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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
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ISBN13 9780554033280
ISBN10 554033283
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 VicePresident 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
PreK12 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 codirected the Bay
Areabased 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 K12 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
HolKwood, 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 SmithHaye
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\'ffrinEasttomi NHddle
School
Berwyn, PA
Maridith Gebhart
Math 'ieacher
Ramay Junior High School
Fayette\ille. AR
Ruth HarbinIVIiles
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 MidHigh
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 twostep 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 twodimensional patterns
(nets) for threedimensional
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 supportsbut is not a substitute for
the development of skills with basic operations, quantitative reasoning, and
problemsolving 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 threedimensional 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 threedimensional shapes to model realworld
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 ProgressPlus 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 openended
responses that are scored with a
2point rubric.
Extended Response — wTite open
ended responses that are scored with a
4point rubric.
J S"''" ^.00 ^"^, ^,
«»■«■
<P 9
Test Tackier
Use the Test Tackier to
become familiar with
and practice testtaking
strategies.
The first page of this
feature explains and
{ shows an example of
a testtaking 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 testtaking 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.
TestTaking 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 testtaking 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. 421
42J
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; 2511 =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
3ft
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)
l2l
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. 415 + 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 onefourth 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 onethird 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 4foottall 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^yi'iiyry
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. stemandleaf 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 sciencefiction. 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 yaxis,
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 xaxis,
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 xaxis,
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 39gallon 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, 12m
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 makeup 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 mangobanana 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 higherorder 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 higherorder
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 higherorder 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 pointtopoint distance between Boston and
Atlanta.
3. Put the information below into a bar graph.
IN37
• Analysis includes classifying, comparing, making asso
ciations, verifying, seeing causeandeffect 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
HigherOrder
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
higherorder 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
MultipleChoice Questions
The most common type of test question is multiple choice.
To answer questions on a multiplechoice 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 multiplechoice 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 multiplechoice 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
GriddedResponse 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
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(f>
(•')
•
f.>
,.,
(•)
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6
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d
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n
6
9
(21
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5
6
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(31
6
7
a
W
2
/
5
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5
m
(/)
(fj
<7>
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1/
(^)
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(•)
(•)
(•)
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«
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®
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5
6
P
J)
4
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e
19)
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4
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(3)
®
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5
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(21
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5
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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
TestTaking 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
11
12
13
7.1.7 14
Patterns and Relationships
Numbers and Patterns
Exponents
Scientific Notation
Scientific Notation with a Calculator
Order of Operations
Explore Order of Operations ^^. . .
15 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
16
17
18
19
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
7.2.1 110 Solving Equations by Adding or Subtracting 48
7.2.1 111 Solving Equations by Multiplying or Dividing 52
Ready to Go On? Quiz 56
RealWorld Connection: Illinois 57
Study Guide; Preview 4
Reading and Writing Math 5
Game Time: Jumping Beans 58
It's in the Bag! StepbyStep 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
KnowIt 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:<tVU. 'I V4W i lM,*W^
IN44
■ •*,
Integers and
Rational lUumbers
Are You Ready? 69
Integers
7.1.6 21 Integers 72
EXT Negative Exponents 76
fj^^ Model Integer Addition 78
22 Adding Integers 80
^03 Model Integer Subtraction 84
23 Subtracting Integers 86
^^^ Model Integer Multiplication and Division 90
7.1.7 24 Multiplying and Dividing Integers 92
^^3 Model Integer Equations 96
7.2.1 25 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 26 Prime Factorization 104
7.1.3 27 Greatest Common Factor 108
28 Least Common Multiple 112
Ready to Go On? Quiz 116
Focus on Problem Solving: Look Back 117
Rational Numbers
29 Equivalent Fractions and Mixed Numbers 118
210 Equivalent Fractions and Decimals 122
7.1.6 211 Comparing and Ordering Rational Numbers 126
Ready to Go On? Quiz 130
RealWorld 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
KnowIt 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. C2C3 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 C2C3 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
RealWorld 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
KnowIt 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 41 Ratios 214
7.1.9 42 Rates 218
43 Identifying and Writing Proportions 222
44 Solving Proportions 226
Ready to Go On? Quiz 230
Focus on Problem Solving: Make a Plan 231
Measurements
45 Customary Measurements 232
46 Metric Measurements 236
47 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 48 Similar Figures and Proportions 248
7.3.5 49 Using Similar Figures 252
7.3.5 410 Scale Drawings and Scale Models 256
Make Scale Drawings 260
Ready to Go On? Quiz 262
RealWorld 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
KnowIt 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
SlopeIntercept Form 308
Direct Variation 313
Inverse Variation 318
Ready to Go On? Quiz 320
RealWorld 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 FoldABooks 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
KnowIt 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
61 Percents 336
^3 Model Percents 339
62 Fractions, Decimals, and Percents 340
63 Estimating with Percents 344
64 Percent of a Number 348
65 Solving Percent Problems 352
Ready to Go On? Quiz 356
Focus on Problem Solving: Make a Plan 357
Applying Percents
66 Percent of Change 358
67 Simple Interest 362
Ready to Go On? Quiz 366
RealWorld 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
KnowIt 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 C2C3 for text
B
7.4.4
7.4.3
7.4.4
7.4.4
7.4.4
71
72
73
74
75
7.4.4 76
IW
7.4.1
77
11^)
7.4.4
78
7.4.4
79
@
7.4.4
710
Collecting, Displaying,
and Analyzing Data
Are You Ready? 377
Organizing and Displaying Data
Frequency Tables, StemandLeaf Plots, and Line Plots 380
Mean, Median, Mode, and Range 385
Bar Graphs and Histograms 390
Reading and Interpreting Circle Graphs 394
BoxandWhisker Plots 398
Explore BoxandWhisker 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
RealWorld 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
KnowIt 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
81
82
7.3.1 83
84
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
RealWorld 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
85
86
87
88
.3.4
89
.3.2
810
lUJ
EXT
.3.4
811
^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
KnowIt 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 C2C3 for te>.i
Measurement: Two
Dimensional Figures
Are You Ready? 521
Perimeter, Circumference, and Area
91 Accuracy and Precision 524
^^p Explore Perimeter and Circumference 528
92 Perimeter and Circumference 530
^^p Explore Area of Polygons 534
93 Area of Parallelograms 536
94 Area of Triangles and Trapezoids 540
^^3 Compare Perimeter and Area of Similar Figures 544
95 Area of Circles 546
96 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
97 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
RealWorld 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
98
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
KnowIt 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 ThreeDimensional Figures from Different Views .... 586
101 Introduction to ThreeDimensional Figures 588
EXT Cross Sections 592
^^3 Explore the Volume of Prisms and Cylinders 594
102 Volume of Prisms and Cylinders 596
103 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
104 Surface Area of Prisms and Cylinders 607
^^J Explore the Surface Area of Pyramids and Cones 612
105 Surface Area of Pyramids and Cones 614
^^p Explore the Surface Areas of Similar Prisms 618
106 Changing Dimensions 620
Explore Changes in Dimensions ^g^ 625
Ready to Go On? Quiz 626
RealWorld Connection: Kentucky 627
Study Guide: Preview 584
Reading and Writing Math 585
Game Time: Blooming Minds 628
It's in the Bag! CD 3D 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
KnowIt 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 C2C3 for text.
Countdown to Testing Weeks 22, 23
Test Prep and Spiral Review 591,
599,503,511,517,524
ISTEPi Test Prep 534
7.FPC.7: Probability
See pp C2C3 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
RealWorld 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
Knowit 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
_>^
Multistep Equations
and Inequalities
U Are You Ready? 69i
MultiStep Equations
Model TwoStep Equations 694
7.2.2 121 Solving TwoStep Equations 696
7.2.1 122 Solving MultiStep Equations 700
7.2.1 123 Solving Equations with Variables on Both Sides 704
Ready to Go On? Quiz 708
Focus on Problem Solving: Solve 709
Inequalities
124 Inequalities 710
7.2.1 125 Solving Inequalities by Adding or Subtracting 714
7.2.1 126 Solving Inequalities by Multiplying or Dividing 718
7.2.2 127 Solving MultiStep Inequalities 722
EXT Solving for a Variable 726
Ready to Go On? Quiz 728
RealWorld Connection: New Hampshire 729
Study Guide: Preview 692
Reading and Writing Math 693
Game Time: Flapjacks 730
It's in the Bag: Wired for MultiStep 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
KnowIt 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.^nU4.:<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 stepbystep
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
11
Numbers and Patterns
12
Exponents
7.1.2
13
Scientific Notation
7.1.1
LAB
Scientific Notation with
a Calculator
14
Order of Operations
7.1.7
LAB
Explore Order of
Operations
15
Properties of Numbers
IB
Algebraic Thinking
16
Variables and Algebraic
Expressions
7.2.3
17
Translating Words
into Math
7.2.1
18
Simplifying Algebraic
Expressions
7.2.3
19
Equations and Their
Solutions
7.2.1
LAB
Model Solving Equations
110
Solving Equations by
Adding or Subtracting
7.2.1
111
Solving Equations by
Multiplying or Dividing
7.2.1
■f^fg^^s^^rrj^fv^^
£?.
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'^*'^ ^^^^
tjf^!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. 222 14.9999 15.141414 16.10101010
17. 3355 18.2257 19. 33 11 11 20.5101010
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 18, 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
11
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
42 42 42 42 42 42
A pattern is to divide each number by 2 to get the next number.
842 = 4 442 = 2 242=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 "'" "' ^
lll2(3l4l5 16 17
A pattern is to add one more than you did the time before.
12 + 5=17 1716 = 23 2317 = 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
11 Numbers and Patterns
11
ZI3.
/
Homework Help Online go.hrw.com,
[goI
keyword ■mBiliUBM
Exercises 114, 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 40yearold athlete? a
65yearold 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
77 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 • 33 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
12 Exponents 11
12
lifi^jjii
GUIDED PRACTICE
See Example 1 Find each value.
u 1. 2^ 2. 3^
3. 6^
4. 9'
keyword ■mbiiwbjM ®
Exercises 130, 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 oneyearold 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 11)
66. 100, 91, 82, 73, 64, . . . 67. 17, 19, 22, 26, 31, . . . 68. 2, 6, 18, 54, 162, . . .
12 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
13 Scientific Notation
15
[•JllllK
^ Homework Help Online go.hrw.com,
keyword MMtllBcM
Exercises 128, 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 myapresent)
Mesozoic
(248 mya55 mya)
Paleozoic
(540 mya248 mya)
Period
Quaternary (1.8 myapresent)
Holocene epoch
(1 1,000 yrs agopresent)
Pleistocene epoch
(l,8mya11,000yrsago)
Tertiary (65 mya 1.8 mya)
Pliocene epoch {5.3 mya1 .8 mya}
Miocene epoch (23.8 mya5.3 mya)
Oligocene epoch (33.7 mya23.8 mya)
Eocene epoch (54.8 mya33.7 mya)
Paleocene epoch (65 mya54.8 mya)
Cretaceous (144 mya65 mya)
Jurassic (206 mya 144 mya)
Triassic (248 mya206 mya)
Permian (290 mya248 mya)
Pennsylvanian (323 mya290 mya)
Mississippian (354 mya323 mya)
Devonian (41 7 mya354 mya)
Silurian (443 mya417 mya)
Ordovician (490 mya443 mya)
Cambrian (540 mya 490 mya)
Proterozoic (2,500 mya540 mya)
Archean (3,800 mya2,500 mya)
Hadean (4,600 mya3,800 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 11)
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
13 Scientific Notation 17
;. Scientific Notation witli
LAB/\ a Calculator
Use with Lesson 13
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
14
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
273
24
B 3618H23 + 8
36  18 H 2 3 + 8
3693 + 8
3627 + 8
9 + 8
17
C 5 + 6^10
5 + 6" • 10
5 + 3610
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
14 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
364
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 parttime 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
Exercises118, 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 43^6 + 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
INDEPENgNjLPRACTICE
See Example 1 Simplify each expression. Use the order of operations to justify your answer.
L 8. 3 + 751 9. 593 10. 32 + 62"
See Example 2 11.(333)^ + 3 + 3 12. 2''  (4  5 + 3)
13. (3 + 3) + 3 • (3^ 3)
L
14. 4^ + 82
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.
19.9036x2 20.16+14 + 27 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 + 62 18 + (6  2)
30. (18  14) + (2 + 2) 1814 + 2 + 2
27. 12 + 3 4 12 + (3  4)
29. [6(83) +2] 6(83) +2
Critical Thinking Insert grouping symbols to make each statement true.
