 

CS^> /_GFH is a central angle of the circle. 

C£i /.GFH and /LJFI are supplementary angles. 

18. Gridded Response The diameter of the circle is perpendicular 
to chord HF. What is the measure of AHFI in degrees? 

Estimate. (Lesson 6-3) 

19. 28% of 150 20. 21% of 90 21. 2% of 55 

Use the alphabet at right. (Lesson 8-3) 

23. ldentif\' the letters that appear to have parallel lines. 

24. Identify' the letters that appear to have perpendicular lines. 




22. 53% of 72 

ABCDEFGH 

IJKLMN 

OPQRST 

UVWXYZ 



8-4 Properties of Circles 471 



UVBI^ Construct Circle Graphs 



Use with Lesson 8-4 



REMEMBER 

• There are 360° in a circle. 

• A radius is a line segment with one endpoint at the center of a circle and 
the other endpoint on the circle. 



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A circle graph can be used to compare data that are parts of a whole. 



Activity 



You can make a circle graph using information from a table. 

At Booker Middle School, a survey was conducted to find the percent of 
students who favor certain types of books. The results are shown in the 
table below. 

To make a circle graph, you need to find the size of each part of your graph. 
Each part is a sector. 

To find the size of a sector, you must find the measure of its angle. You do 
this by finding what percent of the whole circle that sector represents. 



Find the size of each sector. 

a. Copy the table at right. 

b. Find a decimal equivalent for 
each percent given, and fill in the 
decimal column of your table. 

c. Find the fraction equivalent for 
each percent given, and fill in the 
fraction column of your table. 

d. Find the angle measure of each 
sector by setting up a proportion 
with each fraction. 



Students' Favorite Types of Bool<s 


Type of Book 


Percent 


Decimal 


Fraction 


Degrees 


Mysteries 


35% 








Science Fiction 


25% 


0.25 


1 

4 




Sports 


20% 








Biographies 


15% 








Humor 


5% 









■T 

360° 



4x = 360° 
X = 90° 



The measure of a sector that is I of a circle is 90° 



Fill in the last column of your table. Use a calculator to check 
by multiplying each decimal by 360°. 



472 Chapter 8 Geometric Figures 



Fiir<j:t^^rmp*«r;!^^W':wr!^-:rsr:'^7''rtr^---'^-i*f>'' 



Follow the steps below to draw a circle graph. 

a. Using a compass, draw a circle. Using 
a straightedge, draw one radius. 



r. 




b. Use a protractor to measure the angle 
of the first sector. Draw the angle. 



Mysteries 




c. Use a protractor to measure the angle 
of the next sector. Draw the angle. 



Mysteries 




Science 
fiction 



d. Continue until your graph is complete. 
Label each sector with its name and 
percent. 



Mysteries 

35% 



Humor 

5% 



Biographies 

15% 




Science fiction 

25% 



Sports 

20% 



Think and Discuss 



1. Total each column in the table from the beginning of the activity. 
What do you notice? 

2. What type of data would you want to display using a circle graph? 

3. How does the size of each sector of your circle graph relate to the 
percent, the decimal, and the fraction in your table? 



Try This 



1. Complete the table below and use the information to make a circle graph. 



How Alan Spends His Free Time 


Activity 


Percent 


Decimal 


Fraction 


Degrees 


Playing sports 


35% 








Reading 


25% 








Working on computer 


40% 









2. Ask your classmates a survey question. Organize the data in a table, and 
then use the data to make a circle graph. 



8-4 Hands-On Lab 473 



8-5 



assiTymg Koiygons 



Vocabulary 

polygon 
regular polygon 



eadmdMath 



Vertices is the plural 
form of vertex. 



^1 



From the earliest recorded 
time, geometric shapes, 
such as triangles and 
rectangles, have been used 
to decorate buildings and 
works of art. 

Triangles and rectangles 

are examples of polygons. 

A polygon is a closed plane 

figure formed by three or 

more line segments. Each 

line segment forms a side 

of the polygon, and meets, but does not cross, another line segment 

at a common point. This common point is a vertex of a polygon. 

Side 




The Kalachakra sand mandala is made entirely of colored sand. 




The polygon at left has six sides 
and six vertices. 



Vertex 



EXAMPLE [jj Identifying Polygons 



Determine whether each figure is a polygon. If it Is not, explain 
why not. 



The figure is a polygon. 

It is a closed figure with 5 sides. 



[^^ 



The figure is not a polygon. 
Not all of the sides of the figure 
are line segments. 




The figure is not a polygon. 
It is not a closed figure. 




The figure is not a polygon. 
There are line segments in the 
figure that cross. 



474 Chapter 8 Geometric Figures 



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Polygons are classified by the number of sides and angles they have. 




Triangle 

3 sides 
3 angles 




Heptagon 

7 sides 
7 angles 



Quadrilateral 

4 sides 
4 angles 




Octagon 

8 sides 
8 angles 




Pentagon 

5 sides 
5 angles 



Nonagon 

9 sides 
9 angles 




Hexagon 

6 sides 
6 angles 

Decagon 

10 sides 
10 angles 



EXAMPLE [?] Classifying Polygons 

Name each polygon. 




10 sides, 
10 angles 



Decagon 



6 sides, 
6 angles 



Hexagon 



A regular polygon is a polygon in which all sides are congruent 
and all angles are congruent. If a polygon is not regular, it is 
called irregular. 

EXAMPLE [3] Identifying and Classifying Regular Polygons 



Name each polygon, and tell whether it is a regular polygon. If it 
is not, explain why not. 



Caution! 



7////y 

A polygon with 
congruent sides is not 
necessarily a regular 
polygon. Its angles 
must also be 
congruent. 




3 m 




3 m 



The figure has congruent 
angles and congruent sides. 
It is a regular triangle. 



The figure is a quadrilateral. It is 
an irregular polygon because not 
all of the angles are congruent. 



^^^^^^^^■^^^^^^^^^^^^^^^^B 


TftiHk and Discuss 




1. Explain why a circle is not a polygon. 




2. Name three reasons why a figure would not be 


a polygon. 



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8-5 Classifying Polygons 475 



■:y«V ' ■-:•.. -Sfjll 



8-5 




keyword ■BHIilA-M ® 
Exercises 1-18, 21, 23 



GUIDED PRACTICE 



Determine whether each figure is a polygon. If it is not, explain why not. 



See Example 1 1. 




3. 




See Example 2 Name each polygon. 

4. / ^ 5. 






See Example 3 Name each polygon, and tell whether it is a regular polygon. If it is not, explain 
why not. 



7. 24 in. 

24 in. 



8. 



24 in. 



Til 



D H a 



18 cm 



12.3 cm 




24 in. 



18cm 



INDEPENDENT PRACTICE 



See Example 1 Determine whether each figure is a polygon. If it is not, explain why not. 
10. A 11. / 7 12. 





See Example 2 Name each polygon. 

13. V^ ^^^ 14. 




15. 




See Example 3 Name each polygon, and tell whether it is a regular polygon. If it is not, explain 

I why not. 
^' 16. ^A^^ 17. 5ft3^ft 18. 12in. 

See page EP21. I.Y \) 3 ft 

12 in. 




9 in. 

{ 
9 in. 



9 in. 

t 

9 in. 



476 Chapter 8 Geometric Figures 



Quilting is an art form that has existed in many 
countries for hundreds of years. Some cultures 
record their histories and traditions through the 
colors and patterns in quilts. 

19. The design of the quilt at right is made 
of triangles. 

a. Name two other polygons in the pattern. 

b. Which of the polygons in the pattern 
appear to be regular? 



Use the photograph of the star quilt for Exercises 20 and 21. 

20. The large star in the quilt pattern is made of smaller 
shapes stitched together. These smaller shapes are 
all the same type of polygon. What type of polygon 
are the smaller shapes? 

21. A polygon can be named by the number of its sides 
followed by -goii. For example, a polygon with 

14 sides is called a 14-gon. What is the name of 
the large star-shaped polygon on the quilt? 



22. ^p Challenge The quilt at right has a modern design. 
Find and copy one of each type of polygon, from a triangle 
up to a decagon, onto your paper from the design. Write 
the name of each polygon next to its drawing. 




i 



Test Prep and Spiral Review 



23. Multiple Choice What is true about the figure? 
CS) It is a polygon. CO It is a quadrilateral. 
(X* It is a regular polygon. CD) It is a nonagon. 

24. Short Response Draw an example of a figure that is NOT a polygon. 
Explain why it is not a polygon. 



Write a function that describes each sequence. (Lesson 5-4) 
25. 4,7, 10, 13,... 26. -1, 1,3,5,... 



27. 2.3,3.3,4.3,5.3, 



Solve. Round answers to the nearest tenth, if necessary. (Lesson 6-5) 

28. 8 is what percent of 15? 29. What is 35% of 58? 

30. 63 is 25% of what number? 31. 22 is what percent of 85? 



8-5 Classifying Polygons 477 



8-6 



Classifying Triangles 



Vocabulary 

scalene triangle 
isosceles triangle 
equilateral triangle 
acute triangle 
obtuse triangle 
right triangle 



A harnessed rider uses the 
triangle-shaped control bar 
on a hang glider to steer. The 
framework of most hang gliders 
is made up of many types of 
triangles. One way to classify 
triangles is by the lengths of 
their sides. Another way is by 
the measures of their angles. 



^^fc., 





A scalene triangle 

has no congruent sides. 




In an acute triangle, all 

of the angles are acute. 



Triangles classified by sides 





An isosceles triangle 

has at least 2 congruent sides. 



In an equilateral triangle 

all of the sides are congruent. 



Triangles classified by angles 





An obtuse triangle 

has exactly one obtuse angle. 



A right triangle 

has exactly one right angle. 



XAMPLE [T] Classifying Triangles 

Classify each triangle according to its sides and angles. 




scalene No congruent sides 
obtuse One obtuse angle 

This is a scalene obtuse triangle. 




isosceles Two congruent sides 

right One right angle 

This is an isosceles right triangle. 



478 Chapter 8 Geometric Figures 



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EXAMPLE 



Classify each triangle according to its sides and angles. 

C A D 





scalene No congruent sides 
right One right angle 

This is a scalene right triangle. 



isosceles Two congruent sides 
obtuse One obtuse angle 

This is an isosceles obtuse triangle. 



Identifying Triangles 

Identify the different types of triangles in the figure, and 
determine how many of each there are. 




Type 


How 
Many 


Colors 


Type 


How 
Many 


Colors 


Scalene 


4 


Yellow 


Right 


6 


Purple, yellow 


Isosceles 


10 


Green, pink, 
purple 


Obtuse 


4 


Green 


Equilateral 


4 


Pink 


Acute 


4 


Pink 



Think and Discuss 

1. Draw an isosceles acute triangle and an isosceles obtuse triangle. 

2. Draw a triangle that is right and scalene. 

3. Explain why any equilateral triangle is also an isosceles triangle, 
but not all isosceles triangles are equilateral triangles. 



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8-6 Classifying Triangles 479 




[•Til 1 1 II 

Homework Help Online go.hrw.com, 



keyword HBIil^Sl ® 
Exercises 1-8, 9, 11, 13, 15, 17, 
19, 21 



GUIDED PRACTICE 



See Example 1 Classify each triangle according to its sides and angles. 

1. 2. V 3. 




See Example 2 



4. Identify the different types of 
triangles in the figure, and 
determine how many of each 
there are. 




INDEPENDENT PRACTICE 



See Example 1 Classify each triangle according to its sides and angles. 

5. A 6. /I 7. 



See Example 2 





8. Identify the different types of triangles in 
the figure, and determine how many of 
each there are. 





Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EP21. 



Classify each triangle according to the lengths of its sides. 

9. 6 ft, 9 ft, 12 ft 10. 2 in., 2 in., 2 in. 11. 7.4 mi, 7.4 mi, 4 mi 

Classify each triangle according to the measures of its angles. 

12. 105°, 38°, 37° 13. 45°, 90°, 45° 14. 40°, 60°, 80° 

1 5. Multi-step The sum of the lengths of the sides of AA5C is 25 inches. 
The lengths of sides AB and BC are 9 inches and 8 inches. Find the length 
of side AC and classify the triangle. 

16. Draw a square. Divide it into two triangles. Describe the triangles. 



480 Chapter 8 Geometric Figures 




Classify each triangle according to its sides and angles. 
17. A 18. 15 cm 19. 



100 ft 




100 ft 



sAS" 35" 

3.7 cm\ 100°// 10.8 cm 3 mi 




103 ft 



4 mi 



The Waslnington 
IVlonument opened 
in 1888— 105 years 
after Congress pro- 
posed a memorial 
to honor the first 
president of the 
United States. 



20. Geology Each face of a topaz crystal is a triangle whose sides are all different 
lengths. Wliat kind of triangle is each face of a topaz ciystal? 

•^fl Architecture The Washington Monument is an obelisk, the top of which 
^ is a pyramid. The pyramid has four triangular faces. The bottom edge of 
each face measures 10.5 m. The other edges of each face measure 17.0 m. 
What kind of triangle is each face of the pyramid? 

22. Critical Thinking A line segment connects each vertex of a regular octagon to 
the vertex opposite it. How many triangles are within the octagon? Wliat type of 
triangles are they? 

23. Choose a Strategy How many triangles are in the figure? 
C£) 6 CD 9 CD 10 'CD) 13 

^ 24. Write About It Is it possible for an equilateral triangle to be 
obtuse? Explain yoiu- answer. 

25. Challenge The centers of circles^, B, C, D, and £ 
are connected by line segments. Classify each triangle 
in the figure, given that the diameter of circle D is 4 and 
DE = 5, BD = 6, CB = 8, and AC = 8. 




Test Prep and Spiral Review 



26. Multiple Choice Based on tiie angle measures given, which triangle is NOT 
acute? 




CK> 60°, 60°, 60° d:) 90°, 45°, 45° CO 54°, 54°, 72° CD) 75°, 45°, 60° 

27. Multiple Choice Which of the following best describes the triangle? 
CD Scalene, right triangle CH) Isosceles, obtuse triangle 
CG)> Isosceles, acute triangle CT) Equilateral, acute triangle 

28. Order the numbers |, -0.4, 2.3, and \j^ from least to greatest. (Lesson 2-11) 

Name each polygon, and tell whether it is a regular polygon. If it is not, explain 
why not. (Lesson 8-5) 

30. rn h 





"•/^^ 



8-6 Classifying Triangles 481 



8-7 



Classifying 



fit 



Vocabulary 

parallelogram 

rectangle 

rhombus 

square 

trapezoid 



Interactivities Online ► 



College campuses are often 
built around an open space 
called a "quad" or "quadrangle. 
A quadrangle is a four-sided 
enclosure, or a quadrilateral. 

Some quadrilaterals have 
properties that classify them 
as special quadrilaterals. 



Parallelogram /s ^ 

A — ^ — 




The Liberal Arts Quadrangle at 

the University of Washington, Seattle 

Opposite sides are parallel and 
congruent. Opposite angles are 
congruent. 



Rectangle 






J " L 
"> 11 ■" 


Parallelogram with four right angles. 














Rhombus 


/ / 


Parallelogram with four congruent 
sides. 









Square 



n I n 



Parallelogram with four congruent 
sides and four right angles. 



Trapezoid 



Exactly one pair of opposite 
sides is parallel. 



Quadrilaterals can have more than one name because the special 
quadrilaterals sometimes share properties. 



EXAMPLE [Tj Classifying Quadrilaterals 



Give all of the names that apply to each quadrilateral. Then give 
the name that best describes it. 



The figure lias opposite sides thiat are 
parallel, so it is a parallelogram. It has 
four right angles, so it is also a rectangle. 




Rectangle best describes this quadrilateral. 



482 Chapter 8 Geometric Figures 



[71ilbu| Lesson Tutorials Online inv.hrw.com 



Give all of the names that apply to each quadrilateral. Then give 
the name that best describes it. 



J 



The figure has exactly one pair of opposite 
sides that is parallel, so it is a trapezoid. 



Trapezoid best describes this quadrilateral. 



5 cm 



5 cm 




5 cm 



5 cm 



The figure has two pairs of opposite sides 
that are parallel, so it is a parallelogram. 
It has four right angles, so it is also a 
rectangle. It has four congruent sides, 
so it is also a rhombus and a square. 
Square best describes this quadrilateral. 

The figure has two pairs of opposite sides 
that are parallel, so it is a parallelogram. It 
has four congruent sides, so it is a rhombus. 
It does not have four right angles, so it is 
not a rectangle or a square. 
RJioiubiis best describes this quadrilateral. 



EXAMPLE 



[ 2 J Drawing Quadrilaterals 

Draw each figure. If it is not possible to draw, explain why. 
A a parallelogram that is not a rhombus 




The figure has two pairs of parallel sides, 
but all sides are not congruent. 



