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Full text of "Mathematics Course 2"

f^ HOLT McDOUGAL ml I II Indiana *: Course 2 ^■%A THIS BOOK IS THE PROPERTY STATE Xnii^W. PROVINCE PARISH COUNTY i^_ .' SCHOOL DISTRICT OF: M Book No. 1 Enter information in spaces to the OTHER left as instructed YEAR ISSUED TO USED U^/k\ C&sseUn Qd^9M CONDITION ISSUED RETURNED £)U Me^Jt^ . PUPILS to whom this textbook is issued must not write on any page or mark any part of it in any way, consumable textbooks excepted. 1 Teachers should see that the pupil's name is clearly written in ink in the spaces above in every book issued. 2 The following terms should be used in recording the condition of the book: New. Good, Fair; Poor; Bad. Holt McDougai Online Learning All the help you need, any time you need it. yo.hrw.com Lesson Tutorial Videos feature entertaining and enlightening videos that illustrate every example in your textbook! Log on to www.go.hrw.com to access Holt's online resources. T Premier Online Edition • Complete Student Edition • Lesson Tutorial Videos for every example > Course 1 : 287 videos > Course 2: 317 videos > Course 3: 341 videos • Interactive practice with feedback Extra Practice • Homework Help Online • Intervention and enrichment exercises • State test practice Online Tools • Graphing calculator • TechKeys "How-to" tutorials on graphing calculators • Virtual Manipulatives • Multilingual glossary For Parents • Parent Resources Online Digitized by tine Internet Arciiive in 2011 witii funding from Lakewood Parl< Christian Scliool 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 '""■'■'•■'—' \.^u.\<^\ Y j.^^.«|.^l '■ 1 i.nr*oi -TirU...vlp.^n-U4.:<r..idl 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 yVld'j Lesson Tutorials Online mv.hrw.com 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 'Mbii Lesson Tutorials Online mv.hrw.com 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. VjiJaii LESson Tutorials Online mv.hrw.com 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 y'l'h'j Lesson Tutorials Online mv.hrw.com I a:iQ j(3fe3£ „.„^ tJ Homework Help Online go.hrw.com. 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. yjdiu Lesson Tutorials OnlinE mv.hrw.com 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? ,r^ Learn It Online ^^ 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. Vliliij Lesson Tutorials Online my.lirw.com 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 y'l'Juu Lesson Tutorials OnllnE my.hrw.com 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. ^Mbd Lesson Tutorials OniinE my.hrw.com 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 y'l'ld'j Lesson Tutorials Online mv.hrw.com 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. l/jjiiDJ Lesson Tutorials OnllnE mv.hrw.com 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 f\tiliu Lesson Tutorials OnllnE my.hrw.com ^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. ^Mhu Lesson Tutorials Online 7-70 Solving Equations by Adding or Subtracting 49 1-10 !tiUjj;djii^ i <w£^ili3uf')<iJ-l^ '; 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 ^J'* State Test Practice go.hrw.com, IWl.y.ll VlSIOTestPrep^J 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^^ "^W*^ '.:^_ "^ sri Learn It Online Chapter Project Online qo.hrw.com, keyword tlllWiM ® ^ziiA 68 Chapter 2 ■^ £^'i ;P^%;;.S r^ ; Are You Ready? 7 .^P Learn It Online ** 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. ^Mb'j Lesson Tutorials OnlinE mv.hrw.com 2-1 Integers 73 ^^ 2-1 i<aa»<t&«.v..w«ii.-<tifai«jaiJ«a>iWff>riiilMh»^^^ •1', /fr ^'-"v^l^^lM^O ^.lijjC^-s: Homework Help Online go.hrw.com, 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. ViiliLi Lesson Tutorials Online my.hrw.com 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. ^Mb'j Lesson Tutorials OnlinE mv.hrw.com 2-7 Greatest Common Factor 109 2-7 i Homework Help Online go.hrw.com, 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. Vjiliii Lesson Tutorials Online mv.hrw.com 2-8 Least Comnnon Multiple 113 2-8 See Example 1 See Example 2 See Example 3 L Homework Help Online go.hrw.com, 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? if^P Learn It Online 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 yjiJay Lesson Tutorials Online my.hrw.com 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. '•Mb'j Lesson Tutorials OnlinE mv.hrw.com 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. Interactivities Online ► 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 yjiJii; Lesson Tutorials Online mv.hrw.com 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 !/i:l3ii Lesson Tutorials Online mv.hrw.com 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. ^M-i'j\ Lesson Tutorials Online my.hrw.com 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 VjdaiJ Lesson Tutorials OnlinE my.hrw.com 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 Ready To Go On? <£t Learn It Online Resources Online go.hrw.com IBTOBIms'i rtgobbk go; 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. Nv^^S^W^v^'^^'^'^'fSSS^'^S^'S^^^^ Learn It Online Game Time Extra go.hrw.com, keyword IJiWMcEBBffl \ 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) ® ® ® • (1) • ® @ ® @ (2) • ® ® ® ® ® ® ® ® (4) @ ® ® ® (5/ ® ® re- '61 '6 '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 /^ Learn It Online 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? .