31.483=20 32.5 + 93+2 = 8 33. 12  2" + 5 = 20
34. 42 + 6= 32 35. 4 + 63+7=1 36. 986 + 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.
14 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. MultiStep 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 1819^3 + 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 12)
46. 8'' 47. 9^
Multiply. (Lesson 13)
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 14
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. 32' + 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
14 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 14 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
81=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. =2107
= 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
15 Properties of Numbers 25
15
.iiij'3}^3^
HomeworkHelpOnlinego.hrw.com, j
keyword ■BHIiBBiM ®
Exercises136, 41,47,49,51,53
GUIDED PRACTICE
See Example 1 Tell which property is represented.
1. 1+ (6 + 7) = (1 + 6) +7 2. 110=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 • +21
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
200102
27
55
200203
17
65
200304
43
39
6 • (9 + 1Z) = 69 + 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 + 53
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 14)
65. 25 + 5  (6  7) 66. 3^  (6 + 3)
67. (4^ + 5) ^ 7
68. (53)^ (3^7)
15 Properties of Numbers
27
CHAPTER
Ready To Go On?
,r^ Learn It Online
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1!HW!B  M s 1 n RTfi0 1 a kgo;
SECTION 1A
Quiz for Lessons 11 Through 15
^f 11 ] 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
12 ] 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.
13 ] 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.
14 ] Order of Operations
Simplify each expression.
18. 8 14 ^ (9 2) 19.5463 + 4" 20. 4  24 H 2''
15 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
16
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
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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
q2
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.
152 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
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16 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. 5a3b + 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 13)
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
16 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 realworld 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
477 +2
477 +2
When solving realworld 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 RealWorld Problems into Algebraic Express
ions
Jed reads p pages each day of a 200page 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
by2y.
2. Explain how you would determine which operation to use to find
the number of chairs in 6 rows of 100 chairs each.
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17 Translating Words into Math 35
:i3.^s:^^3^
keyword MteMMM ®
Exercises 113, 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 5foot 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. MultiStep 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(^78)
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.
Reddishbrown
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 14)
38. 6 + 4 H 2 39. 9 • 1  4 40. 5^  3
42. Evaluate b  a for a = 2 and i) = 9. (Lesson 16)
41. 24^3 + 3
3
17 Translating Words into Matli 37
18
Simplifying Algebr
Expressions
7.2.3 Evaluate numerical expressions and simplify algebraic expressions involving
rational and irrational numbers.
Individual skits at the talent show can
last up to X minutes each, and group
skits can last up to y minutes each.
Intermission will be 15 minutes. The
expression 7x + 9y + 15 represents
the maximum length of the talent
show if 7 individuals and 9 groups
perform.
Vocabulary
term
coefficient
Caution!
//////
A variable by itself,
such as y, has a
coefficient of 1.
So y = ly.
In the expression 7.y + 9y + 15, 7.y, 9_y,
and 15 are terms. A term can be a
number, a variable, or a product of
numbers and variables. Terms in an
expression are separated by plus or
minus signs.
In the term 7.v, 7 is called the
coefficient. A coefficient is a
number that is multiplied by a
variable in an algebraic expression.
Like terms are terms with the same variables raised to the same
exponents. The coefficients do not have to be the same. Constants,
like 5, \, and 3.2, are also like terms.
Coefficient
Variable
Like Terms
3xand2x w and ^ 5 and 1.8
Unlike Terms
5x'' and 2x
The exponents
are different.
6a and 6fa
Ttie variables
are different.
3.2 and n
Only one terni
contains a variable.
EXAMPLE [1J Identifying Like Terms
Identify like terms in the list.
So I 3y It X 4z k
HMH
Use different shapes
or colors to indicate
sets of like terms.
Look for like variables with like powers.
3y
7t
x^
4z
Like terms: 5a and a , 7t, and 2t
4.5y 2t ffl
4.5y
2f
3y^ and 4.5y^
38 Cliapter 7 Algebraic Reasoning
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To simplify an algebraic expression that contains like terms, combine
the terms. Combining like terms is like grouping similar objects.
^ .■mi!?..". j«.iiiircv
X
X
X X
X X X X
+
fl.
=
^
X
X
t.jc:;, ■ X X
X X X X
..
' — _:^'j '
. ^.__  ■~^^._..
4x
+
5jc
9x
To combine like terms that have variables, add or subtract the coefficients.
EXAMPLE [2] Simplifying Algebraic Expressions
EXAMPLE
CS
To find the perimeter
of a figure, add the
lengths of the sides.
Simplify. Justify your steps using the Commutative, Associative,
and Distributive Properties when necessary.
7x and 2x are like terms.
Add the coefficients.
Identify like terms.
Commutative Property
Associative Property
Add or subtract the coefficients.
A
7a + 2x
7x + 2x
9x
1
B
Sx^ + 33/ + 7x^  2y  4x2
5x^ + 3y + 7x^  2y  4x
5x^ + 7x^ + 3,y  2y  4x
(5x^ + 7x^) + (3y  2y)  4.x
\2.x^ +y4x
C 2(fl + 2a'') + 2b
2(a + 2a") + 2h
2a + 4a + 2b
There are no like terms to combine.
Geometry Application
Write an expression for the perimeter
of the rectangle. Then simplify
the expression.
Distributive Property
b + h + b + h
{b + b) + (/z + h)
2b + 2h
Write an expression using the side lengths.
Identify and group like terms.
Add the coefficients.
Think and Discuss
1. Explain whether 5x, 5x", and 5x' are like terms.
2. Explain how you know when an expression cannot be simplified.
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18 Simplifying Algebraic Expressions 39
1
(iik^^J>iUiKl£liii£>^i^^
Homework Help Online go.hrw.com,
keyword ■BEiMBJ ®
Exercises 117, 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
bagshaped 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. MultiStep 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?
iS) 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 sixtyeight million dollars. Write this amount in scientific
notation. (Lesson 13)
Evaluate the expression 9y — 3 for each given value of the variable. (Lesson 16)
38. y = 2 39. y=6 40. y=10 41. y=18
18 Simplifying Algebraic Expressions 41
19
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 RealWorld 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
19 Equations and Their Solutions 43
19
.im^i£&
y
keyword MBtaHMBiM ®
Exercises 113, 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, 9,00010,500 ft
PinonJuniper, 7,0009,000 ft
Montane forest, 7,5009,000 ft
Semidesert, 5,5007,000 ft
Foothills, 5,5007,500 ft
'■■"'' — ; •:
Great Plains, 3,0005,500 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 seventhgraders at Pecos Middle School
is 316. This is 27 more than the number of eighthgraders. How many
eighthgraders are enrolled?
CE' 289
CD 291
CH) 299
Write each number in scientific notation. (Lesson 13)
31. 10,850,000 32. 627,000
Tell which property is represented. (Lesson 15)
34. (7 + 5) + 3 = 7 + (5 + 3) 35. 181 + = 181
CD 343
33. 9,040,000
36. be = cb
19 Equations and Their Solutions 45
Model Solving Equations
Use with Lessons 110 and 111
KEY
REMEMBER
miD^
= 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
770 HandsOn Lab 47
11
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
73=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
770 Solving Equations by Adding or Subtracting 49
110
!tiUjj;djii^ i <w£^ili3uf')<iJl^
'; J
□^
Ci3,J^303:
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 9yard 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. Al 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. 1234 S6 7852.
TICKETS, Rafael MencJoza in concert.
EscamihQl^^w*^^BHCT'^gWw^^i]M. two
fsea^, row 17. $75 123567'
J,ES. 2 . Traiect 6 7^
protessibnar mlWf^^^^Rush M8S8
great condition. Must sell last,
SlOOO/best First buyer takes all. Chad,
3213213211
KCV. Ultrasonic 16334578 rpm, S100.
" ■ * ' Rhure V15, 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 Tshirt. 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 17)
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 18)
51. 6(2 I 2/7) I 3» 52. 4x  7v + x
53. 8l3rl2(4f)
770 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.
34 = 12
23 4 = 212
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
52 = 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
77 7 Solving Equations by Multiplying or Dividing 53
keyword ■mBiwiiHiiM @
Exercises 120, 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=g3
24. 7/ = 84
27. /; + 33 = 95
30. 504 = f212
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. r92 = 215
MultiStep 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 110)
53. 76 + H = 115 54. ;  97 = 145
55. t 123 = 455
56. f? + 39 = 86
77 7 Solving Equations by Multiplying or Dividing 55
To Go On?
.^^ Learn It Online
^J* ResourcesOnlinego.hrw.com,
Quiz for Lessons 16 Through 111
(^ 16 ] Variables and Algebraic Expressions
Evaluate each expression for the given values of the variable.
1. 7(.v + 4)forA = 5 2. 11  7? H3for/; = 6 3. /7 + 6r forp = 11 and r = 3
^ 17 ] 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 longdistance 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^ 18 ] 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^ 19 ) 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 110] Solving Equations by Adding or Subtracting
Solve each equation.
17. g 4 =13 18.20 = 7 + ^ 19. r 18 = 6
Qy 111] 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 8ounce 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 14, 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
RealWorld 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 3square by 3square section of the grid.
Place nine beans in the nine spaces, as shown below.
You must move all nine beans to the nine markedoff
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 Lshape. No two
counters may occupy the same square.
A complete copy of the rules and a game board are
available online.
Learn It Online
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^
58 Chapter 1 Algebraic Reasoning
/^^ mfJi
Materials
• I full sheet of
decorative paper
• 3 smaller pieces of
decorative paper
• stapler
• scissors
• markers
■ pencil
PROJECT
StepbyStep
Algebra
This "step book" is a great place to record sample
algebra problems.
Directions
O Lay the ll^by7 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^by7^inch sheet of paper under
the flap of the first piece. Do the same with the
5^by7inch and 3by7inch 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
11 } Numbers and Patterns (pp. 69)
■ 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
123 Exponents (pp. 1013)
■ 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
^13] Scientific Notation (pp. 1417)
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.
14] Order of Operations (pp 1922)
■ Simplify 150  (18 + 6) • 5.
:: 150 (18 + 6) 5 /erro. rr: ■;\.^ ^,^^.
In parentheses.
150245 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
walkathon. 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.
15j 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
16] Variables and Algebraic Expressions (pp. 3033)
■ Evaluate 5a — 6b + 7 for a = 4 and b = 3.
5a 6b + 7
5(4) 6(3) + 7
2018 + 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
17 J Translating Words into Math (pp 3437)
■ 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)
18^ Simplifying Algebraic Expressions (pp. 3841)
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
19) 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
110j Solving Equations by Adding or Subtracting (pp. 4851)
■ 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
111j Solving Equations by Multiplying or Dividing (pp. 5255)
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 babysit. Last
montli she earned $136. How many
hours did Lee babysit 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+1650 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 multiplechoice 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 QV 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 backtoschool 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 14(15 I 27)
® 34 + 4(15 + 27) + 7
CD 4(34+ 15 + 27) + 7
CD 34+15 + 427
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
850page 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
+ 1442 + 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+
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Applied Skills Assessment
Constructed Response
1. Luke can swim 25 laps in one hour.
Write an algebraic expression to show
how many laps Luke can swim in h
hours. How many hours will it take
Luke to swim 100 laps?
2. An aerobics instructor teaches a
45minute 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.
MultipleChoice 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
21
EXT
LAB
22
LAB
23
LAB
LAB
25
2B
26
27
28
2C
29
211
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
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Chapter Project Online qo.hrw.com,
keyword tlllWiM ®
^ziiA
68 Chapter 2
■^ £^'i
;P^%;;.S r^ ;
Are You Ready?
7
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■ IMIilli.lijMSIQAYRT 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 + 952
8. 115  153 + 9(8  2)
10. 300 + 6(5 3)  11
7. 12 3  4 • 5
9. 20 • 5 • 2 (7 + 1) H 4
11. 1413 + 92
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 realworld 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 onetime 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 onetime 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€i5m
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 halfprice 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
21
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.
54321 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
4321 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*
54321 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
21012345678
7 is 7 units from 0, so 1 7 1 = 7.
4
4 units
H — h
^ — h
H — \ — h
65 4321 1 2 3 4
—4 is 4 units from 0, so I 4 1 = 4.
Think and Discuss
1. Tell which number is greater: 4,500 or 10,000.
2. Name the greatest negative integer and the least nonnegative
integer. Then compare the absolute values of these integers.