B a trapezoid that is also a rectangle 

A trapezoid has exactly one pair of opposite sides that is parallel, 
but a rectangle has two pairs of opposite sides that are parallel. 
It is not possible to draw this figure. 



Think and Discuss 

1. Describe how you can decide whether a rhombus is also a square. 
Use drawings to justify your answer. 

2. Draw a Venn diagram to show how the properties of the five 
quadrilaterals relate. 



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8-7 Classifying Quadrilaterals 483 





■ ■tOLlMlI 

! HomeworkHelpOnlinego.hrw.com, 



keyword ■QSEB9I W 
Exercises 1-13, 15, 17, 19, 21, 
23,25 



GUIDED PRACTICE 



See Example 1 Give all of the names that apply to each quadrilateral. Then give the name that 
best describes it. 



1. 



6 yd 



4.5 yd. 



/4.5 yd 



6 yd 




See Example 2 Draw each figure. If it is not possible to draw, explain why. 

4. a rectangle that is not a square 

5. a parallelogram that is also a trapezoid 



INDEPENDENT PRACTICE 



See Example 1 Give all of the names that apply to each quadrilateral. Then give the name that 
best describes it. 



6. 





8. 



11. 



9 m 




See Example 2 Draw each figure. If it is not possible to draw, explain why. 

12. a parallelogram that is also a rhombus 

13. a rliombus that is not a square 



9 m 



Extra Practice 



See page EP21. 



PRACTICE AND PROBLEM SOLVING 



Name the types of quadrilaterals that have each property. 

14. four right angles 15. two pairs of opposite, parallel sides 

16. four congruent sides 17. opposite sides that are congruent 



18. Describe how to construct a 
parallelogram from the figure 
at right, and then complete the 
construction. 



4 cm 




10 cm 



484 Chapter 8 Geometric Figures 




Tell whether each statement is true or false. Explain your answer. 

19. All squares are rhombuses. 20. All rectangles are parallelograms. 

21. All squares are rectangles. 22. All rhombuses are rectangles. 

23. Some trapezoids are squares. 24. Some rectangles are squares. 

^^ Social Studies Name the polygons made by each 
^ color in the flag of Tanzania. Give the specific names 
of any quadrilaterals you find. 



Located in north 
eastern Tanzania, 
Mount Kilimanjaro 
is an inactive 
volcano and 
includes the 
highest peak in 
Africa. 



26. 



27. 



Graph the points ^(-2, -2),B(4, 1), C(3,4),and 
D(- 1, 2), and draw line segments to connect the points. 
What kind of quadrilateral did you draw? 

Bandon Highway is being built 
perpendicular to Avenue A and 
Avenue B, which are parallel. WTiat 
kinds of polygons could be made by 
adding a fourth road? 





^y 28. Write a Problem Draw a design, 
or find one in a book, and then write 
a problem about the design that 
involves identif\'ing quadrilaterals. 

1^ 29. Write About It Quadrilaterals can 

be found on many college campuses. Describe two special 
quadrilaterals that you commonly find in the world around you. 

^ff 30. Challenge The coordinates of three vertices of a parallelogram are 
(-1, 1), (2, l),and (0, -4). \Vliat are the coordinates of the fourth ver 



rtex? 



Test Prep and Spiral Review 



31. Multiple Choice Which statement is NOT true? 

CS) All rhombuses are parallelograms. CCJ Some trapezoids are rectangles. 

CD All squares are rectangles. CS^ Some rhombuses are squares. 

32. Extended Response Graph the points /1(-1, 5), B(4, 3), C(2, -2), and 
D(-3, 0). Draw segments AB, BC, CD, and AD, and give all of the names 
that apply to the quadrilateral. Then give the name that best describes it. 

Use the data set 43, 28, 33, 49, 18, 44, 57, 34, 40, 57 for Exercises 33 and 34. (Lesson 7-1) 

33. Make a stem-and-leaf plot of the data. 

34. Make a cumulative frequency table of the data. 

Classify each triangle according to the measures of its angles. (Lesson 8-6) 

35. 50°, 50°, 80° 36. 40°, 50°, 90° 37. 20°, 30°, 130° 38. 20°, 60°, 100° 



8-7 Classifying Quadrilaterals 485 



8-8 




Vocabulary 

diagonal 



180 



If you tear off the corners of a 
triangle and put them together, 
you will find that they form a 
straight angle. This suggests that 
the sum of the measures of the 
angles in a triangle is 180°. 



JANGLE SUM RULE 



The sum of the measures 
of the angles in a triangle 
is 180°. 




mZl + mZ2 + mZ3 = 180° 



EXAMPLE [Ij Finding an Angle Measure in a Triangle 

Find the unknown angle 
measure in the triangle. 



25° + 37° + .v= 180° 

62° + .V = 180° 
- 62° - 62° 




The sum of the angle measures in a 
triangle is 180°. 
Combine like terms. 
Subtract 62° from both sides. 



x= 118° 



The unknown angle measure is 118° 



Interactivities Online ^ The sum of the angle measures in any four-sided 
figure can be found by dividing the figure into two 
triangles. You can divide the figure by drawing a 
diagonal. A diagonal is a line segment that connects 
two non-adjacent vertices of a polygon. Since the 
sum of the angle measures in each triangle is 180°, 
the sum of the angle measures in a four-sided figure 
is 2- 180°, or 360°. 




Diagonal 



486 Chapter 8 Geometric Figures 



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SUM OF THE ANGLES OF A QUADRILATERAL 



The sum of the measures of 
the angles in a quadrilateral 
is 360°. 




m/11 + mzl2 + m^3 + mZ.4 = 360° 



EXAMPLE [2] Finding an Angle Measure in a Quadrilateral 



Find the unknown angle measure 
in the quadrilateral. 

98° + 137° + 52° + .V = 360° 




287° + x = 360° 

- 287° - 287° 

A- = 73° 



The sum of the 

angle measures is 3b0 . 

Combine like terms. 
Subtract 287° from both sides. 



The imknowTi angle measure is 73°. 



In a convex polygon, all diagonals can be drawn within the interior of 
the figure. By dividing any convex polygon into triangles, you can find 
the sum of its interior angle measures. 

EXAMPLE [3] Drawing Triangles to Find the Sum of Interior Angles 

Divide the polygon into triangles to find the sum of its 
angle measures. 

There are 5 triangles. 

5 • 180° = 900° 

The sum of the angle measures 
of a heptagon is 900°. 




TftiHk and Discuss 

1. Explain how to find the measure of an angle in a triangle when 
the measures of the two other angles are known. 

2. Determine for which polygon the sum of the angle measures is 
greater, a pentagon or an octagon. 

3. Explain how the measure of each angle in a regular polygon 
changes as the number of sides increases. 



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8-8 Angles in Polygons 487 



8-8 



p^ 



Homework Help Online go.hrw.com, 



keyword ■«iMM;g;M ® 
Exercises 1-18, 19, 21, 23, 25 



GUIDED PRACTICE 



See Example 1 Find the unknown angle measure in each triangle. 
1. /\ 2. 

\40° 






See Example 2 Find the unknown angle measure in each quadrilateral . 
4. Q 



127° 



135° 





See Example 3 Divide each polygon into triangles to find the sum of its angle measures. 
7. / \ 8. / -^ 9. 




INDEPENDENT PRACTICE 



See Example 1 Find the unknown angle measure in each triangle. 
10. 11. 





12. 




See Example 2 Find the unknown angle measure in each quadrilateral. 

14. 




15. 




135° 



105° 




See Example 3 Divide each polygon into triangles to find the sum of its angle measures. 
16. ~-x 17. / \ 18. 




Extra Practice 



PRACTICE AND PROBLEM SOLVING 



See page EP21. 



19. Earth Science A sundial consists of a circular base and a right triangle 
mounted upright on the base. One acute angle in the right triangle is 52° 
What is the measure of the other acute angle? 



488 Chapter 8 Geometric Figures 




Find the measure of the third angle in each triangle, given two angle measures. 
Then classify the triangle. 



56°, lor 



21. 18°, 63° 



22. 62°, 5^ 



23. 41°, 49° 



Multi-Step Each outer wall of the Pentagon in Washington, D.C., 
measures 921 feet. Wliat is the measure of each angle made by the 
Pentagon's outer walls? 



The Pentagon, 
headquarters 
of the U.S. 
Department of 
Defense, has 
about 23,000 
employees and 
17.5 miles of 
hallways. 



/Ki\717l\ 




25. Critical Thinking A truss bridge is 
supported by triangular frames. If 
every triangular frame in a truss bridge 
is an isosceles right triangle, what is 
the measure of each angle in one of the 
frames? (Hint: Two of tlie angles in 
each frame are congruent.) 

26. Make a Conjecture Use what you 
have learned to write a formula for I^E 
finding the sum of interior angle 

measures in polygons with five or more sides. 

^ 27. What's the Error? A student finds the sum of the angle measures in an 
octagon by multiplying 7 • 180°. What is the student's error? 

28. Write About It Explain how to find the sum of the angle measures in a 
quadrilateral by dividing the quadrilateral into triangles. 

q9 29. Challenge The angle beUveen the lines of sight 

from a lighthouse to a tugboat and to a cargo ship Cargo ship 

is 27°. The angle benveen the lines of sight at the 2x 

cargo ship is twice the angle between the lines of ^ 27Y 

sight at the tugboat. What are the angles at the Tugboat Lighthouse 

tugboat and at the cargo ship? 



i 



Test Prep and Spiral Review 



30. Multiple Choice A triangle has three congruent angles. What is the 
measure of each angle? 



(X) 50° 



CD 60° 



(X> 75° 



CE) 100° 



31. Gridded Response Two angles of a triangle measure 58° and 42°. 
What is the measure, in degrees, of the third angle of the triangle? 



Solve each proportion. Lesson 4-4) 

8 _ 24 
P 27 



32. ^ = 30 
3 18 



33. 



34. 



35. 



0.5 



Name the types of quadrilaterals that have each property. ( Lesson 8-7) 

36. two pairs of opposite, congruent sides 37. four congruent sides 



8-8 Angles in Polygons 489 




To Go On? 



.^pLearn It Online 

t* RP< 



Resources Online go.hrw.com, 

IBBIWIIm^i RTGosB^ Go; 



& 



Quiz for Lessons 8-4 Through 8-8 

8-4 ] Properties of Circles 
Name the parts of circle B. 



1. radii 



2. diameter 



3. chords 



4. A circle is divided into 6 equal sectors. Find the 
measure of the central angle of each sector. 




Qj 8-5 j Classifying Polygons 



Name each polygon, and tell whether it is a regular polygon. If it is not, 
explain why not. 




6. 





8. 



(^ 8-6 ] Classifying Triangles 

Classify each triangle according to its sides and angles. 



9. P 




10. 




11. 




(vj 8-7 j Classifying Quadrilaterals 



Give all of the names that apply to each quadrilateral. Then give the name 
that best describes it. 




15. 



16. 



(v) 8-8 J Angles in Polygons 



Find the unknown angle measure in each figure. 




18. 




19. 




490 Chapter 8 Geometric Figures 




Focus on Problem Solving 



v< 





Understand the Problem 

• Understand the words in the problem 

Words that you do not understand can sometimes make a simple 
problem seem difficult. Some of those words, such as the names of 
things or persons, may not even be necessary to solve the problem. 
If a problem contains an unfamiliar name, or one that you cannot 
pronounce, you can substitute another word for it. If a word that 
you don't understand is necessary' to solve the problem, look the 
word up to find its meaning. 




Students in a physics class use wire and 
resistors to build a VVheatstone bridge. Each 
side of their rhombus-shaped design is 2 cm 
long. What angle measures would the design 
have to have for its shape to be a square? 



Read each problem, and make a list of unusual or unfamiliar words 
If a word is not necessary to solve the problem, replace it with a 
familiar one. If a word is necessary, look up the word and write its 
meaning. 

O Using a pair of calipers, Mr. Papadimitriou 
measures the diameter of an ancient Greek 
amphora to be 17.8 cm at its widest point. 
Wliat is the radius of the amphora at 
this point? 



Q Joseph wants to plant gloxinia and 
hydrangeas in two similar rectangular 
gardens. The length of one garden is 5 ft, 
and the width is 4 ft. The other garden's 
length is 20 ft. What is the width of the 
second garden? 

O Mr. Manityche is sailing his catamaran 
from Kaua'i to Ni'ihau, a distance of about 
• 12 nautical miles. If his speed averages 
10 knots, how long wdll the trip take him? 

O Aimee's lepidoptera collection includes a 
butterfly with dots that appear to form a 
scalene triangle on each wang. What is the 
sum of the angles of each triangle on the 
butterfly's wings? 




Focus on Problem Solving 491 



8-9 







7.3.4 



Recognize, describe, or exten 
words, or symbols 



Vocabulary 

Side-Side-Side Rule 



geometric patterns using tables, graphs. 

Originally rolled and twisted by 
hand, pretzels today are primarily 
manufactured in production 
lines. After the dough is mixed, 
automated machines stamp the 
dough into consistent forms. 
These forms are the same shape 
and size. Recall from Lesson 
8-1 that congruent figures are 
the same shape and size. The 
automation of the production line 
process ensures that the pretzels 
are congruent. 

One way to determine whether figures are congruent is to see 
whether one figure will fit exactly over the other one. 




{ EXAMPLE [1j Identifying Congruent Figures in the Real World 

Identify any congruent figures. 



^n^BHEIil 



The squares on a 
checkerboard are 
congruent. The checkers 
are also congruent. 




The rings on a target are 
not congruent. Each ring 
is larger than the one 
inside of it. 



If all of the corresponding sides and angles of two polygons are 
congruent, then the polygons are congruent. For triangles, if the 
corresponding sides are congruent, then the corresponding angles will 
always be congruent. This is called the Side-Side-Side Rule. Because of 
this rule, when determining whether triangles are congruent, you only 
need to determine whether the sides are congruent. 



492 Chapter 8 Geometric Figures 



Vldau Lesson Tutorials OnlinE mv.hrw.com 



EXAMPLE [2] Identifying Congruent TriangI 



Determine whether the triangles 
are congruent. 



The scale factor of 
congruent figures 
is 1. Notice that in 
Example 2 the ratio 
of corresponding 



4 m 





4 m 



■_] ■ 3 4 

Sides is| = ^ 



AC = 3m DF = 3 m 
AB = 4m DE = 4m 
BC=5m EF=5m 4 3m C f 3m 

By the Side-Side-Side Rule, AABC is congruent to ADEF, or 

A ABC = ADEF. If you flip one triangle, it will fit exactly over the other. 

For polygons with more than three sides, it is not enough to compare 
the measures of their sides. For example, the corresponding sides of 
the figures below are congruent, but the figures are not congruent. 

120 m 120 m 

q 



70 m 



_d 



70 m 




120 m 



120 m 



If you know that two figures are congruent, you can find missing 
measures in the figures. 

EXAMPLE [3] Using Congruence to Find Unknown Measures 

Determine the unknown measure in each set of congruent 
polygons. 







87 

X 


7 


87^ 
930 


7 

118°/ 


118/ 







The corresponding angles 
of congruent polygons 
are congruent. 



The unknown angle measure is 93 
B 2 cm 2 cm 



3 cm 




3 cm 



4 cm 



5 cm 



The corresponding sides 
of congruent polygons 
5 cm 3re congruent. 



4 cm 



The unknown side length is 3 cm. 



Tfimk and Discuss 

1. Draw an illustration to explain whether an isosceles triangle can 
be congruent to a right triangle. 

2. Explain why congruent figures are always similar figures. 



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8-9 Congruent Figures 493 




i3.^^M3^ 



HonieworkHelpOnlinego.hrw.com, " 



keyword BBbiWKflgM <S) 

Exercises 1-14, 15, 17, 19 



GUIDED PRACTICE 



See Example 1 Identify any congruent figures. 

1. mM^sasi^^^ 2. 



M 



'?SIS^ 



j 



See Example 2 Determine whether the triangles are congruent. 




4. ^ A n c 

6 mm^ '-' -^ 5 mm 

C<^/5mm \^^ 

4 mm\/ 5 mm '4 mm 

B 



5. N 

6i 

M 




n- 7 in.\ /8in. 




f 7 in:^-^o R 

See Example 3 Determine the unlcnown measure in each set of congruent polygons. 

6. /\ ^ 7. 3.„^-^2.5 2 

(88 \ /y qfio\ , ^\ 2.5 




2.5 




INDEPENDENT PRACTICE 



See Example 1 Identify any congruent figures. 
8. 9. 



i^ 




10. r. 



/ 



See Example 2 Determine whether the triangles are congruent. 

11- 5 5ft 12. 1^13^ 




5m 



6ft 




14ft 



J 12 m " 13 m 
C 



5 m 



12m >4 



See Example 3 Determine the unknown measures in each set of congruent polygons. 

14. /\^4 In. /"^\4 in. 




4 in 



3 cm 




Bin. 



3 In 




Bin. 



Bin. 