^y Learn It Online *^ 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. VjJcii; Lesson Tutorials OnlinE my.hrw.com 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? ^/ajau Lesson Tutorials Online my.hrw.com 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 'Mh'j Lesson Tutorials Online iny.hrw.com 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. y'niaii Lesson Tutorials Online mv.hrw.com 4-3 Identifying and Writing Proportions 223 4-3 iicioajsaa Homework Help Online go.hrw.com, 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. 'Mill Lesson Tutorials Online mv.hrw.com 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. Tidbii Lesson Tutorials Online my.hrw.com 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 Tldi'j Lesson Tutorials Online mv.hrw.com 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. yVldu Lesson Tutorials Online mv.hrw.com 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 ^fidHu Lesson Tutorials Onlins my.hrw.com 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. Ti'h'j Lesson Tutorials Online mv.j-i rw.com 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 /*9 Learn It Online ^»** 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 l/jdiD Lesson Tutorials Online mv.hrw.com 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. Ii'ldul Lesson Tutorials Online mv.hrw.com 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 VWau Lesson Tutorials OnllnE mv.hrw.com 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. ^fi'h'j Lesson Tutorials Online my.hrw.com 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 y'l&d'j Lesson Tutorials Online mv.hrw.com 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. [ 'Mh'j] Lesson Tutorials Online mv.hrw.com 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? .^y Learn It Online *J^ ResourcesOnlinego.hrw.com, 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. ,^^ Learn It Online A complete copy of the rules and *J* GameT1n1eExir3go.hrw.com, the game pieces are available online. IBMBIm si Games^g! 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 OM-b mi ±Tfi\r\ ^-^s^ 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. Iiii-i'j Lesson Tutorials OnliriE mv.hrw.com 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?, Learn It Online State Test Practice go.hrw.com, IBWBBIm's 1 TestPrep^ 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. .^>^ Learn It Online *"** Chapter Proiect Online go.hrw.com, keyword H9EBI9 ® 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? ,fy Learn It Online Resources Online go.hrw.com, il MS10AYR5 ■Go' 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 1/jiJiLi Lesson Tutorials Online mv.hrw.com 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? Vldi'j Lesson Tutorials Online mv.hrw.com 5-1 The Coordinate Plane 277 5-1 3.. 3 GUIDED PRACTICE [•Jiiiiii-i Homework Help Online go.hrw.com, keyword ■BHIlWH ® 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 'Mdu Lesson Tutorials OnlinE inv.hrw.com 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. 'fulH'j Lesson Tutorials Online my.hrw.com 5-2 Interpreting Graphs 281 ■L'-^ H^^^ ' v;!^l^l5!«i^^sy;ri^W&iS^iGi*ijii^.;*-W4"J;i^^ ^i^?i;'J33^ [•rmiii 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 \ '•J'lLJ-d'j] Lesson Tutorials OnlinE mv.hrw.com 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. 'faih'j Lesson Tutorials Online mv.hrw.com 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 'Mh'j Lesson Tutorials OnlinE mv.hrw.com 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, ... . 'fi'lu'j Lesson Tutorials Online my.hrw.com 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 ,^"y Learn It Online ^^ ResourcesOnlinego.hrw.com, ■W.ll.li,lMSIl.RTG05AMGo: 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 £?. Learn It Online Lab Resources Online go.hrw.com, 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 yjdiii Lesson Tutorials GnlinE mv.hrw.com 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. 'J\<\z'j Lesson Tutorials OnlinE mv.hrw.com 5-5 Graphing Linear Functions 297 5-5 3 iiitojiiiiii Homework Help Online go.hrw.com, 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. yidiv Lesson Tutorials OnlinG mv.hrw.com 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. ,r^ Learn It Online *^ HomeworkHelpOnlinego.hrw.com, 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 Vliiaii Lesson Tutorials Online my.hrw.com 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 y'ni^u Lesson Tutorials Online my.hrw.com 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. y'i&du Lesson Tutorials Online mv.hrw.com 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. Vlda:; Lesson Tutorials Online mv.hrw.com 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. 