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21 Integers 73
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Exercises 130, 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 eastwest 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 13)
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
21 Integers 75
LESSON 21 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
101
1
IQi
^ or 0.1
102
10 10 10^
100°^ 001
103
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
105 = ^05 . 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. 105
4. 10
■10
Write each number in scientific notation or standard form.
6. 0.00000021
10. 0.0009
14. 5.8 X 109
18. 2.77 X 10'
7. 0.00086
11. 0.0453
15. 4.5 X 105
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 105
26. Write About It Explain the effect that a zero exponent has on a power.
21 Extension 77
LABl^ Model Integer Addition
Use with Lesson 22
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)
22 HandsOn Lab 79
22
\
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
54321 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; 86 = 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
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22 Adding Integers 81
^festitij^'
'^Jj^JS3i
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Exercises 132, 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. MultiStep 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
14)
60. 30  5 • (3 +
Compare. Write <, >, or =. (Lesson 21)
62. 14 12 63. 4 3
64. 6
4
H \ \ — \ — h
54321
CD 4+ (6)
61. 15 3 2 + 1
65. 9
■11
22 Adding Integers 83
\  ' ■ "
LABIV^ Model Integer Subtraction
Use with Lesson 23
,.■■■'
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.
83 = 5
O Use integer chips to find each difference.
a. 6 5 b. 6 (5) c. 107
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.
63 = 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 23 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)
23 HandsOn 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 38
8
h
H — \ — \ — I — \ — I
H — \ — \ — h
6 54321 1 2 3 4
3  8 = 5
B 42
, 2
4
H
H — \ — h
H — \ — \ — \ — \ — I — h
654321 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
32101234567
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
59 = 5 + (9)
= 4
B 9 (2)
9 (2) = 9 + 2
C 43
43= 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
ab
— 67 = — 6 + (7) Substitute for a and b. Add the opposite
= 13 of 7.
B rt = 14, /;= 9
ab
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
23 Subtracting Integers 87
23
GUIDED PRACTICE
See Example 1 Use a number line to find each difference.
L 1.47 2.65 3. 2 (4) 4. 8  (2)
See Example 2 Find each difference.
L 5.610 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.
13.712 14. 5 (9) 15. 2 (6)
^ 17. 9 (3) 18. 410 19. 8 (8)
See Example 2 Find each difference.
21. 22  (5) 22.421 23.2719
L 25. 30 (20) 26.1515 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?
. fKAUIKt ANU fKUIS
Ltm M
Simplify.
[Extra Practice J
See page EPS.
36. 28
37.
59
38.
15128
39. 6+ (5) 3
40.
1  8 + (6)
41.
4 (7) 9
42. (2  3)  (5  6)
43.
5 (8)  (3)
44.
1012 + 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 22)
23 Subtracting Integers 89
Model Integer
Multiplication and Division
Use with Lesson 24
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.
42= 8
Q Use integer chips to find each product.
a. 6 5 b. 4 • 6 c. 3 • 4
d. 23
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. 32
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
24 HandsOn Lab 91
24
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.
32 = 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.
109 8 7 6 5 4 3 21 1 Add 3 three times.
3 (3) = 9
B 42
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 21 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
36
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
24 Multiplying and Dividing Integers 93
24
[•Li I hi
Homework Help Online go.hrw.com,
keyword ■MMMKaM @
Exercisesl34,35,37,39,41,
43,45,47
GUIDED PRACTICE
See Example 1 Use a number line to find each product.
_ 1. 5 (3) 2. 5 (2) 3.
35
See Example 2 Find each product.
L 5.5 (3)
6. 25
See Example 3 Find each quotient.
9. 32 H (4) 10. 18 H3
13. 63^ (9)
14. 50 ^ 10
7. 3 • (5)
11. 20 H (5)
15. 63 H
4. 46
8. 7 (4)
12. 49 H (7)
16. 45 f (5)
See Example 4 17. Angelina hiked along a 2,250foot 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) 19.52 20.42 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) 28.1644
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 35foot 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.
35.410 36. 3 H 37. 45 h 15 38. 34 (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?
1210 II 40 + (2) III 5 (2)' IV 4212
(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 23)
69. 3  (2) 70. 56
71. 68
72. 2 (7)
24 Multiplying and Dividing Integers 95
\'\ '
Model Integer Equations
Use with Lesson 25
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. x7 =
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
25 HandsOn 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 25 Solving Equations Containing Integers 99
25
iJ
GUIDED PRACTICE
1 HomeworkHelpOnlinego.hrw.com,
keyword ■MWllKBiM ®
Exercises 120, 23, 25, 31, 33,
35,37,43
See Example 1
See Example 2
Solve each equation. Check your answer.
1. w6= 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 = .vl35
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 crosscountry 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. MultiStep 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 11 )
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 IIOI
(Lessons 21, 22, and 23)
54. 4 4
57. 78 15
55. 7 9
58. 12 10 f (12)
25 Solving Equations Containing Integers 101
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Quiz for Lessons 21 Through 25
Q) 21 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. 1231 6. 17
22 j Adding Integers
Find each sum.
8.6 + 3 9. 5+ (9)
7. IIOI
ST
Evaluate p + t for the given values.
11. ;;= 5, r= 18 12. ^= 4, f = 13
23 ] Subtracting Integers
Find each difference.
14. 21  (7) 15. 9  (11)
10. 7 + (11)
13. p= 37, ^ = 39
16. 617
&
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?
24 j Multiplying and Dividing Integers
Find each product or quotient.
18.73 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
25 ] 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
26
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.
5495
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
26 Prime Factorization
105
26
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 13)
70. 2.45 10^ 71. 58.7 lO'
72. 200 • 10^
Solve each equation. Check your answer. (Lesson 25)
74. 3.Y = 6 75. V  4 = 3 76. z I 4 = 3  5
73. 1,480 • lO''
77. = 4x
26 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==22(3)(5)
45 =(3) 3 •©
35 = 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
25=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
27 Greatest Common Factor 109
27
i
Homework Help Online go.hrw.com,
keyword ■miaiiKBiM ®
Exercises 120, 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
SLXthgraders. 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. MultiStep 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 Signup 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 12)
46. 10^ 47. 13' 48. 6^ 49. 3^
Use a number line to find each sum or difference. (Lessons 22 and 23)
50. 5^(3) 51.27 52. 4 + (8) 53. 3  (5)
Complete the prime factorization for each composite number. (Lesson 26)
54.100= 5 55. 147 = 3 • isBi 56. 270 = 2 • 3^ • 57.140= •57
27 Greatest Common Factor 111
28
, 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 [VJiJjuj 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
28 Least Comnnon Multiple 113
28
See Example 1
See Example 2
See Example 3
L
Homework Help Online go.hrw.com,
keyword MBteinBiB;« ®
Exercises 121, 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. MultiStep 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 13day
cycle and a 20day 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
6day 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 360day 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 18)
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 27)
47. 12,28 48. 16,24 49. 15.75
50. 28, 70
28 Least Common Multiple 115
To Go On?
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ResourcesOnlinego.hrw.com,
gBaMSI0RTGO2B51
Quiz for Lessons 26 Through 28
26 ] 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
27 ) 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 firstgrade
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
28 ] 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 1mile 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 multiplechoice 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 fundraising 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 fundraising 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£
162 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 12H34
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  184 _ 72 25 _ 25 • 3 _ 75
15 154 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 29 Equivalent Fractions and Mixed Numbers 119
F >
29
)jfiU£fcfe
keyword mtlismWEiM ®
Exercises 144, 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 4125i
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 longdistance
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 longdistance
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 c7 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 110 and 111)
72. 5b = 25 73. 6 I y =18 74. ^  57 = 119
Find the least common multiple (LCM). (Lesson 28)
76. 2,3,4 77. 9, 15 78. 15,20
75. y = 20
4
79. 3, 7,
29 Equivalent Fractions and Mixed Numbers 121
210
^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 "thirtysix
thousandths."
0.036 =
36
L
,000
36 ^ 4
1,0004 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
270 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
__ „ 111 111 ■ r 1 ,,, r^ 1 from the floor of a
52. On which date did the price ot stock XYZ change ^^^ji^ 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 24f 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 tickertape 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,
tickertape 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 = 2for3x4= 1 58. .v = 3 for 5x + 4 = 19 59. x = 14 for 9(4 + x) = 162
Write each as an improper fraction. (Lesson 29)
60. 4^
61. 3^
62. If
63. 6:^
270 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 extralarge 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 65 30
J7_ _ 73 _ 21
10 103 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
=
39
59
_ 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
27 7 Comparing and Ordering Rational Numbers 127
211
1!S HomeworkHelpOnlinego.hrw.com,
keyword IKHIiKaiB ®
Exercises 130, 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.
MultiStep Twentyfour karat gold is considered pure.
a. Angie's necklace is 22karat 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 treedwelling animals that live in South and
Central America. They generally sleep about  of a 24hour 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 onehalf 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 21)
48. 14 12 49. 7 8
Simplify. (Lessons 22 and 23)
52. 13 + 51 53. 142  (27)
50.
54. 118  (57)
51. 3
55. 27 + 84
27 7 Comparing and Ordering Rational Numbers 129
CHAPTER
2
SECTION 2C
Ready To Go On?
.^y Learn It Online
Quiz for Lessons 29 Through 211
^^ 29 ] 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^
^andj
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 2IOj 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 30gram 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.
^; 211] 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 35, 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
ZebraTailed Lizard
51
5
RealWorld 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 19.
Modified TicTacToe
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 lMTiOGaniesI
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 flipflopfold 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
Zf] Integers (pp. 7275)
■ Use a number line to order the integers
from least to greatest.
3,4,2,1,3
■ I I l»*l »»» ! ■
EXERCISES
642 2 4 6
3. 2, 1.3,4
22 ) Adding Integers (pp. 8083)
■ 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. 117 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
23] Subtracting Integers (pp. 8689)
E ■ Find the difference.
i 5  (3)
■ — 5 + 3 = — 2 Add the opposite of 3.
Find eacli difference.
17. 82 18. 10  19
19. 6 (5) 20. 54
21. 6 (5) 8 22. 10 (3)  (1)
24] Multiplying and Dividing Integers (pp 9295)
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. 34
28. 45 ^ (15)
25] 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.
26] Prime Factorization (pp. 104107)
Write the prime factorization of 56.
■ 56 = 8 • 7 = 2 • 2 • 2 • 7, or 2' • 7
27] Greatest Common Factor (pp. 108111)
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
28] Least Common Multiple (pp. 112115)
■ 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?
29 j Equivalent Fractions and Mixed Numbers (pp. 118121)
Write 5^ as an improper fraction.
r2 _ 35 + 2 _ 1?
3 3 3
Write ^ as a mixed number.
11= 17 H4 = 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
2I0] Equivalent Fractions and Decimals (pp. 122125)
Write 0.75 as a fraction in simplest form.
r.,nJ5__ 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
211] Comparing and Ordering Rational Numbers (pp. 126129)
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) 8.63 9. 17 (9) 8 10. 102 + ( 97) + 3
11.320 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.
MultipleChoice 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
Dil^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)^ 34.
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 12 139
CHAPTER
3
• i I
niumbers
3A Decimal Operations
and Applications
31 Estimating with Decimals
32 Adding and Subtracting
Decimals
LAB Model Decimal
Multiplication
33 Multiplying Decimals
LAB Model Decimal Division
34 Dividing Decimals
35 Solving Equations
Containing Decimals
3B Fraction Operations
and Applications
36 Estimating with Fractions
LAB Model Fraction Addition
and Subtraction
37 Adding and Subtracting
Fractions
38 Adding and Subtracting
Mixed Numbers
LAB Model Fraction
Multiplication and
Division
Multiplying Fractions and
Mixed Numbers
Dividing Fractions and
Mixed Numbers
39
310
311 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
baseten 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
si
7 64
192
8 21
^ 35
9 ii
^ 99
loi
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. 117 30. 4 (10)
31. 22 + (11) 32. 23 + (30) 33. 3374
34. 62 • (34) 35. 84 + (12) 36.2618
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.,.enamberwbolenumber>lthat^
..octly Z factors: I and itself £ ^ ^^
Composite namberwbole 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 notetaking skills.
Applying Rational Numbers 143
Vocabulary
compatible numbers
Jessie earned $26.00 for babysitting. She
wants to use the money to buy a ticicet to
an aquarium for $14.75 and a souvenir
Tshirt 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
Tshirt, 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
31 Estimating with Decimals 145
31
illiJj'iJEQS
[•Tiiiiii
Homework Help Online go.hrw.com,
keyword ■MBiBKaM ®
Exercises 120, 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. 18.2565.71
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.