494 Chapter 8 Geometric Figures 



PRACTICE AND PROfDEM^OLVING 



Extra Practice 



See page EP22. 



Tell the minimum amount of information needed to determine whether the 
figures are congruent. 

15. two triangles 16. two squares 17. two rectangles 18. two pentagons 



19. Surveying In the figure, trees /I and B are 
on opposite sides of the stream. Jamil wants 
to string a rope from one tree to the other. 
Triangles ABC and DEC are congruent. 
What is the distance between the trees? 

20. Hobbies In the 

quilt block, which 
figures appear 
congruent? 




^i!^n 



48°/ 






® 



Home 



21. Choose a Strategy Anji and her brother Art walked 
to school along the routes in the figure. They started at 
7:40 A.M. and walked at the same rate. Who arrived first? 
CA) Anji CS) Art (c]) They arrived at the same time. 

22. Write About It Are similar triangles always congruent? 
Explain. 



»" 23. Challenge If all of the angles in two triangles have the same measure, 
are the triangles necessarily congruent? Explain. 



Art's route 



' 








Anj 


's 


route 


C,l,«„ 



m 



Test Prep and Spiral Review 



24. Multiple Choice Which figures are congruent 



(S) 



Q 



'^'(•A) 



25. Multiple Choice Determine the unknown measure 
in the set of congruent triangles. 



6 mm 



CD) 



4 mm\y 
B 



CD 4 mm CH) 6 mm 

CS) 5 mm QD Cannot be determined 

Plot each point on a coordinate plane. (Lesson 5-1 ) 

26. A{-4,3) 27. B(l, -4) 28. C(-2, 0) 

Find the measure of the third angle in each triangle, given two angle 
measures. Then classify the triangle. (Lesson 8 ■ 

30. 25°, 48° 31. 125°, 30° 32. 60°, 60° 




Dr..^ 5 mm 



6 mm 4 mm 
F 

29. D(3. 2) 
33. 72°, 18° 



8-9 Congruent Figures 495 



8-1 



Translations, Reflections, 
and Rotations 1 







7.3.2 Identify, describe, and use transformations (translations, rotations, reflections 
and simple compositions of these transformations) to solve problems. 

In the photograph, Sasha Cohen 
is performing a layback spin. She is 
holding lier body in one position while 
she rotates. This is an example of a 
transformation. 



Vocabulary 

transformation 

image 

preimage 

translation 

reflection 

line of reflection 

rotation 



In mathematics, a transformation 
changes the position or orientation of a 
figure. The resulting figure is the image 
of the original figure, called the 
preimage . Images resulting from 
the transformations described below 
are congruent to the preimages. 




Translation 



Types of Transformations 
Reflection 



Rotation 




The figure slides along 
a straight line without 
turning. 





The figure flips across 
a line of reflection. 

creating a mirror image. 




The figure turns 
around a fixed point. 



EXAMPLE 1 



Identifying Types of Transformations 

Identify each type of transformation. 

A ga Ay 




Tlie figure slides along 
a straight line. 
It is a translation. 




7"/ie figure flips across 

ttie X-axis. 

It is a reflection. 



496 Cliapter 8 Geometric Figures 



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In a translation, the preimage slides a 
units right or left and b units up or down. 
A translation to the right or up is positive. 
A translation to the left or down is negative. 



(.V, V) 



(x + a,y+ b) 



EXAMPLE [2] Graphing Translations on a Coordinate Plane 




A' is read "A prime" 
and is used to 
represent the point 
on the image that 
corresponds to 
point A of the 
preimage. 



Graph the translation of hABC 6 units right and 4 units down. 
Write the coordinates of the vertices of the image. 



*y 



/\(-4, 5) 6 units 




B(-4. 3) C(-1,3) 



O 
-2 



right 



Each vertex is moved 6 units 

riglit ="^ ^ unite down. 



4 units 
down 



AABC 


(X + 6, y + (-4)) 


AA'B'C 


A{-4, 5) 


(-4 + 6, 5 + (-4)) 


A'(2, 1) 


S(-4, 3) 


(-4 + 6, 3 + (-4)) 


fi'(2, -1) 


C(-1,3) 


(-1 +6, 3 + (-4)) 


C'(5, -1) 



The coordinates of the vertices of AA'B'C aveA'(2. 1), B'(2. 
C'(5, -1). 



D.and 



In a reflection across the .v-axis, (.v, y) — »- U'. -.v)- 
In a reflection across the y-axis, (.v, y] — *- (-.v, y). 

EXAMPLE [3] Graphing Reflections on a Coordinate Plane 



Graph the reflection of each 
Write the coordinates of the 



figure across the indicated axis, 
vertices of each image. 



.v-axis 



*y 



f(-3, 3) 



G(1,4) 




B y-axls 



♦ y 




x-coordinates —*■ same 


x-coordinates — »- opposites 


y-coordinates — »- opposites 


y-coordinates — »- same 


The coordinates of the 


The coordinates of the 


vertices of AFC H' are 


vertices of the image are 


F(-3, -3),G'(1, -4), 


^'(5, -4),B'(3, -2), 


and//'(3, -1). 


C'(3, -3),D'(1, -3), 




F(l, -5),F(3, -5), 




andG'(3, -6). 



^Mh'jI Lesson Tutorials Online mv.hrw.com 8-10 Translations, Reflections, and Rotations 497 



EXAMPLE [4j Graphing Rotations on a Coordinate Plane 



The point that a 
figure rotates 
around may be on 
the figure or away 
from the figure. 



Triangle JKL has vertices J(0, 0), K{0, -3), and L(4, -3). Rotate 
AJKL 90° counterclocl<wise about the origin. Write the 
coordinates of the vertices of the image. 

— , — ft~ "^ — [ 

The corresponding sides, JK and JK', 
make a 90° angle. 

Notice that vertex K Is 3 units below 
the origin, and vertex K' is 3 units to 
the right of the origin. 



The coordinates of the vertices ofAjK'L' are/(0, 0), A" (3, 0), 
and L' (3, 4). 




Think and Discuss 

1. Explain how a figure skater might perform a translation and 
a rotation at the same time. 




See Example 1 Identify each type of transformation. 



** Hni 



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Exercises 1-14, 17 




See Example 2 Graph each translation. Write the coordinates of the vertices of each image. 

3. 2 units left and 3 units up 4. 3 units right and 4 units down 



*y 




^y 



Y 



I 



F G X 

^ — \ — I — I — I — I — i — *■ 



2 4 6 



498 Chapter 8 Geometric Figures 



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See Example 3 Graph the reflection of each figure across the indicated axis. Write the 
coordinates of the vertices of each image. 



5. A- axis 



6. y-axis 



Ay 




k-y 




See Example 4 7. Triangle LA/Nhas vertices L(0, 0), A/(-3, 0), and N(\, 4). Rotate ALMN 
[ 180° about the origin. Write the coordinates of tlie vertices of the image. 



INDEPENDENT PRACTICE 



See Example 1 Identify each type of transformation. 






See Example 2 Graph each translation. Write tlie coordinates of the vertices of each image. 
10. 5 units right and 1 unit down 11.4 units left and 3 units up 



Ay 




Ay 



-2 



-2 



fH 



X 



y z* 

See Example 3 Graph the reflection of each figure across the indicated axis. Write the 
coordinates of the vertices of each image. 
12. y-axis 13. .v-axis 

Ay 



Extra Practice 



See page EP22. 




K t 




See Example 4 14. Triangle MNL has vertices M(0, 4), Af(3, 3), and L(0, 0). Rotate AM/VL 90° 
counterclockwise about the origin. Write the coordinates of the vertices of 
the image. 



L 



8-10 Translations, Reflections, and Rotations 499 



Social Studies 



The Native American art pieces in tlie photos show 
combinations of transformations. Use the photos for 
Exercises 15 and 16. 



15. 



^) Write About It The Navajo blanket at right 
has a design based on a sand painting. The two 
people in the design are standing next to a stalk of 
corn, which the Native Americans called maize. The 
red, white, and black stripes represent a rainbow. 
Tell how the design shows reflections. Also explain 
what parts of the design do not show reflections. 



16. ^ Challenge What part of the bead design in 
the saddle bag at right can be described as three 
separate transformations? Draw diagrams to 
illustrate your answer. 



« 



I 



ik 







i 



Test Prep and Spiral Review 



LlUUi^^^^^^^^ 



17. Multiple Choice What will be the coordinates of point A' after 
a translation 2 units down and 3 units to the right? 



CAT. (0, 1) 



CD (1.0) 



(T) (-1.0) 



cb:' (0, -1) 



18. Short Response Triangle ABChas vertices/l(-4, 0), B(0, 0), 
and C(0, 5). Rotate AABC 90° clockwise around the origin. Draw 
AABC and its image. Write the coordinates of the vertices of the image 

Use the box-and-whisker plot for Exercises 19 and 20. (Lesson 7-5) 



Ay 




H — I — h 



H \ — \ \ h 



H \ \ h 



33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 

19. Wliat is the median of the data? 20. What is the range of the data? 

Determine the unknown measure in each set of congruent polygons. (Lesson 8-9) 
21. S E 22. N 

6 ir 
4m \ — ■ -■•■/ 4m 

A 3rr\ C 







■"— ~"^" 



500 Chapter 8 Geometric Figures 



tfedjlaJSfaw 



?^LAB7\ Explore Transformations 



Use with Lesson 8-10 



You can use geometry software to perform transformations of 
geometric figures. 



£?., 



Learn It Online 

Lab Resources Online go.hfw.com, 

IWfiffiTM'siO Labs MGoU 



Activity 



Q Use your dynamic geometry software to construct a 5-sided polygon 
like the one below. Label the vertices A. B. C. D, and E. Use the 
translation tool to translate the polygon 2 units right and I, unit up. 





O Start with the polygon from O- Use the rotation tool to rotate tiie 
polygon 30° and then 150°, both about the vertex C. 



1 

1 





Think and Discuss 



1. Rotate a triangle 30° about a point outside the triangle. Can this 
image be found by combining a vertical translation (slide up or down) 
and a horizontal translation (slide left or right) of the preimage? 

2. After what angle of rotation will the rotated image of a figure have the 
same orientation as the preimage? 



Try This 



1 . Construct a quadrilateral ABCD using the geometry software. 

a. Translate the figure 2 units right and 1 unit up. 

b. Rotate the figure 30°, 45°, and 60°. 



8-10 Technology Lab 501 



LESSON 8-10 



EXTENSI 



Dilations 



Vocabulary 

dilation 



EXAMPLE 



Similar figures have 
the same shape but 
not necessarily the 
same size. 



You can use computer software 
to dilate an image, such as a 
photograph. A dilation is a 
transformation that changes the 
size, but not the shape, of a figure. 
After a dilation, the image of a 
figure is similar to the preimage. 




9 



Identifying Dilations 

Tell whether each transformation is a dilation. 





The figures are similar, so 
the transformation is a 
dilation. 



5 m 



The figures are not similar, 
so the transformation is not 
a dilation. 



A dilation enlarges or reduces a figure. The scale factor tells you how 
much the figure is enlarged or reduced. On a coordinate plane, you 
can find the image of a figure after a dilation by multiplying the 
coordinates of the vertices by the scale factor. 



^EXAMPLE [zj Using a Dilation to Enlarge a Figure 



Draw the image of AABC after a dilation by a scale factor of 2. 

Ay 
Write the coordinates of the vertices of 

AABC. Then multiply the coordinates by 

2 to find the coordinates of the vertices 

ofAA'B'C. 

A(l,3)^A' (1 -2,3 -2) =A'(2,6) 

B (4, 3) -> B' (4 • 2, 3 • 2) = B'(8, 6) 

C(4, l)^C'(4-2, 1 -2) = C'(8, 2) 

Plot A',B', and C and draw AA'B'C 




502 Chapter 8 Geometric Figures 



EXAMPLE [3] Using a Dilation to Reduce a Figure 



Draw the image of ADEF after a dilation by a scale factor of |. 



Write the coordinates of the vertices of 
ADEF. Then multiply the coordinates by 
I to find the coordinates of the vertices 
ofAD'FF'. 

D(3, 3)^D'(3-|, 3-|) =D'(1, 1) 
£(9, 6)^f(9-|, 6-^) = F(3,2) 
F(6, 0)^F'(6-|, 0-|) =F'(2,0) 
Plot D',E'. and F' and draw AD'E'F'. 



*y 




i^i 



EXTENSION 



Exercises 



Tell whether each transformation is a dilation 
1. R R' 2. 

9.2 cm 




15 ft 



, 6 ft A'^ r? 

^Z7 \ / 

D^^C D'^ ^C 

3 ft 7.5 ft 



8 cm 



7 cm 



Draw the image of each figure after a dilation by the given scale factor. 

3. scale factor 3 4. scale factor 2 

i^y 

4 



X 

H — t — ^*■ 



A 4 



5. scale factor 77 
*y 




J 



M L X 

H 1 1 1 (-»- 



2 4 



6. scale factor ^ 



t 


ky 




c 






8 










6 













4. 


/ 








2- 


/»' 




s 
















X 


' 


2 


4 ( 


3 


8 




' 


' 











Lesson 8-10 Extension 503 








7.3.4 Recognize, describe, or extend geometric patterns using tables, graphs, 
words, or symbols 

When you can draw a line 
through a plane figure so that 
the two halves are reflections 
of each other, the figure has 
line symmetry. The line of 
reflection is called the 
line of symmetry . The 
reflections you created in 
Lesson 8-10 have line 
symmetiy. 



Vocabulary 

line symmetry 
line of symmetry 
asymmetry 
rotational symmetry 
center of rotation 




Many architects and artists use symmetry in their buildings. The structure 
of the Puerta de Europa towers in Madrid, Spain, is symmetrical. You 
can draw a line of symmetry between the towers. 

When a figure is not symmetrical, it has asymmetry , or is asymmetrical. 



EXAMPLE 




Identifying Line Symmetry 

Decide whether each figure has line symmetry. If it does, draw al 
the lines of symmetry. 





3 lines of symmetry 



4 lines of symmetry 



, EXAMPLE [2j Social Studies Application 

Find all the lines of symmetry in each flag 

A ^ ^ 1 B 





There is 1 line of symmetry. 



There are no lines of symmetry. 



504 Chapter 8 Geometric Figures 



Vjdaii Lesson Tutorials Online my.hrw.com 



EXAMPLE 



When you rotate a figure, you can create 
a figure with rotational symmetiy. 
A figure has rotational symmetry if, 
when it is rotated less than 360° around 
a central point, it coincides with itself. 
The central point is called the 
center of rotation. 

If the stained glass window at right is 
rotated 90°, as shown, the image looks 
the same as the original stained glass 
window. Therefore the window has 
rotational symmetry. 

[ 3 J Identifying Rotational Symmetry 

i Tell how many times each figure will show rotational symmetry 
within one full rotation. 



Center of 
rotation 





The starfish will show 
rotational symmetn,' 5 times 
within a 360° rotation. 



m 




The pinwheel will show 
rotational symmetry' 4 times 
within a 360° rotation. 



Draw lines from thie center of 
tfie figure out tfiroughi identical 
places in the figure. 



Count the number 
of lines drawn. 



Draw lines from the center of 
the figure out through identical 
places in the figure. 



Count the number 
of lines drawn. 



ThiHk and Discuss 

1. Draw a figure that does not have rotational symmetry. 

2. Determine whether an equilateral triangle has rotational 
symmetry. If so, tell how many times it shows rotational symmetry 
within one full rotation. 



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8-11 Symmetry 505 




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keyword IBBilifcaiM ® 
Exercises 1-18, 19, 21 



GUIDED PRACTICE 



See Example 1 Decide whether each figure has line symmetry. If it does, draw all the lines 
j of symmetry. 

1. 





See Example 2 Find all the lines of symmetry in each flag. 
4. ^ 1 5. 



ES 





See Example 3 Tell how many times each figure will show rotational symmetry within one 
full rotation. 




8. ^ 




^ 



INDEPENDENT,RBACTJCE 



See Example 1 Decide whether each figure has line symmetry. If it does, draw all the lines 
I of symmetry. 

10. n n 11. K A 12. 




See Example 2 Find all the lines of symmetry in each flag. 
13. ^^ 14. 




E 



15. 



See Example 3 Tell how many times each figure will show rotational symmetry within one 
full rotation. 



16. 



O 



17. 



18. 



OVO/ 




506 Chapter 8 Geometric Figures 



PRACTICE AND PROBLEM SOLVING 



Extra Practice 



See page EP22. 




19. Critical Thinking Which regular polygon shows rotational symmetry 
9 times vvithin one full rotation? 

20. Nature How many lines of symmetry, if any, 
does the snowflake have? How many times, if 
any. will the snowflake show rotational symmetry 
within one full rotation? 