'fi'^b'j Lesson Tutorials Online mv.hrw.com 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. 7jdaj Lesson Tutorials Online mv.hrw.com 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. 'Mb'j Lesson Tutorials Online my.hrw.com 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 fi'h'j Lesson Tutorials OnliriE my.hrw.com 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. 'Mb'j Lesson Tutorials OnliriE mv.hrw.com 6-4 Percent of a Number 349 6-4 i»irinTn Homework Help Online go.hrw.com, 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 y'niau Lesson Tutorials Online my.hrw.com 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. 'faib'j Lesson Tutorials OnlinE my.hrw.com 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? .^^ Learn It Online *■** ResourcesOnlinego.hrw.com, llWllffi | M';inRTGn6ALc°l 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 yjday Lesson Tutorials OnlinE my.hrw.com 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? 'Mb'j Lesson Tutorials Online mv.hrw.com 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 y]'hD Lesson Tutorials OnlinE mv.hrw.com 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. ^^^ . 'Mb'j Lesson Tutorials Online mv.hiw.com 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? ^^^ Learn It Online ^** 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 ,^^ Learn It Online «* 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? 'fl'J-i'j Lesson Tutorials Online my.hrw.com 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 .r^ Learn It Online *•** SlateTestPracticego.hrw.com, ■W.li.U.lMsiQTestPrepiGoJ 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 hrzing Dat 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? ^£V Learn It Online ^** ResourcesOnlinego.hrw.com, 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. ,^^ Learn It Online *^ HomeworkHelpOnlinego.hrw.com, keyword ■miawwl ® 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>^.^^ \3Z}^JA1^^ 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? VldiLi Lessod Tutorials OnllriE my.hrw.com 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 f s \^XfM r=^^ry i/HrT I^Jk 1 ^^^^^a ^ _=^ 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. £?. Learn It Online Lab Resources Online go.hrw.com, keyword IBEIMeig;! m 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 [VjJ3!;j Lesson Tutorials Online my.hrw.com 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. iU:iU. Lesson Tutorials OnlinE my.hrw.com 8-3 Line and Angle Relationships 461 ;i(^2B333 Homework Help Online go.hrw.com, keyword ■aMM;aeM ® 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. <f?, Learn It Online Lab Resources Online go.hrw.com, ■!imii.iMsi.H.h8"mr 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? ^^^ Learn It Online ResourcesOnlinego.hrw.com, ■Pffl||vlSlilkl(308AtGoB 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 Vlilaj Lesson Tutorials OnlinE my.hrw.com 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°. 'Ii'h<j Lesson Tutorials Online mv.hrw.com 8-4 Properties of Circles 469 8-4 Homework Help Online go.hrw.com, * 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. „„«,-.,„ Learn It Online Lab Resources Online go.hrw.com, ■am MS10 LJb8 Baj S"^.':r^:5W.%V^^-'^^>?^^N^7Ttw^T« 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 'Mbu Lesson Tutorials Online my.hrw.com 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. 'faib'j Lesson Tutorials OnlinE mv.hrw.com 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 l/JUaij Lesson Tutorials OnlinE mv.hrw.com 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. 'Mb'j Lesson Tutorials OnlinE mv.hrw.com 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. 'Mb'j Lesson Tutorials Online n-iy.lirw.com 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 VJ]i\='j\ Lesson Tutorials Online mv.hrw.com 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. Mbii Lesson Tutorials Online my.hrw.com 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. 'Mb'j Lesson Tutorials Online mv.hrw.com 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 LESSon Tutorials Online mv.hrw.com 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 Learn It Online Homework Help Online go.hrw.com, keyword lAHMabl ® 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 H'j Lesson Tutorials OnlinE mv.hrw.com 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. '.''jt^E'j Lesson Tutorials Online mv.hrw.com 8-11 Symmetry 505 •ruiiii Homework Help Online go.hrw.com, 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 Learn It Online LabResourcesOnlinego.hrw.com, | keyword IBEIWBlg;! ® 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? .£*P Learn It Online ^J" ResourcesOnlinego.hrw.com, ■l|miiJMSIORTG08CKGog 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 Homework Help Online go.hrw.com, 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