14.51.384.33 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. MultiStep 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 iceskating 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 23 and 24)
49. 5 + 42 50. 16 • (3) I 12
52. 90 (6) • (8)
53. 731
51. 28 (2) • (3)
54. 10 • (5) I 2
31 Estimating witli Decimals 147
L
32
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
32 Adding and Subtracting Decimals 149
32
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 127, 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. 4.169.043 30. 2.0915.271
31. 5.23  (9.1) 32.1232.55 33.5.293.37
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. MultiStep 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
Eggdrop
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 200meter dash was won
in 22.20 seconds. In 2000, the 200meter 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 25)
50. A  8 = 22 51. 3j' = 45
52. f = 8
Estimate. (Lesson 31)
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
32 Adding and Subtracting Decimals 151
CKV\6S>0
p
Model Decimal
Multiplication
KEY
Use with Lesson 33
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 baseten
blocks, always use
the largest value
block possible.
You can use baseten blocks to model multiplying decimals by
whole numbers.
Activity 1
O Use baseten blocks to find 3 • 0.1.
Multiplication is repeated addition, so 3 • 0.1 = 0.1 10.1 ) 0.1.
^^
^.m^
F^
lEspsr
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
30.1 =0.3
Q Use baseten 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 baseten blocks to find each product.
1. 40.5 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
33 HandsOn Lab 1 53
33
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 60.1
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
12 = 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
3000.3 = 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
33 IVIultiplying Decimals 155
33
liM^i^iiMiiiaMiMMMSiisii^
keyword MteiWKflM ®
Exercises 127, 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. 1.71.2 6. 2.60.4 7. 1.5 (0.21) 8. 0.41.17
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.
18.2.43.2 19.2.81.6 20.5.34.6 21.4.020.7
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. 9.62.05 29. 0.070.03 30. 44.15
31. 1.08 • (0.4)
34. 325.9 1.5
37. 7.02  (0.05)
32. 1.46 (0.06)
35. 14.70.13
38. 1.104 (0.7)
33. 3.2 0.9
36. 28.5 • (1.07)
39. 0.072  0.12
40. MultiStep 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
19941995
9992000
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
19992000 than in
19941995. According to
the graph, is this claim
reasonable?
b. Suppose a future survey
shows that 6 times as many
people enjoyed kayaking
in 20162017 than in
19992000. About how
many people reported that
they enjoyed kayaking in 2016201"
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 0.755.6
(^ 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 26)
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. 32.0812.37 64. 196.92 65. 75.25 + 6.382
33 Multiplying Decimals 157
Model Decimal
Division
Use with Lesson 34
£?,
Learn It Online
Lab Resources Online go.hrw.com,
KEY
p.^f.i^::.=m(^r!b:..}r..[ [5
rn
Ti r
f\
li=! ■ ■ .. ■ ,■ '■
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
34 HandsOn 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
358.8mile drive.
To find the number of miles per gallon
the car got, you will need to divide a
decimal by a decimal.
.ULULiUuuwiiiiii 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
34 Dividing Decimals 161
34
A.i3j'i}:ii}3.
y
Homework Help Online go.hrw.com,
keyword ■WHIiKgM ®
Exercises 127, 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 tfctfe**.****,**.*.*** **.«*.«.
%= 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 + 62
51. 10  (5 3) + 4 + 2
49. 3^80
52. 2^+ (7+ 1)
50. (2  D' + 32
53. 623 + 5
Multiply. Estimate to check whether each answer is reasonable. ison 33)
54. 2.756.34 55. 0.2  (4.6)  (2.3) 56. 1.3  (6.7)
57. 6.87 (2.65)
58. 94.26
59. 7.13 (14)
34 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 40yard 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.
mi
Lesson Tutorials Onlin€ mv.hrw.com 35 Solving Equations Containing Decimals 165
35
;
U£5
[•Jllllll
Homework Help Online go.hrw.com,
keyword ■BHIifcBiM ®
Exercises 123, 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 36
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. w9.0\ = 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 13)
52. 340,000 53. 6,000,000
Simplify each expression. (Lesson 34)
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)
35 Solving Equations Containing Decimals 167
To Go On?
,1^ Learn It Online
^^'* ResoLircesOnlinego.hrw.com,
Quiz for Lessons 31 Through 35
Q) 31 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 32 ] 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)
^ 33 ] Multiplying Decimals
Multiply.
10. 3.4 9.6
11. 2.660.9
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.
^) 34 ] 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 7.4gram gold necklace for $162.18. How much was the
necklace worth per gram? Round your answer to the nearest tenth.
Q<) 35 ] 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 100yard 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
ii1
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.
614)
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.
^Mi'j\ Lesson Tutorials Online my.hrw.com
36 Estimating with Fractions 171
■? Homework Help Online go.hrw.com,
keyword ■MiaiiKgiM ®
Exercises 126, 27, 29, 31, 35,
37,39,43
GUIDED PRACTICE
See Example 1 1. The length of a large SUV is 18j 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. 2li7i
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 + lr)
3 I 6j
1MS
16. 15i10
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.
^■ll
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 110 and 111)
49. A + 16 = 43 50. V  32 = 14 51. 5??? = 65
Solve. (Lesson 35)
53. 7.1.v= 46.15
54.
.7 = 1'+ (4.6)
55.
(] _
5.4
3.6
52. f = U
56. r 4 = 31.2
36 Estimating with Fractions 173
ih
Model Fraction Addition
and Subtraction
Use with Lesson 37
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 + 25
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 niLxednumber 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
di + f
You can use fraction bars to find 14.
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
624
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?
37 HandsOn Lab 175
37
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
14
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\!ii!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 _33, 52
8 12 8 • 3 122
_ 9 , 10 _ 19
24 24 24
Estimate i + i = i
B
1 5
10 8
1 5 _ 1 • 4 55
10 8 104 85
_ 4 25 _ 21
40 40 40
Estimate o  ^ = ^^
fc
3^8
2,7_ 28,73
3 8 38 83
_ 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 35 53
= 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
37 Adding and Subtracting Fractions 177
Zi3l^JS3^
diiictrniiii
(P^ HomeworkHelpOnlinego.hrw.com,
keyword ■BHMcaa ®
Exercises 127, 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. t4
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
201 + !
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 5153.
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. MultiStep 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 36)
66.
7 9
"• "s  4
68.7i(3)
65. 12,36.50
69. 6^ + 2
o /
37 Adding and Subtracting Fractions 179
38
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«i4
10^
4 = 6^
9 9
B I 2^  S^
12^
5I2 = 1221 _ 5I7
^24 ^24 24
' 1 1
C 72  63
5 5
7263^ = 7l63
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
38 Adding and Subtracting Mixed Numbers 181
38
GUIDED PRACTICE
Homework Help Online go.hrw.com,
keyword ■««!««;< ®
Exercises 126, 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
166^ + 4
17.
^h^fo
18.
^5 ^ ^4
See Example •
t Subtract. Write each answer in simplest
brm
19 2—  1~
14 14
20 4—  1 —
21.
82f
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 + 89i
Compare. Write <, >, or =.
39. 12i  lOf
2 10
40.
4^ + 3^
^2 ^^5
4f
.31
41. 13f2
■^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 6inchlong 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
® 5Jf3^
CS:> 63^
CE) li+li
52. Short Response Wliere Maddie lives, there is a S^cent state sales tax, a
lcent 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 22)
53.
3 + 9
54.
6+ (
■15)
55.
4 + (
8)
56. 
11+5
Find each sum or difference. Write
your
answer in simplest
form.
(Lesson 37)
57.
1 + ^
58.
3 1
7 3
59.
3 + _L
4 18
^«l
4
5
38 Adding and Subtracting Mixed Numbers 183
^m<^<>S
Model Fraction
Multiplication and Division
Use with Lessons 39 and 310
You can use grids to model fraction multiplication and division.
JT?.
Learn It Online
Lab Resources Online go.hrw.com,
■lMM510Lab3tGoM
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
39 HandsOn 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
13,
= 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 ,45
_ 1
5
(4)
3 • 1
42
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 82^
o3 _ 8 11
4 1 4
_' X 11
Write mixed numbers as improper fractions.
Simplify.
= =Y= = 22 Multiply numerators. Multiply denominators.
B i4i
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
md
Lessor Tutorials Online
39 IVIultiplying Fractions and IVIixed Nunnbers 187
39
»rtTiiih'iii>7^iV{ri'Viitiiitiiiiiiiii«i^^^^^
Homework Help Online go.hrw.com,
keyword MHIiltgjM ®
Exercises 127, 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. U1
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. U1
24. 75
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. 68
b 3
31.
35.
2i
'^ 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?
® 84
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 21 )
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 38)
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?
39 Multiplying Fractions and Mixed Numbers 189
310
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
25
3 1
3^
^3
Multiply by ttie reciprocal of ^
B
3
5
46
3
5
6 = ^1
5 6
'31
562
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
=
V7
650,0
—
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.
_ 247
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
310 Dividing Fractions and Mixed Nunnbers 191
310
GUIDED PRACtltE
Homework Help Online go.hrw.com,
keyword laailfcaill ®
Exercises 127, 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 2pint 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
38J^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. MultiStep 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.
MultiStep 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 8inch 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 thirdperiod 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 39)
57. 
iL. 5
15 8
58. l,.6
59. l24
60.
62^
3W 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 18karat 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 311 Solving Equations Containing Fractions 195
311
;iIJjr'3Jd33
GUIDED PRACTICE
Homework Help Online go.hrw.com.
keyword MiBifcaiM ^
Exercises 120, 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. ri = j
12 " + 10=11)
0.
,^ = 1
24 3
3.
y + 5 = 19
■'6 20
6.
^yfo
9.
h = a
See Example 3 20. Earth Science Carbon 14 has a halflife 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.
jl = \^.
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 IBj^ 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, 17891845
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. MultiStep 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 3z = 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 211)
50. 0.61, , , 1.25
5 3
51. 3.25,3^,3,3.02
Estimate. (Lesson 31)
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 36 Through 311
(v) 36 j Estimating with Fractions
Estimate each sum, difference, product, or quotient.
1.4^ 2.1 + 5^
3. 4fk • 3i
15 4
er
37 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 38] 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?
39] 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) 310] 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?
^) 311] 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 selfguided 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
CarverPrice
^
.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
RealWorld 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 16 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 3inch 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
31j Estimating with Decimals (pp. 144147)
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?
32 ) Adding and Subtracting Decimals (pp. 148151)
■ 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
33] Multiplying Decimals (pp. 154157)
■ 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.
4.39
17.
7.88 • 7.65
19.
9.9 • 1.9
202 Chapter 3 Applying Rational Numbers
EXAMPLES
EXERCISES
34] Dividing Decimals (pp. 160163)
■ 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?
35 ) Solving Equations Containing Decimals (pp 164167)
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?
36j Estimating with Fractions (pp. 170173)
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. llflli
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
37 ! Adding and Subtracting Fractions (pp. 176179)
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
38] Adding and Subtracting Mixed Numbers (pp. 180183)
■ 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
39] Multiplying Fractions and Mixed Numbers (pp. 186189)
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. l4i
58. 4 31
57.
5 10
59. 34 • l4
3IOJ Dividing Fractions and Mixed Numbers (pp. 190193)
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 21inch long loaf of bread is cut into
3
inch slices. How many slices will
there be?
311] Solving Equations Containing Fractions (pp. 194197)
■ 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
66i + 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. 5l3\
17 ^ + ^
4 8
19. 6l2
/ 9
20. 8l + 3A
Add or subtract. Write each answer in simplest form.
21.
10 "^ 5
22.
23. 7^ + 5i
24. 93^
Multiply or divide. Write each answer in simplest form.
25. 54:1
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.
3MI
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
•
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(0)
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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
•
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®
®
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®
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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
©
®
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@
(D
®
(D
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@
d;
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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
•
©
©
©
©
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(0)
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7
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8. What answer is shown in the grid?
9. Explain why you cannot show a
mixed number in a grid.
10. Write two equivalent forms of the
answer 1^ that co
the grid correctly.
answer 1^ that could be filled in
Test Tackier 207
CHAPTER
3
ISTEP+
Test Prep
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■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 triplejump
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?
MultipleChoice 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?
lm
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 13 209
'k
4A
41
42
43
44
4B
45
46
47
4C
LAB
48
49
410
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.
£?.
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apter
• Use proportionality to
solve problems, including
problems involving
similar objects, units of
measurement, and rates.
apter 4
^
ESSit
piiii ~im
L' ■"'■iic ' *saffl»«S?W«!c! V
m^f,
;'M!£*Mii
Are You Ready?