21. Fold a piece of paper in half vertically and then 
in half horizontally. Cut or tear a design into 
one of the folded edges. Then unfold the paper. 
Does the design have a vertical or horizontal 
line of symmetry'? rotational symmetry? Explain. 

22. Art Tell how many times the stained glass 
design shows rotational symmetn,' in one full 
rotation if you consider only the shape of the 
design. Then tell how many times the image 
shows rotational symmetry if you consider 
both the shape and the colors in the design. 

D 23. What's the Question? Maria drew a square 
on the chalkboard. As an answer to Maria's 
question about symmetiy, Rob said "90°." 
What question did Maria ask? 

^P 24. Write About It Explain why an angle of rotation must be less than 360° 
for a figure to have rotational symmetry. 

^p 25. Challenge Print a word in capital letters, using only letters that have 

horizontal lines of symmetr\'. Print another word using only capital letters 
that have vertical lines of symmetiy. 



^V 




\r 






X 






gOsv 


i\) 


(M 






K 


?7\ 


yi 





i 



Test Prep and Spiral Review 



26. Multiple Choice How many lines of symmetry does the figure have? 

C£) None CD 1 CD 2 CD 4 

<(■■)> 



# 



27. Gridded Response How many times will the figure 
show rotational symmetry within one full rotation? 



28. A bridge in an architectural model is 22 cm long. The model scale 
is 2 cm:30 m. Find the length of the actual bridge. (Lesson 4-10) 

Triangle /AT has vertices /( -3, -1), A'(-l, -1), and L{ — 1, —4). Write the coordinates 
of the vertices of the triangle after each transformation. (Lesson 8-10) 

29. Translate the triangle 4 units right and 2 units down. 

30. Reflect the triangle across they-axis. 



8-11 Symmetry 507 



Bl^ Create Tessellations 



Use with Lessons 8-10 and 8- 1 1 



£?., 



Tessellations are patterns of identical shapes that completely 
cover a plane with no gaps or overlaps. The artist M. C. Escher 
created many fascinating tessellations. 



Activity 



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Q Create a translation tessellation. 

The tessellation by M. C. Escher shown 
at right is an example of a translation 
tessellation. To create your own 
translation tessellation, follow the 
steps below. 










a. Start by drawing a square, rectangle, or 
other parallelogram. Replace one side 
of the parallelogram with a curve, 
as shown. 



b. Translate the curve to the opposite side 
of the parallelogram. 




c. Repeat steps a and b for the other two 
sides of your parallelogram. 





The figure can be translated to create an 
interlocking design, or tessellation. You can 
add details to your figure or divide it into 
two or more parts, as shown below. 




508 Chapter 8 Geometric Figures 



Q Create a rotation tessellation. 

The tessellation by M. C. Escher shown 
at right is an example of a rotation 
tessellation. To create your own rotation 
tessellation, follow the steps below. 

a. Start with a regular hexagon. Replace one 
side of the hexagon with a curve. Rotate the 
curve about point B so that the endpoint at 
point A is moved to point C. 





b. Replace side CD with a new curve, and rotate 
it about point D to replace side DE. 





Replace side fFwith a new curve, and rotate it 
about point f to replace side FA. 



The figure can be rotated and fitted together with 
copies of itself to create an interlocking design, or 
tessellation. You can add details to your figure, if desired. 




Think and Discuss 



1. Explain why the two t\'pes of tessellations in this activity' are known as 
translation and rotation tessellations. 



Try Til is 



1. Create your own design for a translation or rotation tessellation. 

2. Cut out copies of your design from 1 and fit them together to fill 
a space with your pattern. 



8-n Hands-On Lab 509 



CHAPTER \ 




Ready To Go On? 



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Quiz for Lessons 8-9 Through 8-11 

(^ 8-9 j Congruent Figures 

Determine whether the triangles are congruent. 

D 2. 



25 ft 25 ft 





20 ft 




C F 

3. Determine the unknown measure in 
the pair of congruent polygons. 



1/1/ A^X 



17 m, 
Z' 



\17m 17 m'' 



'17m 



20 m 



8 m 



(v) 8-10] Translations, Reflections, and Rotations 

Graph each transformation. Write the coordinates of the vertices of each image. 



4. Translate triangle 
RST5 units down. 



*y 



5. Reflect the figure 
across the x-axis. 



6. Rotate triangle JKL 90° 
clockwise about die origin. 





*y 




Q) 8-11] Symmetry 



7. Decide whether the figure has line 
symmetry. If it does, draw all the 
lines of symmetry. 



8. Tell how many times the figure will 
show rotational symmetry within 
one full rotation. 




510 Chapter 8 Geometric Figures 



CONNECTIONS 




Piscataqua River Bridge The first bridge over the 

Piscataqua River, buih in 1794, was the longest bridge in the 
world. The modern bridge, completed in 1971, is not the 
world's longest, but it is well known for its elegant symmetric 
design. The bridge connects Kittery, Maine with Portsmouth, 
New Hampshire. 

1 . Does the Piscataqua River Bridge have any lines of symmetry? 
If so, make a simple sketch of the bridge and draw all of its lines 
of symmetry. 

For 2-7, use the diagram. 

2. Zl and L2 are supplementary. Given that m_l is 78°, what 
is m^2? 

3. Classify AAEF according to its angles. Then measure the sides 
with a ruler, and classif\' the triangle according to its sides. 

4. Quadrilateral AEFD is a trapezoid. What can you conclude about 
AD and IF? 

5. What can you say about Z 1 and ^EAUi Why? 

6. Find m/_£4D. 

7. Given that m /lDFE is 96°, find mZ3. 
Explain how you found the angle measure. 



MAINE 




Kittery 




jonnecTions 




aiffijiMe 



Networks 



A network is a figure that uses vertices and 
segments to show how objects are connected. 
You can use a network to show distances 
between cities. In the network at right, tlie 
vertices identify four cities in North Carolina, 
and the segments show the distances in miles 
between the cities. 

You can use the network to find the shortest 
route from Charlotte to the other three cities 
and back to Charlotte. First find all the possible 
routes. Then find the distance in miles for each 
route. One route has been identified below. 



Greensboro 




Wilmington 



CGWRC 



94 + 215 + 127 + 98 = 534 



Which is the shortest route, and what is the distance? 



Color Craze 



You can use rhombus-shaped tiles to build 
a variety of polygons. Each side of a tile 
is a different color. Build each design by 
matching the same-colored sides of tiles. 
Then see if you can create your own designs 
with the tiles. Try to make designs that have 
line or rotational symmetry. 

A complete set of tiles is available online. 





Learn It Online 

Game Time Extra go.hrw.coiti, 



x-i,-^>\<fi<S:^^s<>>*.*i5,^^ 



512 Chapter 8 Geometric Figures 




Materials 

• 6 sheets of 
construction 
paper 

• card stock 
■ scissors 

• hole punch 

• 4 electrical ties 

• white paper 

• markers 



' » &\ 



PROJECT 



Brochure Book of 
Geometric Figures 

Make an organizer to hold brochures that summarize 
each lesson of the chapter. 

Directions 

" Start with sheets of construction paper that are 
12 inches by 18 inches. Fold one sheet in half to 
make it 12 inches by 9 inches and then in half again 
to make it 6 inches by 9 inches. Figure A 

" Hold the paper with the folds at the bottom and on 
the right-hand side. Turn the top left-hand corner 
back and under to form a pocket. Figure B 

^ Turn the whole thing over and fold the top right-hand 
corner back and under to form a pocket. Repeat steps 
1-3 with the other sheets of construction paper. 

^ Cut out two pieces of card stock that are 6 inches by 
9 inches. Punch four equally spaced holes down the 
length of each piece. Similarly, punch four equally 
spaced holes on each pocket as shown. Figure C 

" Stack the six pockets and put the card stock r 
covers on the front and back of the stack. 
Insert electrical ties into the holes to hold 
everything together. 

Taking Note of the Math 

Fold sheets of plain white paper into thirds like a 
brochure. Use the brochures to take notes on the 
lessons of the chapter. Store the brochures in the 
pockets of your organizer. 



■> 



-^■5i' 






It's in the Bag! 513 




Vocabulary 

acute angle 454 

acute triangle 473 

adjacent angles 450 

angle 454 

arc 468 

asymmetry 504 

center of a circle 453 

center of rotation 595 

central angle 459 

chord 468 

circle 453 

complementary angles 454 

congruent 449 

diagonal 436 

diameter 453 

equilateral triangle . . . 473 

image 495 

isosceles triangle 473 

line 448 



line of reflection 495 

line of symmetry 504 

line segment 449 

line symmetry 594 

obtuse angle 454 

obtuse triangle 473 

parallel lines 450 

parallelogram 432 

perpendicular lines . . . 450 

plane 443 

point 443 

polygon 474 

preimage 495 

radius 453 

ray 449 

rectangle 432 

reflection 495 

regular polygon 475 

rhombus 432 



right angle 454 

right triangle 473 

rotation 495 

rotational symmetry . . 505 

scalene triangle 473 

sector 469 

Side-Side-Side Rule . . . 492 

skew lines 46O 

square 432 

straight angle 454 

supplementary angles 454 

transformation 496 

translation 495 

transversal 46I 

trapezoid 432 

vertex 454 

vertical angles 451 



Complete the sentences below with vocabulary words from the list above. 

1. Every equilateral triangle is also a(n) ? triangle. 

2. Lines in the same plane that do not intersect are ? . 

3. A line segment whose endpoints are any two points on a circle is acn) !_ 



EXAMPLES 



EXERCISES 



8-1] Building Blocks of Geometry (pp. 448-451) 

Identify the figures in the diagram. Identify the figures in the diagram. 

; ■ points: A, B. C m lines: AB 4. points 5. lines £ 

I planes: ABC ■ rays:BA'AB Ay\ ^- planes 7. rays 

I line segments: AB; EC ^__ _.__, 8. line segments 




514 Chapter 8 Geometric Figures 



EXAMPLES 



EXERCISES 



8-2J Classifying Angles (pp. 454-457) 

I Tell whether the angle is 
acute, right, obtuse, or 
straight. 

The angle is a right angle. 



Tell whether each angle is acute, right, 
obtuse, or straight. 

9. \ 10. 




8-3] Line and Angle Relationships (pp. 460-463) 



Tell whether the lines 
appear parallel, 
perpendicular, or skew. 



Line a 1 1 line b. Find 
the measure of Z.4. 
Corresponding angles 
are congruent. 
mZ4 = 74° 



perpendicular 



Tell whether the lines appear parallel, 
perpendicular, or skew. 



11, 




12. 



r 



/ 




For Exercises 13-16, use the figure at left. 
Find the measure of each angle. 

14. A3 
16. Z6 



8-4J Properties of Circles (pp. 468-471) 



Name the parts of circle D. 

■ radii: DB, DC, DE 

■ diameter: EB 

■ chords: AB, M £F 




Name the parts of circle F. 

17. radii 

18. diameter 

19. chords 




8-5J Classifying Polygons (pp. 474-477) 

■ Tell whether the figure is a regular 
polygon. If it is not, explain why not. 

il No, all the angles in the \;^^^V? 

polygon are not congruent. /--sj 



Tell whether each figure is a regular 
polygon. If it is not, explain why not. 

20. pr+T] 21. 



O 



8-6] Classifying Triangles (pp 478-481) 

I Classify the triangle 
according to its sides 
and angles. 



Isosceles right 




Classify each triangle according to its 
sides and angles. 

22. A 23. 




!/Jd=K Lesson Tutorials OnlinE mv.hrw.com 



Study Guide: Review 515 



EXAMPLES 



8-7 ] Classifying Quadrilaterals (pp. 482-485) 

■ Give all of the names that apply to the 
quadrilateral. Then give the nanie that 
best describes it. 



EXERCISES 



Give all of the names that apply to each 
quadrilateral. Then give the name that 
best describes it. 



trapezoid; trapezoid 



24. r-+ 



U 



25. 



8-8 j Angles in Polygons (pp. 485-489) 

■ Find the measure 
.^ of the unknovm angle. 

62° + 45° + .V = 180° 
107° + A- = 180° 
A- = 73° 




Find the measure of each unknown angle. 
26. /\ 27. 




8-9] Congruent Figures (pp. 492-495) 



Determine the unknown ^ 
measure in the set of N 
congruent polygons. 

The angle measures 53°. 



M 



^\37° 


L 


53°\N 


\^ 


n 37°^ 


\^ 



28. Determine the unknown measures in 
the set of congruent polygons. 

10 cm 10 cm 

4cm /i3F 47°/ 4cm/i33^ 



1335 



4 cm 



■^4 cm 



10 cm 



8-10] Translations, Reflections, and Rotations (pp 496-500) 



Graph the translation. 
Write the coordinates of 4 
the vertices of the image. 

Translate AABC 1 unit 
right and 3 units down. c 

A/l'B'C has vertices 

^'(3, 1),B'(5, 3),andC'(5, 1). 



A 



H \ 1 1 »■ 



Graph the translation. 
Write the coordinates of 
the vertices of the image. 

29. Translate A BCD 
2 units left and 
4 units down. 



8-11] Symmetry (pp. 504-507) 

■ Find all the lines of 
symmetry in the flag. 

The flag has four lines of 
p symmetry. 




30. Find all the lines of 
symmetry in the flag. 



516 Chapter 8 Geometric Figures 



Chapter Test 



CHAPTER 



Identify the figures in the diagram. 
1. 4 points 2. 3 lines 

4. 5 line segments 5. 6 rays 




3. a plane 



Line AB || line CD in the diagram. Find the measure of each angle 
and tell whether the angle is acute, right, obtuse, or straight. 

6. AABC 7. ABCE 8. ADCE 




Tell whether the lines appear parallel, perpendicular, or skew 
9. M/VandPO 10. LM and PO 11. NO and MN 

Name the parts of circle E. 

12. radii 13. chords 14. diameter 



kew. " Mi Am 

', ,, V ■« • II- 



P O 

A, 




Tell whether each figure is a regular polygon. If it is not, explain why not. 

16. y -X 17. 



,s.^ 



Classify each triangle according to its sides and angles. 

19. 



Give all the names that apply to each quadrilateral. Then give the name that best 
describes it. 



21. 



22. 



-+- 



23. 



Find the measure of each unknown angle 
24 



75^ 





27. Determine the unknown measure 8 in. 

in the set of congruent polygons. 6 in./ 



8 in. 



6 in. 



6 in. 



8 in. 



8 in. 



28. The vertices of A/IBC have the coordinates /1(-1, -3), B(-4, -1), 
and C(-l, -1). Graph the triangle after a translation 3 units left. 
Write the coordinates of the vertices of the image. 



Find all the lines of symmetry in each flag. 



29. 



B 



30. 



Chapter 8 Test 517 



CHAPTER 



8 



^ ISTEP+ 
^ Test Prep 



£t. 



Learn It Online 

State Test Practice go.hrw.com, 

IW'lliMSKilostPrep^GoJ 



Applied Skills Assessment 

Constructed Response 

1. Triangle ABC, with vertices A{2, 3), 
e(4, 0), and C(0, 0), is translated 2 units 
left and 6 units down to form triangle 
A'B'C. 

a. On a coordinate plane, draw and 
label triangle ABC and triangle 
A'B'C. 

b. Give the coordinates of the vertices 
of triangle /\'e'C'. 

2. Taylor's goal is to spend less than 35% 
of her allowance each month on cell 
phone bills. Last month, Taylor spent 
$45 on cell phone bills. If she gets $120 
each month as her allowance, did she 
achieve her goal? Explain your answer. 

3. Consider the sequence 4, 8, 12, 16, 
20 

a. Write a rule for the sequence. Use 
n to represent the position of the 
term in the sequence. 

b. What is the 8th term in the 
sequence? 

Extended Response 

4. Four of the angles in a pentagon 
measure 74°, 111°, 145°, and 95°. 

a. How many sides and how many 
angles does a pentagon have? 

b. Is the pentagon a regular 
pentagon? How do you know? 

c. What is the sum of the angle 
measures of a pentagon? Include a 
drawing as part of your answer. 

d. Write and solve an equation to 
determine the missing angle 
measure of the pentagon. 



Multiple-Choice Assessment 

5. Which angle is a right angle? 
A. _ C. 




D. 



6. What is the number 8,330,000,000 
written in scientific notation? 

A. 0.83 X 10^° C. 83.3 x 10^ 

B. 8.33 X 10'' D. 833 x 10^ 

7. If point A is translated 5 units left and 
2 units up, what will point A's new 
coordinates be? 



-4 


-2 


o 




2 


4 


^1 ...J._.... 
















! ~ 
i 


■2 














1 










A 




i 
1 




p4 


! 









X 



A. (-2, -2) 

B. (8, -2) 



C. (-2, -6) 

D. (0, 1) 



8. Nolan spent ^ hour traveling to his 
orthodontist appointment, | hour at 
his appointment, and ~ hour traveling 
home. What is the total amount of 
time Nolan spent for this appointment? 



11 



hour 



B. 1^ hour 

bU 



C. 1^ hours 



D. ^ hours 



518 Chapter 8 Geometric Figures 



9. A store sells two dozen rolls of toilet 
paper for $4.84. What is the unit rate 
for one roll of toilet paper? 

A. $0.13/roll of toilet paper 

B. $0.20/roll of toilet paper 

C. $0.40/roll of toilet paper 

D. $1.21/roll of toilet paper 

10. Which of the following best describes 
the triangle below? 




A. Acute isosceles triangle 

B. Equilateral triangle 

C. Obtuse right triangle 

D. Obtuse scalene triangle 



11. Which expression represents "twice the 
difference of a number and 8"? 



A. 2(x + 8) 

B. 2x - 8 



C. 2(x - 8) 

D. 2x + 8 



12. For which equation is x = 1 NOT the 
solution? 

A. 3x + 8 = 1 1 

B. 8 - X = 9 

C. -3x + 8 = 5 

D. 8 + X = 9 

13. Which ratios form a proportion? 



A. I and I 



T2^^<s 



^^"< 



D. fandf 



14. The graph shows how Amy spends her 
earnings each month. Amy earned $100 
in May. How much did she spend on 
transportation and clothing combined? 



How Amy Spends Her Earnings 



Savings 

Entertainment ^°°''° 
25% 



I 
Transportation 

15% 




Miscellaneous 
20% 



Clothing 

30% 



A. $15 

B. $30 



C. $45 

D. $55 



# 



Once you have answered a short- or 
extended-response question, check to 
make sure you have answered all parts 
of the question. 



Gridded Response 

15. What is the unknown angle measure 
in degrees? 




16. A figure has vertices /\(-4, -4), 
e(-3, -2), and C(-3, -6). What will 
the x-coordinate of point A' be after 
the figure is reflected across the y-axis? 

17. An antiques dealer bought a chair for 
$85. The dealer sold the chair at her 
shop for 45% more than what she paid. 
To the nearest whole dollar, what was 
the price of the chair? 

18. What is the value of the expression 
-4x^y - y for x = -2 and y = -5? 



Cumulative Assessment, Chapters 1-8 519 



CHAPTER 



Tififo-Dimensional 
Figures 



9A Perimeter, 

Circumference, 
and Area 

9-1 Accuracy and Precision 

LAB Explore Perimeter and 
Circumference 

9-2 Perimeter and 
Circumference 

LAB Explore Area of 
Polygons 

9-3 Area of Parallelograms 

9-4 Area of Triangles and 
Trapezoids 

LAB Compare Perimeter and 
Area of Similar Figures 

9-5 Area of Circles 

9-6 Area of Irregular Figures 

9B Using Squares and 
Square Roots 

LAB Explore Square Roots and 
Perfect Squares 

9-7 Squares and Square 
Roots 

EXT Identifying and Graphing 
Irrational Numbers 

LAB Explore the Pythagorean 
Theorem 

9-8 The Pythagorean Theorem 







apter 



7.1.5 



• Solve problems involving 
area and circumference of 
circles. 

• Investigate the areas of similar 
figures. 



i^i 









Why Learn Th 



The perimeter and area of garden beds car 
be determined by measuring their lengths 
and widths and then using a formula. 



.a 



Learn It Online 

Chapter Project Online go.hrw.com, 



keyword MBIaillMiBM ® 






r:-w^. 



520 Chapter 9 



..;«'■ 




« ./^^Ik'v/r^'^m^-r^' \ 



^j 



^n 




Are You Ready 



^^P Learn It Online 



Resources Online go.hrw.com. 



ST Vocabulary 

Choose the best term from the list to complete each sentence. 

1 . A (n) ? is a quadrilateral with exactly one pair of 
parallel sides. 

2. A(n) ? is a four-sided figure with opposite sides 
that are congruent and parallel. 

3. The ? of a circle is one-half the 



of the circle. 



diameter 

parallelogram 

radius 

right triangle 

trapezoid 



Complete these exercises to review skills you will need for this chapter. 

(v; Round Whole Numbers 

Round each number to the nearest ten and nearest hundred. 

4. 1,535 5. 294 6. 30,758 7. 497 

(v; Round Decimals 

Round each number to the nearest whole number and nearest tenth. 
8. 6.18 9. 10.50 10. 513.93 11. 29.06 

^j Multiply with Decimals 

Multiply. 

12. 5.63 • 8 13. 9.67 • 4.3 14. 8.34 • 16 

16. 0.82-21 17. 2.74-6.6 18. 40-9.54 

Order of Operations 

Simplify each expression. 

20. 2 -9 -I- 2 - 6 21. 2(15 + 8) 

23. 14(25.9+13.6) 24. (27.3 + 0.7) -^ 2" 

26. (63 + 7) -4- 27. 1.1 +3-4.3 



15. 6.08 - 0.56 
19. 0.33 - 0.08 



22. 4 • 6.8 + 7 - 9.3 

25. 5 • 3-* - 8.02 

28. 66 - [5 + (3 + 3)' 



Identify Polygons 

Name each figure. 
29. 




31. 



0easurement: Two-Dimensional Figures 521 




study G 



Where You've Been 



Previously, you 

• found the perimeter or 
circumference of geometric 
figures. 

• explored customary and metric 
units of measure. 

• used proportions to convert 
measurements witiiin the 
customary system and within 
the metric system. 



Key 

Vocabulary /Vocabulario 



area 


area 


circumference 


circunferencia 


hypotenuse 


hipotenusa 


perfect square 


cuadrado perfecto 


perimeter 


perimetro 


Pythagorean Theorem 


Teorema de Pitagoras 


significant digits 


digitos significativos 



square root 



raiz cuadrada 



In This Chapter 



You will study 

• comparing perimeter and 
circumference with the area 
of geometric figures. 

• finding the area of 
parallelograms, triangles, 
trapezoids, and circles. 

• finding the area of irregular 
figures. 

• using powers, roots, and the 
Pythagorean Theorem to find 
missing measures. 

Where You're Going 

You can use the skills 
learned in this chapter 

• to create an architectural 
floor plan. 

• to design a building access 
ramp that meets government 
regulations. 



Vocabulary Connections 

To become familiar with some of the 
vocabulary terms in the chapter, consider the 
following. You may refer to the chapter, the 
glossary, or a dictionary if you like. 

1 . The square root of a number is one of 
the two equal factors of the number. 
For example, 3 is a square root because 
3-3 = 9. How might picturing plant roots 
help you remember the meaning of 
square root ? 

2. The word perimeter comes from the Greek 
roots peri, meaning "all around," and 
inetrou, meaning "measure." What do the 
Greek roots tell you about the perimeter 
of a geometric figure? 

3. To square a number means "to multiply the 
number by itself," as in 2 • 2. Keeping this 
idea of square in mind, what do you think 

a perfect square might be? 

4. The word circumference comes from the 
Latin word circumferre, meaning "to carry 
around." How does the Latin meaning help 
you define the circumference of a circle? 



522 Chapter 9 Measurement: Two-Dimensional Figures 




Reading / 
a"d WriMtta 
MathX -^ 



CHAPTER 




Reading Strategy: Read and Interpret Graphics 

Figures, diagrams, tables, and graphs provide important data. Knowing how 
to read these graphics will help you understand and solve related problems. 



Similar Figures 



AABC and A/AX are similar 




Double-Bar Graph 



udent Enrollme 



2,000 




2004 2005 2006 2007 



I Seventh-graders Year 

I Eighth-graders 



How to Read 



D 



Read all labels. 

AB = 8cm;^C= 16cm;BC= 12 cm; 
]K = 28 cm; ]L — 56 cm; KL = x cm; 
/.A corresponds to AJ. 

Be careful about what you assume. 
You may think AB corresponds to 
LK, but this is not so. Since Z./1 
corresponds to /.J, you know 
AB corresponds to JK. 



How to Read 



m-jJiA' • 



Read the title of the graph and any 
special notes. 

Blue indicates seventh-graders. 
Purple indicates eighth-graders. 

Read each axis label and note the 
intervals of each scale. 
.v-axis — year increases by 1. 
y-axis — enrollment increases by 
400 students. 

Determine what information is presented, 
student enrollment for seventh- 
and eighth-graders per year 



"ny This 



Look up each graphic in your textbook and answer the following questions. 

1. Lesson 4-8 Exercise 1; Which side of the smaller triangle corresponds to BC? 
Which angle corresponds to /LEDFl 

2. Lesson 7-3 Example 1: By what interval does the .v-axis scale increase? About 
how many people speak Hindi? 



Measurement: Two-Dimensional Figures 523 



u 



9-1 



CIS10 



Vocabulary 

precision 
accuracy 



Ancient Greeks used measurements taken during lunar eclipses to 
determine that the Moon was an average distance of 240,000 miles 
from Earth. Modern astronomers place the average distance at 
238,855 miles. 

Although the measurements are relatively close, modern astronomers 
measure with greater precision. Precision is the level of detail an 
instrument can measure. 

The smaller the unit an instrument 
can measure, the more precise its 
measurements will be. For example, a 
millimeter ruler has greater precision 
than a centimeter ruler. 




EXAMPLE [l] Judging Precision of Measurements 

Choose the more precise measurement in each pair. 



25 in. An inch is a 


B 4 qt One tenth of a 


2 ft smaller unit 


4.3 qt quart is a smaller 


than a foot. 


unit than a quart 


25 in. is the more 


4.3 qt is the more 


precise measurement. 


precise measurement. 



You can measure 
length only to the 
precision level of the 
tool you are using. 



In the real world, no measurement is exact and all measurements are 
approximations. Accuracy is the closeness of any given measurement or 
value to the actual measurement or value. 



EXAMPLE [2] Measuring to Varying Degrees of Accuracy 



Measure the length of the paper clip to the nearest half, fourth, 
and eighth inch. Which measurement is the most accurate? 
Explain. 



Length to the nearest half inch: 1 in. 
n[([lllll[l[r|I|l]i|||n Length to the nearest fourth inch: 7| 



1 



Length to the nearest eighth inch: 1~ in. = 1^ in. 



Measuring to the nearest fourth and to the nearest eighth both result 
in 1 1 in. Although measuring to the nearest eighth involves greater 
precision, both measurements are equally accurate because they are 
equally close to the actual value. 



524 Chapter 9 Measurement: Two-Dimensional Figures 



1/JiJiL/ Lesson Tutorials Online 



EXAMPLE 



Since measurements are only as precise as the tool being used, in some 
cases you may need to estimate measurements beyond the level of 
precision provided by tlie instrument. 



[Sj Estimating Measures 

Estimate each measurement. 




10 - 5 = 5 



1 



5 = 2.5 



5 + 2.5 = 



lb 



The weight of the potatoes is 
halfway between the 5 lb and 
10 lb mark. 



Find the difference between the marks. 
Find half of 5 lb. 

Add the two weights together to find 

the weiaht of the potatoes. 



The weight of the potatoes is about 7.5 lb. 




The amount of juice in the cup is about a 
fourth of the way between 2 fl oz and 
4 fl oz. 



Find the difference between the marks. 
Find one fourth of 2 fl oz. 



4-2 = 2 

i.2 = i 

4 2 

2+;^ = 2irflOZ 

The amount of liquid in the cup is about 2 ^ 11 oz 



Add the two amounts together to 
find the number of fl oz of juice. 



^^^^^^^^^^^^^^^^^^^^^^^^^^^^^B 


Think and Discuss 




1. Find the most precise measurement for the pape 
Example 2. 


r clip in 


2. Explain whether measuring to the nearest ^ in. or to the 

nearest ^ in. would give the more accurate measurement for a 
nail that is 3| inches in length. 



[fi'Jb'j] Lesson Tutorials OnliriE mv.hrw.com 



9-1 Accuracy and Precision 525 



9-1 



iiii<«riiiiii 

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keyword MMIlfcHM (S) 

Exercises 1-13, 15, 19, 21 



GUIDED PRACTICE 



See Example 1 Choose the more precise measurement in each pair. 

, 1. 5,281 yd 2. 1.05 g 3. 205 1b 

1 3 mi 1.1 g 205.5 1b 

See Example 2 5. Measure to the nearest half, fourth, 

and eighth inch. Which measurement 
is the most accurate? Explain. 



See Example 3 6. Estimate the mass of the baclcpack. 



4. 1ft 
5 in. 



"% 




INDEPENDENT PRACTICE 



See Example 1 Choose the more precise measurement in each pair. 

7. 1.2 mm 8. 15floz 9. 5Jrft 



1 mm 



Uc 



5ift 



10. 300g 
13 kg 



See Example 2 11. Measure to the nearest half, fourth, and eighth inch. Wliich measurement is 
the most accurate? Explain. 



See Example 3 Estimate each measurement. 
12. 




13. 



8ft 




Extra Practice 



See page EP23. 



PRACTICE AND PROBLEM SOLVING 



Choose the more precise unit in each pair. 

14. liter or millimeter 15. ounce or pound 



16. quart or fluid ounce 



17. Critical Thinking The prefix <Yec(7 means ten. The prefix rfra means 
tenth. Would the length of an object be more accurate if measured in 
decameters or decimeters? Explain. 



526 Chapter 9 Measurement: Two-Dimensional Figures 



Estimate the measure of each angle. 
18. 




19. 




V Z B C 

20. Estimation Estimate and then measure the width of a hallway at your 
school. Give your answer measured to the nearest meter and centimeter. 

Find the greatest precision for each scale shown. 

21. _Cd_I,' 7 22. 

s - 4 — 

2— 



23. Critical Thinking Rita wants to center a poster on the wall of her 
room. The tools available to her to help her measure include a ruler, a 
measuring tape, and a meter stick. Which tool or tools should she choose? 
Explain. 

© 24. What's the Error? Shia says that 4.25 m is a more precise measure than 
4.2 mm. Wliat is his error? 

^ 25. Write About It Give an example of when an accurate measurement is 
important and when an estimate will do. 

^ 26. Challenge The weight limit for vehicles on a bridge is 40 tons. The 
weight of a loaded truck is estimated at 40 tons. Should the truck be 
allowed to cross the bridge? Explain. 



i 



Test Prep and Spiral Review 



27. Multiple Choice Which is the most precise measurement? 
C£) 1 mile CI) 1,758 vards CD 5,281 feet 



CD 63,355 inches 



28. Short Response Kylie is measuring the thickness of a nickel. Which unit, 
inches or millimeters, would give her the more precise measurement? Explain. 

For Exercises 29-30, tell whether you would expect a positive correlation, 
a negative correlation, or no correlation. (Lesson 7 9) 

29. the price of a car and the number of windows it has 

30. the speed a car travels and the amount of time it takes to go 1 00 miles 

Determine whether each figure is a polygon. If it is not, explain why. (Lesson 8-5) 
31. 



33 



<^ 



5-7 Accuracy and Precision 527 



k^ 



Explore Perimeter & 
Circumference 



Use with Lesson 9-2 



The distance around a figure is its perimeter. You can use a 
loop of string to explore the dimensions of a rectangle with 
a perimeter of 18 inches. 



M 



Learn It Online 

Lab Resources Online go.hrw.com, 



keyword MHMBlBsl ® 



Activity 1 



O Cut a piece of string that is slightly longer than 
18 inches. Tie the ends together to form an 
18-inch loop. 

Make the loop into a rectangle by placing it around 
four push pins on a corkboard. Both the length and 
the width of the rectangle should be a whole number 
of inches. 

Q Make different rectangles with whole- number lengths 
and widths. Record the lengths and widths in a table. 



Length (in.) 



Width (in.) 



O Graph the data in your table by plotting points on a 
coordinate plane like the one shown. 



Think and Discuss 



1. What pattern is made by the points on your graph? 

2. How is the sum of the length and width of each rectangle 
related to the rectangle's perimeter of 18 inches? 

3. Suppose a rectangle has length < and width w. Write a 
rule that you can use to find the rectangle's perimeter. 




9 
8 

7 

6 

5 

4 

3 

2 

1 + 

<■ 




T — r 



X 



12 3 4 5 6 7 
Length (in.) 



8 9 



Try This 



Use the rule you discovered to find the perimeter of each rectangle. 

1. I 1 7 9 ft 



4 in. 



3ft 



6 in. 



3. 5 '-''^ 



5 cm 



528 Chapter 9 Measurement: Two-Dimensional Figures 



The perimeter of a circle is called the circumference. You can 
explore the relationship between a circle's circumference and 
its diameter by measuring some circles. 



Activity 2 



O Four students should stand in a circle wath their arms 
outstretched, as showTi in the diagram. 

Q Another student should find the diameter of the circle 
by measuring the distance across the middle of the 
circle with a tape measure. 

Q The student should also find the circumference of the circle 
by measuring the distance around the circle from fingertip 
to fingertip across the backs of the students. 

Q Record the diameter and circumference in a table 
like the one below. 