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■B«lMS10AYR4W^
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 sicjii
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 seventhgrade teachers. Write the ratio of students to
teachers in simplest form.
students
teachers
252
9
= 252 ^9
9^9
Write the ratio as a fraction.
Simplify.
For every 28 students, ttiere is 1 teacher.
The ratio of students to teachers is 28 to 1.
To compare ratios, vwite them as fractions with common denominators.
Then compare the numerators.
EXAMPLE
O
Comparing Ratios
Tell whether the wallet size photo or the portrait size photo has
the greater ratio of width to length.
Width (in.)
Length (in.)
Wallet
3.5
5
Personal
4
6
Desk
5
7
Portrait
8
10
Wallet:
Portrait:
width (in.) _ 3.5
length (in.) 5
width (in.) _ 8 _ 4
length (in.) ~ TO ~ 5
Write the ratios as fractions
with common denominators.
Because 4 > 3.5 and the denominators are the same, the portrait size
photo has the greater ratio of width to length.
Tfiink and Discuss
1. Explain why the ratio ^ in Example IB is not written as a mixed
number.
2. Tell how to simplify a ratio.
3. Explain how to compare two ratios.
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41 Ratios 215
41
;i3:?aB33
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keyword MiTiHIlEBW ®
Exercises 110, 11, 15, 17, 19
See Example 1
[.
See Example 2
L
See Example 3
GUIDED PRACTICE
SunLi 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 40gallon 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 sixthgraders, 30 seventhgraders, and 15 eighthgraders.
Write each ratio in all three forms.
5. 6thgraders to 7thgraders
7. 7thgraders to 8thgraders
6. 6thgraders to total students
8. 7th and 8thgraders to 6thgraders
See Example 2
9. Thirtysix 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 1113
1 1 . Tell whether group
1 or group 2 has the
greater ratio of the
number of people
for an opencampus
lunch to the number of
people with no opinion.
Opinions on OpenCampus 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
opencampus 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 1420.
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 sixthgraders, 150
seventhgraders, and 100 eighthgraders. Which statement is NOT true?
CS) The ratio of sixthgraders to seventhgraders is 5 to 6.
CE) The ratio of eighthgraders to seventhgraders is 3:2.
C£) The ratio of sbcthgraders to students in all three grades is 1:3.
CE) The ratio of eighthgraders 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 35)
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 37)
41 Ratios 217
^'^■^''"■''■*'**siua;OT
Vocabulary
rate
unit rate
7.1,9 Solve problems involving ratios and proportions. Express one quantity
as a fraction of another, given their ratio, and vice versa. Find how...
The Lawsons are going
camping at Rainbow Falls,
which is 288 miles from their
home. They would like to
reach the campground in
6 hours. What should their
average speed be in miles
per hour?
... many times one quantity
is as large as another,
given their ratio, and
vice versa. Express one
quantity as a fraction of
another given the two
quantities. Find the whole,
or one part, when a whole
is divided into parts in
a given ration. Solve
proljlems involving two
pairs of equivalent ratios.
In order to answer the question
above, you need to find the
family's rate of travel. A rate is a
ratio that compares two quantities
measured in different units.
The Lawson familv's rate is
288 miles
6 hours
A unit rate is a rate whose denominator is 1 when it is written as
a fraction. To change a rate to a unit rate, first write the rate as a
fraction and then divide both the numerator and denominator by the
denominator.
EXAMPLE lli Finding Unit Rates
Interactivities Online ►
A During exercise, Sonia's heart beats 675 times in 5 minutes.
How many times does it beat per minute?
675 beats
5 minutes
675 beats ^ 5
5 minutes ^ 5
135 beats
Write a rate that compares heart beats and time.
Divide the numerator and denominator by 5.
Simplify.
1 minute
Sonia's heart beats 135 times per minute.
B To make 4 large pizza pockets, Paul needs 14 cups of broccoli
How much broccoli does he need for 1 large pizza pocket?
14 cups broccoli
4 pizza pockets
14 cups broccoli ^ 4
4 pizza pockets ^ 4
3.5 cups broccoli
Write a rate that compares cups to pocl<ets.
Divide the numerator and denominator by 4.
Simplify.
1 pizza pocket
Paul needs 3.5 cups of broccoli to make 1 large pizza pocket.
218 Chapter 4 Proportional Relationships
y'liiBU] Lesson Tutorials Online mv.hrw.com
An average rate of speed is the ratio of distance traveled to time. The ratio
is a rate because the units being compared are different.
EXAMPLE [2] Finding Average Speed
!; The Lawsons want to drive 288 miles to Rainbow Falls in
6 hours. What should their average speed be in miles per hour?
288 miles
6 hours
288 miles ^ 6 _ 48 miles
Write the rate as a fraction.
Divide tlie numerator and denominator
by the denominator.
6 hours ^ 6 1 hour
ti
I Their average speed sliould be 48 miles per hour.
A unit price is the price of one unit of an item. The unit used depends
on how the item is sold. The table shows some examples.
Type of Item
Examples of Units
Liquid
Fluid ounces, quarts, gallons, liters
Solid
Ounces, pounds, grams, kilograms
Any item
Bottle, container, carton
EXAMPLE r 3J Consumer Math Application
The Lawsons stop at a roadside farmers'
market. The market offers lemonade in
three sizes. Which size lemonade has the
lowest price per fluid ounce?
Divide the price by the number of fluid
ounces (fl oz) to find the unit price of each
size.
$0.89 ^ $0.07 $1.69 _ $0.09
ISfloz floz
Size
Price
12 fl oz
$0.89
18 fl oz
$1.69
24 fl oz
$2.09
$2.09 ^ $0.09
24 fl oz fl oz
12 floz floz
Since $0.07 < $0.09, the 12 tl oz lemonade has the lowest price
per fluid ounce.
Think and Discuss
1. Explain how you can tell whether a rate represents a unit rate.
2. Suppose a store offers cereal with a unit price of $0.15 per ounce.
Another store offers cereal with a unit price of $0.18 per ounce.
Before determining which is the better buy, what variables must
you consider?
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Exercises 18, 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 2,748mile 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 afterschool 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 100meter race in 12.61 seconds.
Shawn won the 200meter 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 16ounce 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 38)
42 Rates 221
43
Vocabulary
equivalent ratios
proportion
' j^/ Si
ReaiiiaMgii
Read the proportion
f = li by saying
"six is to four as
twentyone is to
fourteen."
Identifying and Writ
Proportions ^
Students in Mr. Howell's
math class are measuring
the width w and the length
( of their faces. The ratio of
€ to w is 6 inches to 4 inches
for Jean and 21 centimeters
to 14 centimeters for Pat.
These ratios can be written as t and
Since both ratios simplify to ^, they are
equivalent. Equivalent ratios are ratios
that name the same comparison.
An equation stating that two ratios are
equivalent is called a proportion . The
equation, or proportion, below states that
the ratios  and y^ are equivalent.
Round face, f = *
6 = 21
4 14
If two ratios are equivalent, they are said to be proportional,
or /;; proportion.
EXAMPLE lj Comparing Ratios in Simplest Form
Determine whether the ratios are proportional.
2 ^
7' 21
2
7
21
21 H 3
^ is already in simplest form.
Simplify jj.
Since 5 = ^, the ratios are proportional.
A _6_
24' 20
24
20
8 f 8
24 H 8
6^2
20 H 2
1
3
3_
10
Simplify
Simplify
24'
20
1 q
Since ^ ^ tt^. the ratios are tiot proportional.
222 Chapter 4 Proportional Relationships
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EXAMPLE [2] Comparing Ratios Using a Common Denominator
Use the data in the table to
determine whether the ratios of
oats to water are proportional for
both servings of oatmeal.
Write the ratios of oats to water
for 8 servings and for 12 servings.
Ratio of oats to water, 8 ser\'ings: 
Ratio of oats to water, 12 semngs: 
Servings of
Oatmeal
Cups of
Oats
Cups of
Water
8
2
4
12
3
6
Write the ratio as a fraction.
Write the ratio as a fraction.
2
9
6
12
4
4
6
24
3
3
4
12
6
6
4
24
Write t/ie fractions witli a common
denominator, sucii as 24.
Since both ratios are equal to }^, they are proportional
You can find an equivalent ratio by multiplying or dividing both
terms of a ratio by the same number.
EXAMPLE [3] Finding Equivalent Ratios and Writing Proportions
.♦Qlli.
The ratios of the sizes
of the segments of a
nautilus shell are
approximately equal
to the golden ratio,
1.618. ...This ratio
can be found in many
places in nature.
Find a ratio equivalent to each ratio. Then use the ratios to write
a proportion.
» 8
14
8
14
=
8
14 •
20
20
_ 160
280
8
14
=
160
280
4
IB
4
18
=
4 4
18
2
r 2
2
~ 9
4
18
=
2
9
IVIultiply both terms by any number,
such as 20.
Write a proportion.
Divide both terms by a common factor,
such as 2.
Write a proportion.
Think and Discuss
1. Explain why the ratios in Example IB are not proportional.
2. Describe what it means for ratios to be proportional.
3. Give an example of a proportion. Then tell how you know it is
a proportion.
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43 Identifying and Writing Proportions 223
43
iicioajsaa
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Exercises 128, 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 34)
50. 14.35 H 0.7 51. 9^2.4 52. 12.505^3.05 53. 427 H (5.6)
Compare. Write <, >, or =. (Lesson 41)
54. 3:5 12:15 55. 33:66 1:3
56. 9:24 3:8
57. 15:7 8:3
43 Identifying and Writing Proportions 225
44
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
^Md'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 =
092 g
1 mL
Make a Plan
Set up a proportion using the given information. Let ni represent the
mass of 3 mL of ice.
092 g ^ _m_ *_ mass
1 mL 3 mL < — volume
•HI Solve
Solve the proportion.
^^^pX^ Write the proportion.
/// • 1 = 0.92 • 3 The cross products are equal,
in = 2.76 Multiply.
The mass of 3 mL of ice is 2.76 g.
Q Look Back
Since the density of ice is 0.92 g/mL, each milliliter of ice has a mass
of a little less than 1 g. So 3 mL of ice should have a mass of a little
less than 3 g. Since 2.76 is a little less than 3, the answer is reasonable.
Think and Discuss
1. Explain how the term cross product can help you remember how
to solve a proportion.
2. Describe the error in these steps:  = j^; 2x = 36; .v = 18.
3. Show how to use cross products to decide whether the ratios
6:45 and 2:15 are proportional.
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44 Solving Proportions 227
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11 ai^jQJSQi
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Exercises 115, 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 onedollar bills weighs 5 pounds. How much does a stack
L of 1,470 onedollar 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
121 =
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 oneeuro coins is
21.25 millimeters tall. How tall would a stack of 45 oneeuro 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 50gram 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 . MultiStep 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
rereleased 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 31)
46. 16.21  14.87
47. 3.82 • (4.97;
48. 8.7 (20.1)
Find each unit rate. (Lesson 42)
49. 128 miles in 2 hours 50. 9 books in 6 weeks
51.
;114 in 12 hours
44 Solving Proportions 229
CHAPTER
4
SECTION 4A
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Quiz for Lessons 41 Through 44
er
41 ] 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
42 ] 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
43 ] 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.
(^ 44 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 problemsolving 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 yearold
brother gets $1 1 .25, and his 9yearold
brother gets $9.75. How much money
does his 7yearold 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 2ton 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
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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
_ I98floz
1
Nineteen cups is equal to 152 fluid ounces.
EXAMPLE [3J Adding or Subtracting Mixed Units of Measure
A carpenter has a wooden post that is 4 ft long. She cuts 17 in. off
the end of the post. What is the length of the remaining post?
First convert 4 ft to inches.
inches
feet
\2 _x
1 4
Write a proportion using 1 ft = 12 in.
X = 48 in.
The carpenter cuts off 17 in., so subtract 17 in.
4 ft  17 in. = 48 in.  17 in.
= 31 in.
Write the answer in feet and inches.
31 in. X
12 in. 12
Multiply by a ratio equal to 1.
= 2^ ft, or 2 ft 7 in.
Tfimk and Discuss
1. Describe an object that you would weigh in ounces.
2. Explain how to convert yards to feet and feet to yards.
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45 Customary Measurements 233
i
45
dL
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keyword MblliEHiM ®
Exercises 118, 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.
MultiStep 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 milelong 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 35)
Determine whether the ratios are proportional. (Lesson 43)
46.
20 8_
45' 18
47.
6 5
5' 6
48.