Circumference 



Diameter 




Circumference 





Q Add one or more students to the circle and repeat the 
process. Record the diameter and circumference for at 
least five different circles. 

Q Graph the data in your table by plotting points on a 
coordinate plane like the one shown. 



Thinic and Discuss 



1. Make a Conjecture In general, what do you notice about 
the points on your graph? Wliat shape or pattern do they 
seem to form? 

2. Calculate the ratio of the circumference to the diameter for 
each of the data points. Then calculate the mean of these 
ratios. For any circle, the ratio of the circumference to the 
diameter is a constant, known as pi (n). Give an estimate 
for ;r based on your findings. 



E 



H 1 1 1 1 1- 



Diameter (ft) 



Try This 



1. For a circle with circumference Cand diameter d, the ratio of the 
circumference to the diameter is |^ = .r. Use this to write a formula 
that you can use to find the circumference of a circle when you 
know its diameter. 

2. Use your estimate for the value of ;rto find the approximate 
circumference of the circle at right. 




9-2 Hands-On Lab 529 



9-2 



Vocabulary 

perimeter 

circumference 

pi 



Perimeter and 
Circumference 



In volleyball, the player serving must hit 
the ball over the net but keep it within the 
court's sidelines and end lines. The two 
sidelines on a volleyball court are each 
18 meters long, and the two end lines are 
each 9 meters long.Together, the four lines 
form the perimeter of the court. 

Perimeter is the distance around a geometric 
figure. To find the perimeter P of a rectangular 
volleyball court, you can add the lengths of its 
sides. Perimeter is measured in units of length. 



V 







EXAMPLE [1| Finding the Perimeter of a Polygon 

Find the perimeter. 

P = 9 + 12 + 11 Use the side lengtlis. 



9 cm 




12 cm 



P = 32 



Add. 



11 cm 



The perimeter of the triangle is 32 cm. 



Since opposite sides of a rectangle are equal in length, you can find the 
perimeter of a rectangle by using a formula. 



Interactivities Online ► 



PERIMET 


^d :'-•-: 


^^^^^ 


CTANqLE 1 


PPi| 


The perimeter P of a rectangle 
is the sum of twice its length ( 


P = 2€ + 2w 












IV 


V 










€ 





EXAMPLE 



? 



Using Properties of a Rectangle to Find Perimeter 

Find the perimeter. 

P = 2€ + 2»' Use tiie formula. 

15 m P=(2 • 32)+(2 • 15) Substitute for / and w. 

Multiply. 



[ 



32 m 



P = 64 + 30 
P=94 



Add. 



The perimeter of the rectangle is 94 m. 



530 Chapter 9 Measurement: Two-Dimensional Figures 



y'ld-dpl Lesson Tutorials Online 



The distance around a circle is called circumference . For every circle, the 
ratio of circumference Cto diameter d is the same. This ratio, ^, is represented 
by the Greelc letter ;r, called pi . Pi is approximately equal to 3.14 ory. By 
solving the equation ^ = ;r for C, you get the formula for circumference. 



CIRCUMFERENCE OF A CIRCLE 



The circumference C 
of a circle is tt times 
the diameter d. or 2/: 
times the radius r. 



C = mi 

or 
C = 2n:r 



Radius 
Diamete 



Circumference 




EXAMPLE [3] Finding the Circumference of a Circle 



.jJjJyiJJ 



If the diameter or 
radius of a circle is a 
multiple of 7, use y 
for K. 



22 . ^ 
,T 1 



56 



EXAIVIPLE 



Find the circumference of each circle to the nearest tenth, if 
necessary. Use 3.14 or y for ;r. 

C = /rd You know the diameter. 

C « 3. 1 4 • 8 Substitute 3. U for n and 8 for d. 

C'^IS.U IVIultiply. 

The circumference of the circle is about 25.1 in. 





You know ttie radius. 
Substitute y for k and 14 for r. 



C = Ztti- 

C^2-"4- 14 

C « 88 IVIultiply. 

The circumference of the circle is about 



cm. 



? 



Design Application 

Lily is drawing plans for a circular fountain. The circumference of 
the fountain is 63 ft. What is its approximate diameter? 



C= Ttd 
63 « 3.14 



You know the circumference. 
Substitute 3. 14 for k and 63 for C. 



63 

3.14 



3.14 ■ d 



3.14 



Divide both sides by 3. 14 to isolate the variable. 
20 «f/ 
The diameter of the fountain is about 20 ft. 



Think and Discuss 

1. Describe two ways to find the perimeter of a volleyball court. 

2. Explain how to use the formula C = Ttdto find the circumference 
of a circle if you know the radius. 



'Mbii Lesson Tutorials Online my.hrw.com 



9-2 Perimeter and Circumference 531 



9-2 



Homework Help Online go.hrw.com. 



keyword MiteiniiaiM ® 

Exercises 1-20, 21, 23 



GUIDED PRACTICE 



Find each perimeter. 



See Example 1 1. 




See Example 2 4. 



6 in. 



12 in. 



2. 


7 


in. 






5 in. , 


/ 


/ 


/s 




7 in. 




5. 




8tn 








J 




L 


2r 




1 




r 



3. 8ft 



8 ft 




See Example 3 Find the circumference of each circle to the nearest tenth, if necessary. 
Use 3. 14 or ^ for ;r. 

See Example 4 10. A Ferris wheel has a circumference of 440 feet. What is the approximate 
diameter of the Ferris wheel? Use 3.14 for k. 





INDEPENDENT PRACTICE 



Find each perimeter. 
See Example 1 11. .^^'^"^ , '^^■ 

12cm/ /l2cm 



7 ft 




12 cm 
See Example 2 14. 8 in. 



10ft 



15. 3 ft 



Sin. 



1 ft 



13. '"^ "^ 

8 m 



JOm 



16m 



16. 8 cm 



10.2 cm 



See Example 3 Find the circumference of each circle to the nearest tenth, if necessary. 



Use 3.14 or ^ for ;r. 



17. 




18. 




19. 




See Example 4 20. The circumference of Kayla's bicycle wheel is 91 inches. What is the 
L approximate diameter of her bicycle wheel? Use 3.14 for /r. 



532 Chapter 9 Measurement: Two-Dimensional Figures 



PRACTICE AND PROBLEM SOLVING 



Extra Practice 



See page EP23. 







The U.S. Capitol 
Rotunda Is 96 ft 
in diameter and 
rises 180 ft 3 in. 
to tfie canopy. The 
rotunda contains 
many historical 
paintings, including 
the Frieze of 
American History 
and several 
memorial statues. 



Find each missing measurement to the nearest tenth. Use 3.14 for k. 

22. V = 6.7 yd; ri = ; C = F; 

24. /• = ; ff = ; C = ;r 



21. 
23. 
25. 






d= ;C= 17.8 m 
,-= ;rf=10.6in.; C = 



26. 



< 



28. 



29. 



® 



30. 



31, 



Critical Thinking Ben is placing rope lights around the edge of a circular 
patio with a 24.2 ft diameter. The lights are in lengths of 57 inches. How 
many strands of lights does he need to surround the patio edge? 

Geography The map shows the distances in 
miles between the airports on the Big Island 
of Hawaii. A pilot flies from Kailua-Kona to 
Waimea to Hilo and back to Kailua-Kona. 
How far does he travel? 

Architecture The Capitol Rotunda connects 
the House and Senate sides of the U.S. Capitol. 
The rotunda is 180 feet tall and has a 
circumference of about 301.5 feet. What is its 
approximate diameter, to the nearest foot? 

Describe how you could use a piece of string to 
find the perimeter or circumference of an object. 

Write a Problem Write a problem about finding the perimeter or 
circumference of an object in your school or classroom. 

Write About It Explain how to find the width of a rectangle if you know 
its perimeter and length. 

Challenge The perimeter of a regular nonagon is 25:^ in. Wliat is the 
length of one side of the nonagon? 




m 



Test Prep and Spiral Review 



32. Multiple Choice Which is the best estimate for the circumference of a 
circle with a diameter of 15 inches? 



Ca;- 18.1 inches 



d) 23.6 inches 



CD 32.5 inches 



® 47.1 inches 



33. Gridded Response John is building a dog pen that is 6 feet by 8 feet. How 
many feet of fencing material will he need to go all the way around the pen? 



Solve. (Lesson 6-5) 

34. 18 is 20% of what number? 



35. 78% of 65 is what number? 



Choose the more precise measure in each pair. (Lesson 9-1 

36. 4 ft, 1 yd 37. 2 cm, 21 mm 



38. 5^ in., 5| in. 



39. 37 g, 37.0 g 



9-2 Perimeter and Circumference 533 



Explore Area of Polygons 



Use with Lessons 9-3, 9-4 and 9-5 



You can use a parallelogram to find the area of a triangle or a 
trapezoid. To do so, you must first know how to find the area of 
a parallelogram. 



m^t 



Learn It Online 

Lab Resources Online go.hrw.com, 

l^^!ff j! ]|1^10 Lab9 Mfi^ 



Activity 1 



Q On a sheet of graph paper, draw a parallelogram with a base 
of 10 units and a height of 6 units. 

Q Cut out the parallelogram. Then cut a right triangle off the 
end of the parallelogram by cutting along the altitude. 

Q Move the triangle to the other side of the figure to make 
a rectangle. 

Q How is the area of the parallelogram related to the area of 
the rectangle? 

Q What are the length and width of the rectangle? What is the 
area of the rectangle? 

Q Find the area of the parallelogram. 



' 


■ 1 i 1 1 ! 1 ! 1 i 


, . : 1 1 1 ' ! ■ ' 




l_i . J 


/ 


6 / 








yEZ_ 


„, — ^ 


- i-0-,-^-,---^-^--t-H~J 




Thini< and Discuss 



1. How are the length and width of the rectangle related to the 
base and height of the parallelogram? 

2. Suppose a parallelogram has base h and height h. Write a 
formula for the area of the parallelogram. 



JIL 



Try This 



1 . Does your formula work for any parallelogram? If so, show how 
to use the formula to find the area of the parallelogram at right. 

2. Explain what must be true about the areas of the 
parallelograms below. 





5 in. 



534 Chapter 9 Measurement: Two-Dimensional Figures 



Activity 2 



^ On a sheet of graph paper, draw a triangle with a base of 
7 units and a height of 4 units. 

Q Cut out the triangle. Then use the triangle to trace and 
cut out a second triangle that is congruent to it. 

© Arrange the two triangles to form a parallelogram. 

Q How is the area of the triangle related to the area of the 
parallelogram? 

Q Find the areas of the parallelogram and the triangle. 





Think and Discuss 



1. How are the base and height of the triangle related to the 
base and height of the parallelogram? 

2. Suppose a triangle has base b and height /;. Write a formula 
for the area of the triangle. 




Try Tliis 



1. Find the area of a triangle with a base of 10 ft and a height of 5 ft. 



Activity 3 



^ On a sheet of graph paper, draw a trapezoid with bases 
4 units and 8 units long and a height of 3 units. 

Q Cut out the trapezoid. Then use the trapezoid to trace 
and cut out a second trapezoid that is congruent to it. 

Q Arrange the two trapezoids to form a parallelogram. 

Q How is the area of the trapezoid related to the area of the 
parallelogram? 

Q Find the areas of the parallelogram and the trapezoid. 




Think and Discuss 



1 . What is the length of the base of the parallelogram at 
right? What is the parallelogram's area? 

2. What is the area of one of the trapezoids in the figure? 



b^ 



_tL 



b^ 



Try This 



1 . Find the area of a trapezoid with bases 4 in. and 6 in. and a height 
of 8 in. 



9-3 Hands-On Lab 535 




Vocabulary 



area 



The area of a figure is tlie number 
of unit squares needed to cover the 
figure. Area is measured in units of 
length squared, or square units. For 
example, the area of a chessboard 
can be measured in square inches. 
The area of a lawn cliessboard is much 
larger than a regular chessboard, 
so it can be measured in square 
feet or square yards. 



-,,. , .a 


lRea of a rectangle 


The area A of a rectangle is 
the product of its length ( 
and its width w. 


A = €w 












w 








€ 





EXAMPLE 1 



Finding the Area of a Rectangle 

Find the area of the rectangle. 

A = €w 
A= 10- 7.5 

7.5 ft A =75 



Use the formula. 
Substitute for I and w. 
Multiply. 



The area of the rectangle is 75 ft . 



10 ft 



EXAMPLE [zj Finding Length or Width of a Rectangle 



Bethany and her dad are planting a rectangular garden. The area 
of the garden is 1,080 ft^, and the width is 24 ft. What is the 
length of the garden? 

A = €iu Use the formula for the area of a rectangle. 

1,080 = ( • 24 



,080 _ < . 24 



24 24 

45 = (■ 
Tlie length of the garden is 45 ft. 



Substitute 1,080 for A and 24 for w. 
Divide both sides by 24 to isolate C. 



536 Chapter 9 Measurement: Two-Dimensional Figures 



VliJai;] Lessor Tutorials OnllnE 



The base of a parallelogram is the length of one side. Its height is the 
perpendicular distance from the base to the opposite side. 



"■I 


AREA OF A PARALLELOGRAM J 


f: y 


The area A of a parallelogram 
is the product of its base b 
and its lieight /?. 


A = bh 




A / 

/ n / 






b 

J 



EXAMPLE 



^ 



EXAMPLE 



[3 



Finding the Area of a Parallelogram 

Find the area of the parallelogram. 

A — bh Use the formula. 

Substitute for b and h. 



33 cm 



/l = 6f.3i 



A 



_ 20 10 



Convert to improper fractions. 



6| cm 



A^^or 22| Multiply. 

The area of the parallelogram is 22| cm'^. 



Landscaping Application 

Birgit and Mark are building a rectangular patio measuring 9 yd 
by 7 yd. How many square feet of tile will they need? 

First draw and label a diagram. Look at the units. The patio is 
measured in yards, but the answer should be in square feet. 



9 yd 



7 yd 



f^ = 27 ft 
1 yd 



Convert yards to feet by 
using a unit conversion 
factor. 



3ft _ 
1yd 



21ft 



9 yd 



Now find the area of the patio in square feet. 

A — i w Use the formula for the area of a rectangle. 

/I = 27 • 21 Substitute 27 for i and 21 for w. 

A = 567 Multiply. 

Birgit and Mark need 567 ft*^ of tile. 



7 yd 



Tfiink and Discuss 

1. Write a formula for the area of a square, using an exponent. 

2. Explain why the area of a nonrectangular parallelogram with side 
lengths 5 in. and 3 in. is not 15 in"^. 



'Mb'j Lesson Tutorials Online my.hrw.com 



9-3 Area of Parallelograms 537 



9-3 



-H' 



S3JdM3i5 



Homework Help Online go.hrw.com, 



keyword ■MMiiiiaa ® 
Exercises 1-16, 17, 19 



GUIDED PRACTICE 



See Example 1 Find the area of each rectangle. 

1. 8 ft 2. 3 m 



.2 ft 



7 m 



16.4 cm 



9 cm 



See Example 2 4. Kara wants a rug for her bedroom. She knows the area of her bedroom is 
L 132 ff^. The length of her room is 12 ft. What is the wadtli of Kara's bedroom? 



See Example 3 Find the area of each parallelogram. 
5. / 7 6. 

6 in. 

Jd 



4 cm 



7. 



4.4 m 

n 



8 in. 



2|cm 



6.5 m 



See Example 4 8. Anna is mowing a rectangular field measuring 120 yd by 66 yd. How many 
L square feet will Anna mow? 



INDEPENDENT PRACTICE 



See Example 1 Find the area of each rectangle. 
9. I 1 10. 

7ft 



12 ft 



15j in. 



82 in. 



11. 



9.6 in. 



11.2 in. 



See Example 2 12. James and Linda are fencing a rectangular area of the yard for their dog. 
The width of the dog yard is 4.5 m. Its area is 67.5 m". What is the length 
of the dog yard? 



See Example 3 Find the area of each parallelogram 
13. \ \ 14. 



1.5m;' 
a. 



J2 



21ft 



4 m 



7lft 




8.2 cm 



3.9 cm 



See Example 4 16. Abby is painting rectangular blocks on her bathroom walls. Each block is 
15 in. by 18 in. Wliat is the area of one block in square feet? 



538 Chapter 9 Measurement: Two-Dimensional Figures 



PRACTICE AND PROBLEM SOLVING 



Extra Practice 



See page EP23. 



Find the area of each polygon. 

17. rectangle: ( = 9 yd; w - 8 yd 



18. parallelogram: b = 7 m; /; = 4.2 m 



Graph the polygon with the given vertices. Identify the polygon and then 
find its area. 



19. 
21. 



(2,0),(2, -2),(9,0),(9, -2) 



Art Without the frame, Icarus by Henri Matisse 
measures about 42 cm by 64 cm. The width of 
the frame is 8 cm. 

a. Find the perimeter and area of the painting. 

b. What is the total area covered by the painting 
and the frame? 

i^ 22. What's the Error? Pete says the area of a 3 cm 
by 4 cm rectangle is 12 cm. What is his error? 