11 JL
44' 28
49.
9 27
6' 20
45 Customary Measurements 235
46
Metric Measurements
The Micro Flying Robot II is the world's
lightest helicopter. Produced in Japan
in 2004, the robot is 85 millimeters tall
and has a mass of 8.6 grams.
You can use the following benchmarks
to help you understand millimeters,
grams, and other metric units.
"^.:.
If #r^
IMiIlifiji'
For more on metric
units, see Sl<ills Banl<
p. SB7.
^
rf^'^
Metric Unit
Benchmark
Length
Millimeter (mm)
Thickness of a dime
Centimeter (cm)
Width of your little finger
IVIeter (m)
Width of a doorway
Kilometer (km)
Length of 10 football fields
Mass
Milligram (mg)
Mass of a grain of sand
Gram (g)
Mass of a small paperclip
Kilogram (kg)
Mass of a textbook
Capacity
Milliliter (ml)
Amount of liquid in an eyedropper
Liter (L)
Amount of water in a large water bottle
Kiloliter (kL)
Capacity of 2 large refrigerators
EXAMPLE (l I Choosing the Appropriate Metric Unit
Choose the most appropriate metric unit for each measurement.
Justify your answer.
A The length of a car
Meters — the length of a car is about the width of several
doorways.
B The mass of a skateboard
Kilograms — the mass of a skateboard is about the mass
of several textbooks.
C The recommended dose of a cough syrup
Milliliters — one dose of cough syrup is about the amount
of liquid in several eyedroppers.
236 Chapter 4 Proportional Relationships
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Jii^
;trpjTinjrvVo *■ '  ? The table shows how metric units are based on powers of 10.
Prefixes:
Milli means
"thousandth"
Centi means
"hundredth"
Kilo means
"thousand"
— U
10^ = 1,000
10^ = 100
10^ = 10
10° = 1
^ = 0.1
^=0.01
10^
^0.001
10^
Thousands
Hundreds
Tens
Ones
Tenths
Hundredths
Thousandths
Kilo
Hecto
Deca
Base unit
Deci
Centi
Milli
To convert metric units, multiply or divide by a power of 10. Multiply
to convert to a smaller unit and divide to convert to a larger unit.
EXAMPLE
(B
Converting Metric Units
Convert each measure.
A 510 cm to meters
510 cm = (510 H 100) m 100 cm = 7 m, so divide by 100.
= 5.1 m Since 100 = 10^, move the decimal point
2 places left: 5 1 0.
B 2.3 L to milliliters
2.3 L = (2.3 X 1,000) mL 1 L = 1,000 mL, so multiply by 1,000.
= 2,300 mL Since 1,000 = 10\ move the decimal point
3 places right: 2.300
EXAMPLE [3 J Using Unit Conversion to Make Comparisons
Mai and Brian are measuring the mass of rocks in their earth
science class. Mai's rock has a mass of 480 g. Brian's rock has
a mass of 0.05 kg. Whose rock has the greater mass? How much
greater is its mass?
480 _ 1,000
1
Write a proportion.
The cross products are equal.
480 g = 0.48 kg
480 = l,OOO.v
0.48 = .V
Since 0.48 kg > 0.05 kg, Mai's rock has the greater mass.
0.48  0.05 = 0.43 Subtract to find how/ much greater the
mass of Mai's rock is.
The mass of Mai's rock is 0.43 kg greater than the mass of Brian's rock.
Think and Discuss
1. Tell how the metric system relates to the base 10 number system.
2. Explain why it makes sense to multiply when you convert to a
smaller unit.
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46 Metric Measurennents 237
46
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Exercises 118, 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
SilverHaired 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 12)
45. 9 46. 12°
47. 2'
48. 7'
49. 3"
Use cross products to solve each proportion. (Lesson 44)
50.
80 _ 1000
20
X
51.
5.24
28
2
52.
p_ _ in
25 15
53.
2.4
46 Metric Measurements 239
47
iifietisioiial Analysis
Vocabulary
unit conversion
factor
A unit conversion factor is a fraction in whicli the numerator and
denominator represent the same quantity in different units.
For example, ^ — — is a unit conversion factor. Because 1 mi =
^ 1 nil
5,280 ft, the conversion factor can be simplified to 1.
SlBOft
Imi
§280 ft
S,180
 =1
ft *
Interactivities Online ► You can use a unit conversion factor to change, or convert,
measurements from one unit to another. Choosing an appropriate
conversion factor is called diiiieiisioiial analysis.
EXAMPLE [1] Using Conversion Factors to Solve Problems
Helpful Hint
In Example 1A,
"1 km" appears to
divide out, leaving
"degrees per meter,"
which are the units
asked for. Use this
strategy of "dividing
out" units when
converting rates.
As you go deeper underground, the earth's temperature
increases. In some places, it may increase by 25 °C per kilometer.
Find this rate in degrees per meter.
Convert the rate 25 °C per kilometer to degrees per meter.
To convert the second quantity in a rate,
multiply by a conversion factor with that unit
in the first quantity.
25 °C
I km
1 kni 1000 m
25 °C
1000 ni
0.025 °C
Divide out like units.
C . krrj .
1 m
Divide 25 C by WOO m.
The rate is 0.025°C per meter.
B In the United States in 2003, the average person drank about
22 gallons of milk. Find this rate in quarts per month.
Convert the rate 22 gallons per year to quarts per month.
?? era! 4 qt 1 yr ^ , . , , ■ r
& • r • ^ To convert, multiply by conversion factors
with those units.
lyr
22 • 4 qt
12 mo
88 qt
12 mo
7.3 qt
1 mo
I gal 12 mo
Divide out like units.
Multiply.
Simplify.
J^
J/f _ qt
mo ~ mo
The rate is about 7.3 quarts per month.
240 Chapter 4 Proportional Relationships
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EXAMPLE [2] Converting Between Metric and Customary Units
One inch is 2.54 centimeters. A bookmark has a length of
18 centimeters. What is the length of the bookmark in inches,
rounded to the nearest inch?
Write a proportion using
1 in. = 2.54 cm.
Tiie cross products are equal.
Multiply.
Divide each side by 2.54.
Round to the nearest whole number.
inches
— ►
1 _
2.54
A
18
centimeters
1
• 18 =
18 =
18 _
2.54
2.54
2.54.Y
2.54.Y
2.54
X
7 = X
The bookmark is about 7 inches long
EXAMPLE [3J Sports Application
A football player runs from his team's 9yard line to his team's
44yard line in 7 seconds. Find the player's average speed in yards
per second. Use dimensional analysis to check the reasonableness
of your answer.
Average speed =
total distance
35 yards ^ 7
total time
_ 35 yards
7 seconds
5 yards
The player runs 44
in 7 seconds.
9 = 35 yards
Divide to find yards per second.
7 seconds ^ 7 1 second
The player's average speed is 5 yards per second.
Convert yd/s to mi/h to see if the answer is reasonable.
1 mi . 3 ft _ 3 mi _ 1 mi
5280 ft' 1yd 5280 yd 1760 yd
5 yd 1 mi
Is 1760 yd
_ ^ . 1 mi . 3600 jT
1 X 1 760 iid 1 li
. 3 mi
' 5280 yd
3600 s
Ih
Convert miles to yards.
Set up the conversion factors.
Divide out like units.
_ 5 • 1 mi 3600 _
10.2 mi/h
Multiply. Then simplify.
1 • 1760 Ih
The player's average speed is approximately 10.2 mi/h, which is a
reasonable speed for a football player to run a short distance.
Think and Discuss
1. Tell whether you get an equivalent rate when you multiply a rate
by a conversion factor. Explain.
2. Compare the process of converting feet to inches with the process
of converting feet per minute to inches per second.
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47 Dimensional Analysis 241
47
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tj
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Exercises 18, 9, n, 13, 15
See Example 1
See Example
See Example
GUIDED PRACTICE
1. The maxmimum speed of the TupolevTu144 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 fundraiser 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
200300 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 16)
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 39)
29. 12
30.
(4)
31. 3
3 2
32. I107J
b
47 Dimensional Analysis 243
Ready To Go On?
CHAPTER
4
SECTION 4B
Quiz for Lessons 45 Through 47
(vj 45 ] 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
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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?
46 ] 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 400meter dash and the 800meter run. Hilo ran in the 2kilometer
crosscountry 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
47 ] 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 55liter 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 12yard line to his team's 36yard 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 48
/*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
48 HandsOn 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 FourSided 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
48 Similar Figures and Proportions 249
48
iJ
(•Jiiiiii
Homework Help Online go.hrw.com,
keyword ■mbimebiM ®
Exercises 18, 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 1619, 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 39)
25. 
3
14
26.
(5)
4 "■ "8
28. Tell whether 5:3 or 12:7 is a greater ratio. (Lesson 41)
27.
4 ^8 ^5
48 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
Olympicsized 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.
54«54
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
49 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 18, 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
KSftH
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
'^^^^^ V29°
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 16foot shadow. A 6foot man
standing next to the building casts a 2.5foot shadow. What is the height,
in feet, of the building?
Write each phrase as an algebraic expression. (Lesson 17)
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
49 Using Similar Figures 255
\1
41
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 twodimensional
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
threedimensional 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.
3rf= 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
470 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 1516.
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 MasonDrxon Line was considered the
dividing line between the North and the South. Gettysburg is
about 8.1 miles north of the MasonDixon Line. How far apart in
inches are Gettysburg and the MasonDixon Line on the map?
17.
18.
Mason
MultiStep 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 211)
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 34)
25. 0.32 ^ 5 26. 78.57 ^ 9 27. 40.5 ^ 15 28. 29.68 ^ 28
470 5ca/e Drawings and Scale Models 259
LABF and Models
Use with Lesson 410
<•** Lat
Learn It Online
Lab Resources Online go.hrw.com
lff!By^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
470 HandsOn Lab 261
CHAPTER
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SECTION 4C
Quiz for Lessons 48 Through 410
^) 48 ] 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) 49 ] 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} 410] 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 3ounce glass, a 5ounce glass,
and an 8ounce glass. The 8ounce 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 stepbystep
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 8ounce 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 3ounce glass.)
Next, using 3ounce, 8ounce, 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 8ounce glass.]
Look at the sizes of the glasses in each problem. The volume of
the third glass is the sum of the volumes of the first two glasses:
3 + 5 = 8 and 3 + 8 = 11. Using any amounts for the two smaller
glasses, and starting with the largest glass full, you can get any
multiple of the smaller glass's volume. Try it and see.
Concentration
Each card in a deck of cards has a ratio on one
side. Place each card face down. Each player
or team takes a turn flipping over two cards.
If the ratios on the cards are equivalent, the
player or team can keep the pair, if not, the
next player or team flips two cards. After every card
has been turned over, the player
or team with the most pairs wins.
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264 Chapter 4 Proportional Relationships
Materials
• 2 paper plates
• scissors
• markers
\% %v
PROJECT
Paper Plate
Proportions
Serve up some proportions on this book made from
paper plates.
Q Fold one of the paper plates in half. Cut out a
narrow rectangle along the folded edge. The
rectangle should be as long as the diameter of
plate's inner circle. When you open the plate, you
will have a narrow window in the center. Figure A
Q Fold the second paper plate in half and then
unfold it. Cut slits on both sides of the crease
beginning from the edge of the plate to the inner
circle. Figure B
O Roll up the plate with the slits so that the two slits
touch each other. Then slide this plate into the
narrow window in the other plate. Figure C
O Wlien the rolledup 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
OMb 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
4l3 Ratios (pp. 214217)
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.
42 j Rates (pp. 21 8221)
■ 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
43] Identifying and Writing Proportions (pp 222225)
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
44 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
45] Customary Measurements (pp 232235)
■ 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?
46] Metric Measurements (pp. 236239)
■ 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
47 j Dimensional Analysis (pp. 240243)
■ Amil can run 12 kilometers in 1 hour. How
I many meters can he run at this pace in
I 1 minute?
1,000 m
km to m: =
1 kni
h to min: =
ih
60 mill
2 1trfi . 1,000 m ^ l>f ^ 12 1,000 m ^ 200 m
\M 1 iyl"! 60 min 60 min 1 min
Use conversion factors to find each rate.
30. 162lb/yrtolb/mo
31. 1,232 ft/min to mi/h
32. While driving, Abby passed mile
marker 130 at 3:10 p.m. and mile
marker 170 at 4:00 p.m. Find Abby's
average speed in miles per minute.
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Study Guide: Review 267
EXAMPLES
EXERCISES
48 ] Similar Figures and Proportions (pp. 248251
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
49] Using Similar Figures (pp 252255)
■ 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.
410] Scale Drawings and Scale Models (pp. 256259)
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
484inch 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 worldclass 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
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Applied Skills Assessment
Constructed Response
1. Jana begati the month with $102.50
in her checking account. During the
month, she deposited $8.50 that she
earned from babysitting, 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.
MultipleChoice 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 14 271
CHAPTER
5
I! n'
ii
ctio
5A
Tables and Graphs
51
The Coordinate Plane
52
Interpreting Graphs
53
Functions, Tables, and
Graphs
54
Sequences
5B
Linear Functions
LAB
Explore Linear
Functions
55
Graphing Linear
Functions
EXT
Nonlinear Functions
56
Slope and Rates of
Change
7.3.6
LAB
Generate Formulas to
Convert Units
57
SlopeIntercept Form
7.2.6
58
Direct Variation
7.2.7
EXT
Inverse Variation
Why Learn this?
You can use linear equations to repre
sent how far a sailboat moving at a con
stant rate has traveled after a certain
amount of time.
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apter
Graph linear relationships
and identify the slope of
the line.
Identify proportional
relationships {y = kx).
X=^
 ♦
J«*dV
>y ((
i.V
Are You Ready?
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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. II 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
Xaxis
ejex
yaxis
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 Xaxis and yaxis cross is called the
origin ?
4. Quadrupeds are animals with four feet,
and a quadrilateral is a foursided 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
Xaxis, yaxis
origin
quadrant
ordered pair
A coordinate plane is
a plane containing a
horizontal number line,
the .vaxis , and a vertical
number line, the yaxis .
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
65432
Quadrant III
1?
2
3
4
5
6
y
yaxis
Quadrant I
xaxis
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 .vaxis, between
Quadrants II and III.
Ay
P»
R
• — h
32
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)
ATcoordinate
Units right
or left from
ycoordinate
Units up
or down from
AV
2 units up
3 units
432lP
4T 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_2j0
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
4321P
1
2
3
— 4
X
12 3 4
• J
Start at the origin. Point L is 3 units left on the xaxis.
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 xcoordinate of a point on the yaxis. Name the
ycoordinate of a point on the xaxis.
3. Suppose the equator represents the xaxis on a map of Earth and
a line called the prime meridian, which passes through England,
represents the yaxis. 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
51 The Coordinate Plane 277
51
3..
3
GUIDED PRACTICE
[•Jiiiiiii
Homework Help Online go.hrw.com,
keyword ■BHIlWH ®
Exercises 126, 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
14
S
■< 1 — •
(3f
'2
1
54321 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 '
5432JlO
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 Acoordinate and the ycoordinate are both negative.
30. The Acoordinate is negative and the ycoordinate is positive.
31. The Acoordinate is positive and the ycoordinate 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 Mulrl 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 xaxis 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 310)
47. 8 + 1
1
48.
6^
15
49 2 + 1
46.
50.
55 +(32)
5 . 3
8 ■ 4
57 The Coordinate Plane 279
52
Interpreting
Graphs
You can use a graph to show
the relationship between speed
and time, time and distance, or
speed and distance.
The graph at right shows the
varying speeds at which Emma
exercises her horse. The horse
walks at a constant speed for the
first 10 minutes. Its speed
increases over the next 7 minutes,
and then it gallops at a constant
rate for 20 minutes. Then it slows
down over the next 3 minutes
and then walks at a constant pace
for 10 minutes.
20 30 40 50
Time (min)
EXAMPLE
Relating Graphs to Situations
Jenny leaves home and drives to the beach. She stays at the
beach all day before driving back home. Which graph best shows
the situation?
Graph A
Graph B
Time
Time
Graph C
Time
As Jenny drives to the beach, her distance from home increases.
While she is at the beach, her distance from home is constant. As
she drives home, her distance from home decreases. The answer
is graph B.
280 Chapter 5 Graphs and Functions
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EXAMPLE
(3
PROBLEM
SOLVING
PROBLEM SOLVING APPLICATION
Maili and Katrina traveled 10 miles from Maili's house to the
movie theater. They watched a movie, and then they traveled
5 miles farther to a restaurant to eat lunch. After eating they
returned to Maili's house. Sketch a graph to show the distance
from Maili's house compared to time. Use your graph to find the
total distance traveled.
^ Understand the Problem
The answer will be the total distance that Katrina and Maili traveled.
List the important information:
• The friends traveled 10 miles from Maili's house to the theater.
• They traveled an additional 5 miles and then ate lunch.
• They returned to Maili's house.
Make a Plan
Sketch a graph that represents the situation. Then use the graph to
find the total distance Katrina and Maili traveled.
•U Solve
The distance from Maili's house
increases from to 10 miles
when the friends travel to the
theater. The distance does not
change while the friends watch
the movie and eat lunch. The
distance increases from 10 to 15
miles when they go to the
restaurant. The distance
decreases from 15 to miles
when they return home.
■=■ 20
E
0)
o
E
o
c
IT!
Ate lunch
Maili and Katrina traveled a
total of 30 miles.
Went to
Maili's
Time
Q Look Back
The theater is 10 miles away, so the friends must have traveled
twice that distance just to go to the theater and return. The answer,
30 miles, is reasonable since it is greater than 20 miles.
Thmk and Discuss
1. Explain the meaning of a horizontal segment on a graph that
compares distance to time.
2. Describe a realworld situation that could be represented by a
graph that has connected lines or curves.
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52 Interpreting Graphs 281
■L'^
H^^^
' v;!^l^l5!«i^^sy;ri^W&iS^iGi*ijii^.;*W4"J;i^^
^i^?i;'J33^
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S Homework Help Online go.hrw.com,
keyword WMMsfliM ®
Exercises 14, 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 3mile 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 6mile 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 21)
14. 9 15. 3 16. 15
17.
18.
Find the greatest common factor. (Lesson 27)
19. 12,45 20. 33,110 21. 6,81 22. 24,36
Divide. Estimate to check whether each answer is reasonable. (Lesson 34)
23. 48.6 ^ 6 24. 31.5 H (5)
25. 8.32 ^4 26. 74.1 h 6
52 Interpreting Graplis 283
53
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
yZx+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'lLJd'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
(xy)
2
i2)'
4
(2, 4)
1
(1)^
1
(1,1)
(0)^
(0,0)
1
(1)^
1
(1,1)
2
(2)2
4
(2,4)
*y
 •(!,
< — I — I — I — i — li — I — f
4 O
(1, 2). ■
(2, 4) . 4
(2,4)
2)
(0,0)
xy
(2, 4)
(1, 1)'
H 1 i h
o
2
' (2, 4)
•(1, 1)
H 1 1 H
X
(0,0)
Think and Discuss
1. Describe how a function works like a machine.
2. Give an example of a rule that takes an input value of 4 and
produces an output value of 10.
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53 Functions, Tables, and Graphs 285
53
•p! HomeworkHelpOnllnego.hrw.com, "
IgoI
keyword ■mBiiiiiaM
Exercises 110, 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 23)
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 311)
 22 4c = ^
7 ^^ ^~ 5 9'
21. iA=6
23. y = 3
20. 48 (3)
24 ^x=^
10 8
53 Functions, Tables, and Graphs 287
54
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 yvalues is arithmetic or geometric.
Then find y when n  S.
n
1
2
3
4
5
y
12
5
2
9
In the sequence 7 is added to each term.
9 + 7 = 16 Add 7 to the fourth term.
The sequence is arithmetic. When u = 5, y = 16.
n
1
2
3
1
4
5
y
4
12
36
I 108
In the sequence each term is multiplied by —3.
108 • (3) = 324 Multiply the fourth term by 3.
The sequence is geometric. When n = 5, y = 324.
288 Chapter 5 Graphs and Functions
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EXAMPLE [2] Identifying Functions in Sequences
Write a function that describes each sequence.
A 2,4,6,8....
Make a function table.
n
Rule
y
1
1 2
2
2
2 2
4
3
3 2
6
4
4 • 2
8
Multiply n by 2.
The function y = 2ii
describes this sequence.
4,5,6,7,...
Make a function table.
n
Rule
y
1
1 + 3
4
2
2 + 3
5
3
3 + 3
6
4
4 + 3
7
/^dd 3 to n.
The function v = » + 3
describes this sequence.
EXAMPLE
C3
Using Functions to Extend Sequences
Sara has one week to read a book. She plans to increase the
number of chapters that she reads each day. Her plan is to read
3 chapters on Sunday, 5 on Monday, 7 on Tuesday, and 9 on
Wednesday. Write a function that describes the sequence.
Then use the function to predict how many chapters Sara will
read on Saturday.
Write the number of chapters she reads each day: 3, 5,7,9,...
Make a function table.
Multiply n by 2. Then add 1.
n
Rule
y
1
12+1
3
2
22+1
5
3
32 + 1
7
4
42 + 1
9
3/ = 2;/ + 1
Write the function.
Saturday corresponds to // = 7. When » = 7, y
Sara plans to read 15 chapters on Saturday.
+ 1 = 15.
Think and Discuss
1. Give an example of a sequence involving addition, and give the
rule you used.
2. Describe how to find a pattern in the sequence 1, 4, 16, 64, ... .
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54 Sequences 289
54
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ZJ
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Exercises 116, 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 yvalues 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 selfsimilar 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. MultiStep 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 24)
38. 16 • 2 39. 40 • (5)
36. 7*
40. 4 • (11)
37. 9^
41. 5 • (21)
54 Sequences 291
To Go On?
Quiz for Lessons 51 Through 54
51] The Coordinate Plane
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Plot each point on a coordinate plane. Then identify the quadrant that
contains each point.
1. IV(1,5)
2. X(5, 3)
3. r(i,
4. Z(8, 2)
■; 52 ] 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 53 ) Functions, Tables, and Graphs
Make a function table, and graph the resulting ordered pairs.
7. v=6x 8. v=4.v3 9. y  4x'' 10. 2a + 4
Q^ 54 j Sequences
Tell whether the sequence of yvalues 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 55
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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 inchlong 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 yvalues?
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::'"
55 HandsOn Lab 295
Vocabulary
linear equation
linear function
The graph below shows how far a kayak
travels down a river if the kayak is
moving at a rate of 2 miles per hour.
The graph is linear because all of the
points fall on a line. It is part of the
graph of a linear equation.
A linear equation is an equation ^
whose graph is a line. The solutions
of a linear equation are the points
that make up its graph. Linear
equations and linear graphs can be
different representations of linear functions.
A linear function is a fimction whose graph
is a nonvertical line.
Only two points are needed to draw the graph
of a linear function. However, graphing a third
point serves as a check. You can use a fimction
table to find each ordered pair.
hy
/ X
i^ — i — I — I — \ >
2 4
Hours
EXAMPLE jlj Graphing Linear Functions
Graph the linear function y = 2x + 1.
Ay
Input
Rule
Output
Ordered Pair
X
2x+ 1
y
(x,y)
1
2(1) + 1
1
(1,1)
2(0) + 1
1
(0,1)
1
2(1)+ 1
3
(1,3)
Place eacli ordered pair on the
coordinate grid. Then connect the
points to form a line.
296 Cliapter 5 Graphs and Functions
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EXAMPLE
Physical Science Application
For every degree that temperature
increases on the Celsius scale,
the temperature increases by
1.8 degrees on the Fahrenheit
scale. When the temperature is °C,
it is 32 °F. Write a linear function
that describes the relationship
between the Celsius and
Fahrenheit scales. Then make a
graph to show the relationship.
Let.v represent the input, which is the temperature in
degrees Celsius. Let y represent the output, which is the
temperature in degrees Fahrenheit.
The function is v = lS.v + 32.
Make a function tahle. hiclude a column for the rule.
The solutions to a
function lie on the
line.
Input
Rule
Output
X
1.8x + 32
y
1.8 (0) + 32
32
15
30
1.8(15) + 32
59
1.8(30) + 32
86
Multiply the input by 1.8
and then add 32.
Grapii the ordered pairs (0, 32), (15, 59),
and (30, 86) from your table. Connect
the points to form a line.
Check
Substitute the ordered pairs into
the function y = l.Sx + 32.
32 I 1.8(0) + 32 59 1 1.8(15) + 32
32 ^ 32 • 59 1 59 •
10 20 30 40 50
Temperature (C)
Since each output y depends on the input x, y is called the dependent
variable and .v is called the independent variable.
Ttimk and Discuss
1. Describe how a linear equation is related to a linear graph.
2. Explain how to use a graph to find the output value of a linear
function for a given input value.