^23. Choose a Strategy Theareaof a parallelogram 
is 84 cm~. If the base is 5 cm longer than the 
height, what is the length of the base? 



20. (4, 1), (4, 7), (8, 4), (8, 10) 




CE) 5 cm 



CE' 7 cm 



(^' i: 



cm 



Icarus by Henri Matisse 

CS:> 14 cm 



^y 24. Write About It A rectangle and a parallelogram have sides that measure 
3 m, 4 m, 3 m, and 4 m. Do the figures have the same area? Explain. 

^5 25. Challenge Two parallelograms have the same base length, but the height 
of the first is half that of the second. What is the ratio of the area of the first 
parallelogram to that of the second? What would the ratio be if both the hei^ 
and the base of the first parallelogram were half those of the second? 



:ht 



Test Prep and Spiral Review 



26. Multiple Choice Find the area of the parallelogram. 



CA) 13 in- CD 26 in" 



(T) 40 in- 



E) 56 in-" 




27. Extended Response Kiana is helping her dad build a deck. The plans they 
have are for a 6-foot-by-8-foot deck, but her dad wants a deck that has twice as 
much area. He suggests doubling the length of each side of the deck. Will this 
double the area? If not, suggest another method for doubling the area of the deck. 

Tell whether each angle is acute, obtuse, right or straight. (Lesson 8-2) 

28. ^ 29. / *■ 30. ^\ 31. 



< 



Find the perimeter of each rectangle, given the dimensions. (Lesson 9 2) 



32. 6 in. bv 12 in. 



33. 2 m bv 8 m 



34. 16 cm by 3 cm 35. 4^ ft by l| ft 



9-3 Area of Parallelograms 539 



9-4 



Area of Triangles 
and Trapezoids 



^M. 



jijj 



An altitude of a 
triangle is a segment 
that represents the 
height. 



The Bermuda Triangle is a triangular 
region of the Atlantic Ocean in 
which a number of aircraft and ships 
have mysteriously disappeared. To 
find the area of this region, you 
could use the formula for the area 
of a triangle. 

'/ The base of a triangle can be any 
I side. The height of a triangle is 
the perpendicular distance from 
the base to the opposite vertex. 




Puerto Rico 



mmm^oFmrtmKfmm 



The area A of a triangle is half 
the product of its base b and 



its height //. 



A = ^bh 




EXAMPLE [T] Finding the Area of a Triangle 

Find the area of each triangle. 




A = ^bh 



Use the formula. 

Substitute 4 for b and 3 for h. 




A = ^{4 -3) 

A = 6 

The area of the triangle is 6 square units. 

A = \bh Use the formula. 

yl = i(6 • 5) Substitute 6 for b and 5 for h. 

A^ 15 

The area of the triangle is 15 square units. 



The two parallel sides of a trapezoid are its 
bases, b^ and /;,. The height of a trapezoid is 
the perpendicular distance between the bases. 



^ 



540 Chapter 9 Measurement: Two-Dimensional Figures 



yidHu Lesson Tutorials OniinE 



WPSPWf 



A OF A TRAPEZOID 



The area of a trapezoid is half 
its height multiplied by the sum 
of the lengths of its two bases. 



A = \h[b^ + b.) 



EXAMPLE [2! Finding the Area of a Trapezoid 

Find the area of each trapezoid. 



ReadindjNalh 



In the term b , 
the number 1 is 
called a subscript. 
It is read as "b-one" 
or "b sub-one." 







A = ^hib^ + bj 


Use the formula. 


5 in. 


A^\-4{\0 + 6) 
.4 = i-4{16) 


Substitute. 


A- \ 


Add. 


10 in. 


.4 = 32 


Multiply. 




The area of the trapezoid is 32 iir. 




A = |/2(Z;, + b) 


Use the formula. 


19 cm 


-^ -\— ^•llfl'i+iq) 




11 cm 


J 


A = \-ll{34) 
A= 187 


Add. 




IS rm 


Multiply. 



The area of the trapezoid is 187 cm" 



E X A IVI P L E [3] Geography Application 



The state of Nevada is shaped 
somewhat like a trapezoid. What 
is the approximate area of Nevada? 



T 



320 mi ► 



200 mi parson 
I ^ City 



NEVADA 



A = \h(b^ + h) 
A^\- 320(200 + 475) 



Use the formula. 

Substitute. 

Add. 



475 mi 



/I = i • 320(675) 

A = 108,000 Multiply. 

The area of Nevada is approximately 108,000 square miles. 



^^^^^^^^^^^^^^^^^^^^^^^^^^^^B 


Think and Discuss 






1. Tell how to use the sides of a right triangle to find its 


area. 


2. Explain how to find the area 


of a trapezoid. 





MzLi Lesson Tutorials Online my.hrw.com 



9-4 Area of Triangles and Trapezoids 541 



9-4 




[733333 



(•runii 

Homework Help Online go.hrw.com. 



keyword ■WBiiwaM ® 
Exercises 1-14, IS, 17, 19, 21 



GUIDED PRACTICE 



See Example 1 Find the area of each triangle. 

^. A 2. 




8 





11.2 



See Example 2 Find the area of each trapezoid. 

4. 2.5 cm 5. 6 m 



1 2 cm 



8 m 



4 cm 



10 m 



12ft 



Tl 

:6ft 



6ft 



See Example 3 7. The state of Tennessee is shaped 
somewhat like a trapezoid. Wliat 
is the approximate area of Tennessee? 



442 mi 



T Nashville* %|^ 

3 rr- 

i 



115mi TENNESSEE '^^»-^' 



T" 



H 350 mi- 



INDEPENDENT PRACTICE 



See Example 1 Find the area of each triangle. 

8. 15 9. 





10. 




See Example 2 Find the area of each trapezoid. 
11. 15yd 12. 



12 yd 



n. 



40 yd 



3 in. 




10 in. 



18 in. 



13. 



10 cm 



5 cm 



See Example 3 14. The state of New Hampshire is shaped somewhat 
hke a right triangle. What is the approximate 
area of New Hampshire? 



NEW 
HAMPSHIRE 



160 mi 



Concord 



85 mi- 



542 Chapter 9 Measurement: Two-Dimensional Figures 



PRACTICE AND PROBLEM SOLVING 



Extra Practice 



See page EP23. 



i 



Find the missing measurement of each triangle. 

15. b = 8 cm 16. b= 16 ft 

/; = /; = 0.7 ft 

/I = 18 cm' A = 



17. b^ ^" 
h = 95 in. 
A = 1,045 iir 



Graph the polygon with the given vertices. Identify the polygon and then 
find its area. 



18. (1,2), (4,5), (8,2), (8,5) 
20. (2, 3), (2, 10),(7, 6),(7, 8) 



19. (l,-6),(5, -1),(7, -6) 
21. (3,0), (3,4), (-3,0) 



22. What is the height of a trapezoid with an area of 9 m- and bases that 
measure 2.4 m and 3.6 m? 



23. Multi-Step The state of Colorado is somewhat 
rectangular in shape. Estimate the perimeter 
and area of Colorado. 

^ 24. What's the Error? 

A student says the 
area of the triangle 
showTi at right is 
33 cm". Explain why 
the student is incorrect. 



276 mi 




6 cm 



365 mi — 

Denver -A' 

COLORADO 



u 
id 



11 cm 



« 



25. Write About It Explain how to use the formulas for the area of a rectangle 
and the area of a triangle to estimate the area of Nevada. 

^ 26. Challenge The state of North Dakota is trapezoidal in shape and has 
an area of 70,704 mi". If the southern border is 359 mi and the distance 
between the northern border and the soutiiern border is 210 mi, what is 
the approximate length of the northern border? 



Test Prep and Spiral Review 



3 cm 




5 cm 



27. Multiple Choice Find the area of the trapezoid. 
C£) 8 cm" CO 17 cm- 
CT) 16 cm" CD 30 cm" 

28. Short Response Graph the triangle with vertices (0, 0), (2, 3), and (6, 0). 
Then find the area of the triangle. 



Find the measure of the third angle in each triangle, given two angle 
measures. (Lesson 8-8) 

29. 45°, 45° 30. 71°, 57° 31. 103°, 28° 32. 62°, 19° 

33. lustin is laying a tile floor in a room that measures 5 yd by 6 yd. How many 
square feet of tile does he need? (Lesson 9-3) 



9-4 Area of Triangles and Trapezoids 543 



Compare Perimeter and 
Area of Similar Figures 



Use with Lesson 9-4 



REMEMBER 

• Two figures are similar when the measures of the corresponding angles 
are equal and the ratios of the corresponding sides are equivalent. 

• A scale factor is the ratio used to enlarge or reduce similar figures. 



/^ Learn It Online 

t* LabResourcesOnlinego.hrw.com, 



keyword BiHIllBiail ® 



Activity 1 



Q On graph paper, use a ruler to 
draw two rectangles. 
Rectangle A: ( = 3 in., w = 2 in. 
Rectangle B: ( = 6 in., ii> — 4 in. 

Q Use rectangles A and B to complete 
the first two columns of the table. 

Complete the third column by 
calculating the ratios for each row. 





Rectangle 
A 


Rectangle 
B 


Rectangle 6 
Rectangle A 


Length (in.) 








Width (in.) 


■ 


t - 


■ 


Perimeter (in.) 








Area (in^) 









Think and Discuss 



1. Identify the scale factor between rectangles A and B. Which ratios in the 
table are the same as the scale factor? 



Try This 



Draw rectangle C with ( = 1.5 in., w = 1 in. 

1. Complete a table similar to the table in Activity 1 for rectangles A and C. 

2. Make a Conjecture Make a conjecture about the relationship between 
scale factor and perimeter for any similar rectangles. 

3. Make a Conjecture Based on the information in the tables, how do you 
think the ratio of the areas of similar rectangles is related to the scale 
factor? [Hint: Area is measured in square units.) 



544 Chapter 9 Measurement: Two-Dimensional Figures 



Activity 2 



O Draw and label two isosceles triangles as shown 
in the diagram. 



Triangle A 




5 in. 



Triangle B 

2.5 in. 




3 in. 



Q Complete the table. Use a ruler to 
measure the height of the triangle. 

Q Complete the third column by 
calculating the ratios for each row. 





Triangle A 


Triangle B 


Triangle B 
Triangle A 


Base length (in.) 






^1 


Side length (in.) 








Height (in.) 








Perimeter (in.) 








Area (in^) 









Think and Discuss 



1. Make a Conjecture Based on your results from Activity 1 and the 
information in the table, make a conjecture about the relationship 
between the perimeters of two similar figures. 

2. Make a Conjecture Based on your results from Activity 1 and the 
information in the table, make a conjecture about the relationship 
between the areas of two similar figures. 

3. Make a Prediction Predict what will happen to the area of a triangle if 
the lengths of all its sides are multiplied by 4. 



Try This 



A rectangle has a perimeter of 30 in. and an area of 50 in^. Find the perimeter 
and area of each similar rectangle with the given scale factor. 



1. scale factor = 6 



2. scale factor - 10 



3. scale factor = ^^ 



4. Critical Thinking Do you think the relationship between the scale factor 
and perimeter and area will be true for ANY Uvo similar polygons? Explain. 



9-4 Hands-On Lab 545 




d 



A circle can be cut into equal-sized 
sectors and arranged to resemble a 
parallelogram. The height Ij of the 
parallelogram is equal to the 
radius r of the circle, and the base 
b of the parallelogram is equal to 
one-half the circumference Cof 
the circle. So the area of the 
parallelogram can be written as 

A = bh, ox A = \Cr. 

Since C = 2nr, A = h2/rr)r — ni". 




AREA OF A CIRCLE 



The area /\ of a circle is the 
product of K and the square 
of the circle's radius r. 



A = n-r 




The order of 
operations calls for 
evaluating the 
exponents before 
multiplying. 



EXAMPLE 1 



]l3jjjaijjij-a,rj 



Finding the Area of a Circle 

Find the area of each circle to the nearest tenth. Use 3.14 for ;r. 

A /^ ~\ A = nr~ Use the formula. 

^ « 3.14 • 3- Substitute. Use 3 for r. 

i4 == 3.14 • 9 Evaluate the power. 

A « 28.26 Multiply. 

The area of the circle is about 28.3 m*^. 





A = nr^ 
/I = 3.14 • 4' 
/1 = 3.14 • 16 
A « 50.24 



Use the formula. 
Substitute. Use 4 for r. 
Evaluate the power 
Multiply. 



The area of the circle is about 50.2 in"^. 



546 Chapter 9 Measurement: Two-Dimensional Figures 



Tiday Lesson Tutorials OnlinE 




Nomads in Mongolia 
carried their l:omes 
wherever they roamed. 
These homes, called 
yurts, were made of 
wood and felt. 



EXAMPLE 



^ 



Helpful Hint 



To estimate the area 
of a circle, you can 
square tfie radius 
and multiply by 3. 



Social Studies Application 

A group of historians are building a yurt to display at a local 
multicultural fair. The yurt has a height of 8 feet 9 inches at its 
center, and it has a circular floor of radius 7 feet. What is the area 
of the floor of the yurt? Use ^ for ;r. 



A = TTV 



r 



A^ 154 



d<) 



Use the formula for the area of a circle. 
Substitute. Use 7 for r 

Evaluate the power Then simplify. 
Multiply. 



The area of the floor of the vtirt is about 1 54 tr . 



Measurement Application 

Use a centimeter ruler to measure the 
radius of the circle. Then find the area of 
the shaded region of the circle. Use 3.14 for 
;r. Round your answer to the nearest tenth. 

First measure the radius of the circle; 
It measures 1.8 cm. 

Now find tlie area of the entire circle. 

A = TH'^ Use the formula for the area of a circle 

/I == 3.14 -1.8- Substitute. Use 1 .8 for r and 3.14 for tt. 

y4 « 3.14 • 3.24 Evaluate the power 

A « 10.1736 Multiply. 

Set up a proportion. 



The shaded area is ~ of the circle. 




1 


= 


.V 


4 


10.1736 


4.V 


= 


10.1736 


4.V 




10.1736 


4 




4 



The cross products are equal. 

Divide each side by 4 to isolate the variable. 

X = 2.5434 
The area of the shaded region of the circle is about 2.5 cm". 



Think and Discuss 

1. Compare finding the area of a circle when given the radius with 
finding the area when given the diameter. 

2. Give an example of a circular object in your classroom. Tell 
how you could estimate the area of the object, and then estimate. 



I/Jilii; Lesson Tutorials OnlinG mv.hrw.com 



9-5 Area of Circles 547 




Homework Help Online go.hrw.com, 



keyword ■mianitJiM ® 

Exercises 1-12, 13, 15, 17, 19, 21 



G0|11IEKPRIWI£E 



See Example 1 Find the area ot each circle to the nearest tenth. Use 3.14 for ;r. 

1. /^ N 2. Z' N 3. /^ X 4. 



See Example 2 







See Example 3 



The most popular pizza at Sam's Pizza is the 14-inch pepperoni pizza. 
What is the area of a pizza with a diameter of 14 inches? Use 4? for n. 

Measurement Use a centimeter ruler to measure the diameter of 
the circle. Then find the area of the shaded region of the circle. Use 
3.14 for n. Round your answer to the nearest tenth. 




INDEPENDENT PRACTICE 



See Example 1 Find the area of each circle to the nearest tenth. Use 3.14 for k. 

7. /^^-^ 8. /-~~x 9. /- \ 10. 

/16 







See Example 2 11. A wheel has a radius of 14 centimeters. Wliat is the area of 
L the wheel? Use 44 for k. 

See Example 3 12. Measurement Use a centimeter ruler to measure the 
radius of the circle. Then find the area of the shaded 
region of the circle. Use 3.14 for n. Round your answer 
to the nearest tenth. 




Extra Practice 



See page EP23. 



aj:MiJIJJJ)I.IJ:U:!BTIg 



OLVING 



13. A radio station broadcasts a signal over an area with a 75-mile radius. 
Wliat is the area of the region that receives the radio signal? 

14. A circular flower bed in Kay's backyard has a diameter of 8 feet. Wliat is 
the area of the flower bed? Round your answer to the nearest tenth. 

15. A company is manufacturing aluminum lids. The radius of each lid is 3 cm. 
'What is the area of one lid? Round your answer to the nearest tenth. 

Given the radius or diameter, find the circumference and area of each circle 
to the nearest tenth. Use 3.14 for n. 

16. ; = 7m 17. r/=18in. 18. c/ = 24ft 19. r= 6.4 cm 

Given the area, find the radius of each circle. Use 3.14 for k. 

20. A = 113.04 cm- 21. yl = 3.14 ft" 22. yl = 28.26 in^ 



548 Chapter 9 Measurement: Two-Dimensional Figures 



23. A hiker was last seen near a fire tower in tlie Catalina Mountains. Searchers 
are dispatched to the surrounding area to find the missing hiker. 

a. Assume the hiker could walk in any direction at a rate of 3 miles per hour 
How large an area would searchers have to cover if the hiker was last 
seen 2 hours ago? Use 3.14 for Ji. Round your answer to the nearest 
square mUe. 

b. How much additional area would the searchers have to cover if the hiker 
was last seen 3 hours ago? 

24. Physical Science The tower of a wind turbine 
is about the height of a 20-story building. Each 
turbine can produce 24 megawatt-hours of 
electricity in one day. Find the area covered by 
the turbine when it is rotating. Use 3.14 for k. 
Round your answer to the nearest tenth. 

25. Critical Thinking Two circles have the same 
radius. Is the combined area of the two circles 
the same as the area of a circle with twice 
the radius? 

What's the Question? Chang painted half of a free-throw circle that has 
a diameter of 12 ft. The answer is 56.52 ft"'. What is the question? 

27. Write About It Describe how to find the area of a circle when given 
only the circumference of the circle. 

^ 28. Challenge How does the area of a circle change if you multiply the radius 
by a factor of n, where n is a whole number? 



@26. 



@ 





Test Prep and Spiral Review 



29. Multiple Choice The area of a circle is 30 square feet. A second circle has 
a radius that is 2 feet shorter than that of the first circle. What is the area, to 
the nearest tenth, of the second circle? Use 3.14 for ;:. 

CK) 3.7 square feet <Cb:> 10.0 square feet CD 38.0 square feet CE' 179.2 square feet 

30. Short Response A pizza parlor offers a large pizza with a 12-inch diameter. 
It also offers a "mega" pizza with a 24-inch diameter. The slogan used to 
advertise the mega pizza is "Twice the pizza of a large, and twice the fun." Is the 
mega pizza twice as big as the large? If not, how much bigger is it? Explain. 



Line a \\ line b. Use the diagram to find each angle measure. (Lesson 8-3) 
31. mZl 32. mZ2 33. mZ3 



135° 



Graph the polygon with the given vertices. Identify the polygon and 3/ 

then find its area. (Lesson 9-4) '^ 

34. (-1,1), (0,4), (4,1) 35. (-3,3), (2,3), (1,-1), (-1,-1) 



a 
b 



9-5 Area of Circles 549 




rea or irregular i-igures 



A composite figure is made up of simple geometric shapes, such 
as triangles and rectangles. You can find the area of composite and 
other irregular figures by separating them into non-overlapping 
familiar figures. The sum of the areas of these figures is the area of 
the entire figure. You can also estimate the area of an irregular figure 
by using graph paper. 

EXAMPLE [lj Estimating the Area of an Irregular Figure 



Vocabulary 

composite figure 



EXAMPLE 



S3 



The area of a 
semicircle is 1 the 
area of a circle. 
A = l(7rr') 



Estimate the area of the figure. Each square represents 1 ft^. 

Count the number of filled or almost- 
filled squares: 35 yellow squares. 
Count the number of squares that 
are about half-filled: 6 blue squares. 
Add the number of filled squares plus 
I the number of half-filled squares: 
35 + (^- 6)^ 35 + 3 = 38. 

The area of the figure is about 38 ft". 






12 m 



Finding the Area of a Composite Figure 

Find the area of the figure. Use 3.14 for x 

Step 1: Separate the figure into 
smaller, familiar figures. 

Step 2: Find the area of each 
smaller figure. 
Area of the square: 

A = s- 

A= 12" = 144 



tL 




12 m 
Area of the semicircle: 

.4«i(3.14-6-) 

^« ^(113.04) « 56.52 



Step 3: Add the areas to find the total area. 
>1 « 144 + 56.52 = 200.52 

The area of the figure is about 200.52 m". 



550 Chapter 9 Measurennent: Two-Dimensional Figures 



l/jdau Lesson Tutorials Online 



EXAMPLE 



C3 



PROBLEM 



SOLVING 



Helpful Hint 



There are often 
several different 
ways to separate an 
Irregular figure into 
familiar figures. 



PROBLEM SOLVING APPLICATION 

Chandra wants to carpet the floor of her 

closet. A floor plan of the closet is shown at ^^ 

right. How much carpet does she need? 



12ft 



3ft 



^* Understand the Problem 

Rewrite the question as a statement: 

• Find the amount of carpet needed to cover the floor of the closet. 

List the important information: 

• The floor of the closet is a composite figure. 

• The amount of carpet needed is equal to the area of the floor. 



Make a Plan 

Find the area of the floor by separating the 
figure into familiar figures: a rectangle and a 4 ^^ 
triangle. Then add the areas of the rectangle 
and triangle to find the total area. 

*ll Solve 

Find the area of each smaller figure. 

Area of the triangle: 



12 ft 



4 ft 



3ft 



Area of the rectangle: 

A = ew 

A =12 -4: 

A = 48 ft- 



A^^bh 

A = U5){3 + 4) 



A = U35) = 17.5 ft' 



Add the areas to find the total area, yl = 48 + 17.5 = 65.5 
Chandra needs 65.5 ft" of carpet. 

EZ Look Back 

The area of the closet floor must be greater than the area of the 
rectangle (48 ft-), so the answer is reasonable. 



Thmk and Discuss 

1. Describe two different ways to find the 
area of the irregular figure at right. 

2. Explain how dividing the figure into 
two rectangles with a horizontal line 
would affect its area and perimeter. 



4 in. 



J 

1 


2in^ 


2 in 




L 

r 



2 in. 



in. 



/yJb'j Lesson Tutorials Online mv.hrw.com 



9-6 Area of Irregular Figures 551 



9-6 



Ci3j'3J33:3 




•LlMlI 

Homework Help Online go.hrw.com, 



keyword BBbiniig^M ® 
Exercises 1-12, 13, 15, 17 



GUIDED PRACTICE 



See Example 1 Estimate the area of each figure. Each square represents 1 ft^. 

1. '--—r-—--.-^-- , 2. 







See Example 2 Find the area of each figure. Use 3.14 for ;r. 
3. \T 



18 ft 



10ft 
10ft 



n_ 




18 ft 



12 m 



5. 




10ft 



See Example 3 



6. Luis has a model train set. The layout of the 
track is shown at right. How much artificial 
grass does Luis need in order to fill the 
interior of the layout? Use 3.14 for n:. 



1 


1 


f2ft 

1 


1 

f 

1 


1 

. n 


1 

n 



4.5 ft 



INDEPENDENT PRACTICE 



See Example 1 Estimate the area of each figure. Each square represents 1 ft^ 

7. ^1 I I I i I ' I I 8. 







See Example 2 Find the area of each figure. Use 3.14 for ;r. 



10. 



4m/ \ 4 m 
4 m 



8 ft 



4 m 

■\Z7 



6 m 



J L 
3ft 


4ft 




n 


L 
4ft 


3 ft 

L 
r 



11. 



3 ft 



2 cm 



3 cm 



4 cm 

4 cm 

CJ" 



5 cm 



4 cm 



5 cm 



See Example 3 12. The figure shows the floor plan for a gallery of a 

museum. The ceiling of the gallen>' is to be covered 
with soundproofing material. How much material 
is needed? Use 3.14 for /r. 




552 Chapter 9 Measurement: Two-Dimensional Figures 



Extra Practice 



See page EP24. 



PRACTICE AND PROBLEM SOLVING 



Find the area and perimeter of each ilgure. Use 3.14 tor k. 
13. 3 ft 14. 5 m 4m 15. 



3ft 



4 ft 
3 ft 



2 ft 




12 m 



lOm, 



8 m 



16. Critical Thinking Will the area and perimeter change for the figure in 
Exercise 14 if the triangle part is reflected to the left side? Explain. 



17. Critical Thinking The figure at right is made up of 
an isosceles triangle and a square. The perimeter of 
the figure is 44 feet. What is the value of .v? 



xft 




@ 



18. Multi-Step Afigurehasvertices/l(-8, 5),B(-4, 5),C(-4, 2),D(3,2), 
£(3, -2), F(6, -2), G(6, -4), and H(-Q, -4). Graph the figure on a 
coordinate plane. Then find the area and perimeter of the figure. 

19. Choose a Strategy A figure is formed by combining a square and a triangle. 
Its total area is 32.5 m". The area of the triangle is 7.5 m". What is the length of 
each side of the square? 



20. Write About It Describe how to find the area of the 
composite figure at right. 

10 cm 

21. Challenge Find the area and 

perimeter of the figure at right. ^ \ ( 1 8 cm 

Use 3.14 for ;r. 



12 in. 




5 in. 



m 



Test Prep and Spiral Review 



22. Multiple Choice A rectangle is formed by two congruent right triangles. 
The area of each triangle is 6 in". Each side of the rectangle is a whole number 
of inches. Which of these CANNOT be the perimeter of the rectangle? 



CA) 26 in. 



(X 24 in. 



Cc:) 16 in. 



CE) 14 in. 



23. Extended Response The shaded area of the garden 
represents a patch of carrots. Veronica estimates that she 
will get about 12 carrots from this patch. Veronica is going to 
plant the rest of her garden with carrots. Estimate the total 
number of carrots she can expect to grow. Show your work. 

Z.1 and Z.2 are complementary angles. Find mZZ. (Lesson 8-2) 

24. mZl = 33° 25. mZl = 46° 26. mZl = 60° 



Given the diameter, find the area of each circle to the nearest tenth. 

Use 3.14 for X (Lesson 9-5) 

28. <^ = 30m 29. rf = 5.5cm 30. r/=18in. 



> 



27. mZl = 25.5° 



31. f/= lift 



9-6 Area of Irregular Figures 553 



CHAPTER 




To Go On? „<Cf 



Learn It Online 

Resources Online go.hrw.com, 

l^ !y) HTTrsiORTG09A IgI 



Quiz for Lessons 9-1 Through 9-6 

Q) 9-1 j Accuracy and Precision 



Choose the more precise measurement in each pair. 
1. 5 in. 2. 6c 



56 ft 



8floz 



& 



9-2 j Perimeter and Circumference 

3. Find the perimeter of the figure at right. 

4. If the circumference of a wheel is 94 cm, what is its 
approximate diameter? 



0.2 m 




3.0 m 



Qj 9-3 ] Area of Parallelograms 



5. The area of a rectangular courtyard is 1,508 m'^, and the length is 52 m. 
What is the width of the courtyard? 

6. lackson's kitchen is 8 yd by 3 yd. Wliat is the area of his kitchen in square 
feet? 



^J 9-4 ] Area of Triangles and Trapezoids 

7. Find the area of the trapezoid at right. 

8. A triangle has an area of 45 cm' and a base of 12.5 cm. 
What is the height of the triangle? 



3 in. 



8 in. 



n 



12 in. 



(2f 9-5 ] Area of Circles 



9. Find the area of the circle to the nearest tenth. Use 3.14 or ^ for tt. 
10. The radius of a clock face is 8t in. Wliat is its area to the nearest 

4 

whole number? 




Q) 9-6 I Area of Irregular Figures 



Find the area of each figure to the nearest tenth if necessary. Use 3.14 for jr. 
11. 21cm 12. /i:::^s^3ft 13. 9 yd 



21 cm 



rr 



EL 



_d 



_xi 

6 cm 



13 ft 



6 cm 



7ft 



■q 



10 ft 




11 yd 



4ft 



554 Chapter 9 Measurement: Two-Dimensional Figures 



X 




Focus on Problem Solving 



,^^^ 




"^ 





Understand the Problem 

• Identify too much or too little information 

Problems involving real-world situations sometimes give too much 
or too little information. Before solving these types of problems, 
you must decide what information is necessary' and whether you 
have all the necessary information. 

If the problem gives too much information, identify which of the 
facts are really needed to solve the problem. If the problem gives 
too little information, determine what additional information is 
required to solve the problem. 




Copy each problem and underline the information you need to solve 
it. If necessary information is missing, write down what additional 
information is required. 



Q Mrs. Wong wants to put a fence around 
her garden. One side of her garden 
measures 8 feet. Another side measures 
5 feet. What length of fencing does Mrs. 
Wong need to enclose her garden? 

Q Two sides of a triangle measure 17 inches 
and 13 inches. The perimeter of the triangle 
is 45 inches. What is the length in feet of the 
third side of the triangle? (There are 
12 inches in 1 foot.) 

O During swim practice, Peggy swims 2 
laps each of freests'le and backstroke. The 
dimensions of the pool are 25 meters by 
50 meters. What is the area of the pool? 

O Each afternoon, Molly walks her dog 
two times around the park. The park is a 
rectangle that is 315 yards long. How far 
does Molly walk her dog each afternoon? 



Q A trapezoid has bases that measure 

12 meters and 18 meters and one side that 
measures 9 meters. The trapezoid has no 
right angles. What is the area of the 
trapezoid? 




Focus on Problem Solving 555 



Explore Square Roots and 
Perfect Squares 



Use with Lesson 9-7 



tan lat 



Learn It Online 

Lab Resources Online go.hrw.com, 

lUBiffifMSl Lab9 KGoJI 



You can use geometric models such as tiles or graph paper to represent 
squares and square roots. 



Activity 1 



O Copy the three square arrangements below on graph paper. 
Continue the pattern until you have drawn 10 square 
arrangements. 



D 



Q Copy and complete the table below. In the first column, write the 
number of small squares in each figure you drew. To complete the 
second column, use a calculator to find the square root. 



(To find the square root of 4, press 



SI 4m B 



Total Number of Small Squares 


Square Root 


1 


1 


4 


2 


9 


3 































Q Shade in one column of each square arrangement that you drew in O- 



D 



556 Chapter 9 Measurement: Two-Dimensional Figures 



Think and Discuss 



1. How does the square root relate to the total number of small squares 
in a figure? 

2. How does the square root in the table relate to the shaded portion of 
each figure? 



Try This 



Use graph paper to find each square root. 
1. 121 2. 144 



3. 196 



Activity 2 



Follow the steps below to estimate V 14. 

O On graph paper, use one color to draw the smallest possible 
square arrangement using at least 14 small squares. 

O On the same arrangement, draw the largest possible square 
arrangement using less than 14 small squares. 

Q Count the number of squares in each arrangement. Notice that 
14 is between these numbers. 

Number in small arrangement Number in large arrangement 

9 < 14 < 16 

O Use a calculator to find \ 14 to the nearest tenth. \ 14 = 3.7. Use 
inequality symbols to compare the square roots of 9, 14, and 16. 



V9 <V14 < 
3<3.7<4 



'16 



The square root of 9 is less than the square root 
of 14, which is less than the sauare root of 16. 



Q Use dashed lines on the figure to sketch a square that is 3.7 units 
on each side. 



M 









t ! 


H ^BCjISill 




i 






- 


L 




: 


l"-^3.7-r-| 





Thinic and Discuss 



1. Describe how to use two numbers to estimate the square roots of 
nonperfect squares without using a calculator. 

2. Explain how you can use graph paper to estimate \ 19. 

3. Name three numbers that have square roots between 5 and 6. 



Try This 



Use graph paper to estimate each square root. Then use a calculator to 
find the square root to the nearest tenth. 



1. 



'19 



2. VIO 



^^28 



^35 



9-7 Hands-On Lab 557 



9-7 







7.1.5 Recognize and use the inverse relationship between squaring and finding the 
square root of a perfect square integer 

A square with sides that measure 
3 units each has an area of 3 • 3, 
or 3". Notice that the area of the square 
is represented by a power in which the 
base is the side length and the 
exponent is 2. A power in whicli the 
exponent is 2 is called a square. 



Base 




Exponent 



EXAMPLE [lj Finding Squares of Numbers 

Vocabulary ^'"'^ ^^'^ ^^"^''^- 

perfect square 
square root 
radical sign 



A 


Method 1 


: Use a model. 




, 














. 












4 = (w 


























\ = 6 • 6 














A = 36 


















The square of 


6 is 36. 



B 14^ 



Method 2: Use a calculator. 



Press 14 



14^= 196 



The square of 14 is 196. 



k!"S;] 



V 16 = 4 is read as 
"The square root of 
16 is4." 



A perfect square is the square of a whole number. The number 36 is a 
perfect square because 36 = 6" and 6 is a whole number. 



The square root of a number is one of the two 
equal factors of the number. Four is a square 
root of 16 because 4 • 4 = 16. The symbol for a 
square root is V , which is called a radical sign 



EXAMPLE r 2 J Finding Square Roots of Perfect Squares 

Find each square root. 



A V 64 

Method 1: Use a model. 



The square root of 64 is 8. 



558 Chapter 9 Measurement: Two-Dimensional Figures [VJii^J^i Lesson Tutorials Online 



Find each square root. 



B V324 

Method 2: Use a calculator. Press 



324 



V324 = 18 



The square root of 324 is 18. 



You can use perfect squares to estimate the square roots of 
nonperfect squares. 



EXAMPLE r3j Estimating Square Roots