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55 Graphing Linear Functions 297
55
3
iiitojiiiiii
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Exercises 18, 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
2x2
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?
MultiStep 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 10year 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 halfhour period, where .v is
the number of halfhour 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 52)
Time
Write a function that describes each sequence. (Lesson 54)
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,
55 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. SB18SB19 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 55 Extension 301
56
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
56 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 realworld situation involving a rate of change.
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Exercises 120, 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 . MultiStep 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?
56 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 12)
36. 3'
37.
38. 4'
39. 10'
Write a rule for each sequence in words. Then find the next three terms
(Lesson 53)
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 56
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
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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 forxcoordinates and picas forycoordinates, 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?
56 HandsOn 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 ^
slopeintercept form g
3
\(iO
V » I » I » T
xoD ■JOO l\00 500
Pistance Tiavelecl (mO
I
boo
I
TOO
The points where the line intersects the axes
can help you understand more about the line.
The .vintercept of a line is the .vcoordinate of
the point where the line intersects the .vaxis.
The ycoordinate of this point is alw^ays 0.
The yintercept of a line is the ycoordinate of
the point where the line intersects the yaxis.
The Vcoordinate of this point is always 0.
7^
6
5
4
3
2
1 I
^,yintercept  —
1 i i
\._J_ :
' 'V..
xmtercept
H 1 h^ii — *
12 3 4 5 6 7
EXAMPLE
Finding x and ylntercepts
Find the x and yintercepts.
%
The line intersects tlie xaxis at (1, 0).
The xintercept is 1.
The line intersects the yaxis at (0, 2).
The yintercept is 2.
The line intersects the xaxis at (4, 0).
The vintercept is 4.
The line intersects the yaxis at (0, 2).
The yintercept is 2.
308 Chapter 5 Graphs and Functions
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If you know the slope of a line and the y intercept, you can write an
equation that describes the line. Recall from Lesson 56 that the slope
of a line is the ratio of rise to run.
The linear equation y = mx + bis written in slopeintercept form,
where ni is the slope and b is the yintercept of the line.
'Slope
y=mx + b
yintercept
EXAMPLE [2 J Graphing by Using Slope and ylntercept
Graph each equation.
j'i3JJi3JJdJJ3Si
Since the yintercept
is 1, tine point (0, 1) is
a point on the line.
A y=.vhl
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 SlopeIntercept Form
Write the equation of the line in slopeintercept form.
))i — Y^ = 4 The line rises from left to right,
so the slope is positive.
The line intersects the yaxis
at (0, 2), so the yintercept is 2.
Substitute for m and b.
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57 Slopeintercept Form 309
EXAMPLE [7] Using SlopeIntercept 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 yintercepts and describe their meanings.
Use the slope and yintercept to graph
the equation.
Plot (0, 350). Use the slope 50 to plot
the line down to the xaxis.
The yintercept 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 yintercept of the line y =
■2x
4.
2. Describe how to graph the equation y =  ~x + 6.
£?.
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Exercises 122, 25, 31, 35
GUIDED PRACTICE
See Example 1 Find the. v and yintercepts.
1.
; i : ! ;
; i i !
See Example 2 Graph each equation.
3. y = ^x  2
4. y + 4 = X
3,.
5. y=x+l 6. y2x=5
310 Chapter 5 Graphs and Functions
See Example 3 Write the equation of each line in slopeintercept 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 yintercepts.
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' = fV5
17. V 2 = 7.V
14. y 4a = 2
18. y+^x=4
15. 1= .vl 3
19. v= 2x 5
See Example 3 Write the equation of each line in slopeintercept 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 yintercepts 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
yintercepts. What does each intercept represent?
57 SlopeIntercept Form 311
Use the following values to write an equation in slopeintercept 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 slopeintercept form. Use the equation to find the slope
and the x and yintercepts.
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 yaxis.
Will the line ever intersect the xaxis? 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 yintercept? If so, what are they, and what does each represent?
41. Make a Conjecture Make a conjecture about the yintercept of a line of
the form ]' = ;»x.
^ 42. What's the Error? For the equation y = 2x I 3, a student says the
yintercept is —2 and the slope is 3. Identify the student's error.
43. Write About It Give a realworld example that could represent a line with
a slope of 2 and ayintercept 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 xintercept 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 yintercepts.
47. A car travels 150 miles in 3 hours. What is the unit rate of speed per hour? (Lesson 41)
48. Tell whether the ratios  and ^ are proportional. (Lesson 42)
312 Chapter 5 Graphs and Functions
7.2.7
Identify situations that
situations, and recogni
'ocabulary
irect variation
onstant of variation
You can read direct
variation as "y varies
directly as x" or "y is
directly proportional
to x" or "y varies
with X."
K= mx where the unit rate
mis the slope of the line.
involve proportional relationships, draw graphs representing these
ze that these situations are described by a linear function in the form
An Eastern box turtle can travel at a
speed of about 18 feet per minute.
The chart shows the distance an
Eastern box turtle can travel when
moving at a constant speed.
The distance traveled is found by
multiplying time by 18. Distance
and time are directly proportional.
Direct variation is a linear relationship
between two variables that can be written in the form )' = kx or A. = j,
where A.' ^ 0. The fixed number k in a direct variation equation is the
constant of variation .
Time (min)
112 3 4
Distance (ft)
18 36 54 72
y = kx k = —
To check whether an ecjuation represents a direct variation, solve
for y. If the equation can be written as j' = kx, then it represents a
direct variation.
EXAMPLE fij Identifying a Direct Variation from an Equation
Tell whether each equation represents a direct variation. If so,
identify the constant of variation.
A 2 y = .V
2 ~ 2
Solve the equation for y. Divide botli sides by 2.
Write f as ^x.
The equation is in the form y = kx, so the original equation
2y = .V is a direct variation. The constant of variation is }j.
y+l = 2x
y + 1 = 2.Y
1
Solve the equation for y Subtract 1 from both sides.
1
y = 2.V  1
The equation is not in the form y = kx, so y + 1 = 2x is not a
direct variation.
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58 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
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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 slopeintercept form with
in  18 and b = 0. The graph shows a
slope of 18 and a j'intercept of 0.
2 3 4 5
Time (min)
c. How long does it take an Eastern box turtle to travel 162 feet?
Find the value of x when y = 162.
y = 18.V Write the equation for the direct variation.
162 = 18.T Substitute 162 for y.
Divide both sides by 18.
9 = A
It will take an Eastern box turtle 9 minutes to travel 162 feet.
162 _ 18.Y
18 ~ 18
Think and Discuss
1. Explain how to use a table of data to check whether the
relationship between two variables is a direct variation.
2. Describe how to recognize a direct variation from an equation,
from a table, and from a graph.
3. Discuss why every direct variation equation is a linear equation,
but not every linear equation is a direct variation equation.
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58 Direct Variation 315
58
Exercises
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Exercises 118, 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 vintercept 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)
58 Direct Variation 317
LESSON 58
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 .vaxis
oryaxis.
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 58 Extension 319
CHAPTER
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SECTION SB
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Quiz for Lessons 55 Through 58
(^ 55 ] 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) 56 ] 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
57 ] SlopeIntercept Form
Write the equation of the line in slopeintercept 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 108meter 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) 58 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
RealWorld 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, flipflops, 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 flipflops 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
FoldABooks
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
fourpage 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 54
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
slopeintercept form . . . 309
term 288
Xaxis 276
Aintercept 308
yaxis 276
yintercept 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
5IIJ The Coordinate Plane (pp 276279)
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. 280283)
■ 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.
53] Functions, Tables, and Graphs (pp 284287)
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
54] Sequences (pp. 288291)
■ 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
55 ] Graphing Linear Functions (pp. 296299)
■ 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.V1
20. y = 3x
21. y = A3
22. y = 2.x + 4
23. y = .V  6
24. V = 3.V  9
56 j Slope and Rates of Change (pp 302306)
■ 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.
57] Slopeintercept Form (pp. 308312)
■ 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
slopeintercept
form.
28. Graph y = i.v + 4.
Ay
./
::k
■rr^ — I — ^
X
58] Direct Variation (pp. 313317)
■ 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.v4 10. j' = .v8 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 slopeintercept form.
15.
16.
Ay
17. Paula walks up a 520meter 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 yintercepts and describe their meanings.
Tell whether each equation represents a direct variation. If so, identify the
constant of variation.
18. 5y = lOx
19. y3 = x
20. X + y = 4
21. 7x^y
Chapter 5 Test 327
Test Tackier
Extended Response: Understand the Scores
Extendedresponse test items usually involve multiple steps and require
a detailed explanation. The items are scored using a 4point 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 10pound bag of apples costs $4. Write and solve a
proportion to find how much a 15pound 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.
4point 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
3point 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.
2point 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, doublecheck 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
Each 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
MultipleChoice Assessment
6. The fraction j^ is found between which
pair of numbers on a number line?
A. ^ and 1
B. Iand
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
Xaxis?
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 babysitting. Which is
the best estimate of the total amount
she makes for 9 hours of babysitting?
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 weightlifting
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 ycoordinate 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 15 331
CHAPTER
Per
c ^1
6A
Proportions and
Percents
61
Percents
LAB
Model Percents
62
Fractions, Decimals, and
Percents
63
Estimating with Percents
64
Percent of a Number
65
Solving Percent Problems
6B
Applying Percents
66
Percent of Change
67
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 32 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,
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Exercises 126, 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
61 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. MultiStep 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 36)
46 ^^
" 8 7
47. 6jL + 5^
48. 5
(!)
Plot each point on a coordinate plane. (Lesson 51)
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 10bylO grids to model percents less than 1 or greater than 100.
Activity 1
O Use 10bylO grids to model 132%.
Think: 132% means 132 out of 100.
Shade 100 squares plus 32 squares to model 132%.
O Use a 10bylO 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 10bylO grid.
2. How can you model 0.7%)? Explain your answer.
Try This
Use 10bylO grids to model each percent.
1. 280% 2. 16^% 3. 0.25%
4. 65%
5. 140.757o
61 HandsOn Lab 339
62
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.tV'"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
420 _
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 — ¥
210123
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.
Twentynine 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
62 Fractions, Decimals, and Percents 341
62
il3C?33333
^iitorniiit
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Exercises 135, 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 themepark 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. MultiStep Onehalf of the 900 students at Jefferson Middle School are
boys. Onetenth of the boys are in the band, and onefifth 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 4245.
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
WallaceRose 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
WallaceRose 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 53)
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
62 Fractions, Decimals, and Percents 343
Estimafi
Percents
7.1.9 Solve problems involving ratios and proportions. Express one quantity as a fraction of
another, given their ratio, and vice versa Find how many times one quantity is as ..
A basketball at Hoops Haven costs $14.99.
Cam's Sports is offering the same basketball
at 20% off the regular price of $19.99.
To find out which store is offering the
better deal on the basketball, you can
use estimation.
The table shows common percents and
their fraction equivalents. You
can estimate the percent of a number
... large as another, given , i ,, .• e ^ *i » • i
their ratio, and vice versa by Substituting a fraction that IS close
Express one quantity as a
fraction of another given
the two quantities. Find the
whole, or one part, when a
whole IS divided into parts
in a given ration. Solve
problems involving two
pairs of equivalent ratios.
.^
Percent
10%
20%
25%
331%
50%
56%
Fraction
1
10
1
5
1
4
1
3
1
2
2
3
EXAMPLE
Compatible numbers
are close to the
numbers in a
problem and help
you use mental math
to find a solution.
Using Fractions to Estimate Percents
Use a fraction to estimate 48% of 79.
48% of 79 « i • 79
1
2
40
80
ihinK: 48" is about 50% and
50% is equivalent to .
Ciiange 79 to a compatible number.
Multiply.
48% of 79 is about 40.
EXAMPLE 12
Consumer Math Application
Cam's Sports is offering 20% off a basketball that costs $19.99.
The same basketball costs $14.99 at Hoops Haven. Which store
offers the better deal?
First find the discount on the basketball at Cam's Sports.
20% of $19.99 = \ • $19.99 Think: 20% is equivalent to .
$20 Change $19.99 to a compatible number.
$4 Multiply.
«1
5
The discount is approximately $4. Since $20  $4 = $16,
the $14.99 basketball at Hoops Haven is the better deal.
344 Chapter 6 Percents
Mbd Lesson Tutorials OnllnE my.hrw.com
Another way to estimate percents is to find 1% or 10% of a number.
You can do this by moving the decimal point in the number.
1% of 45: 45.0
= 0.45
To find 7% of a number,
move the decimal point
two places to the left.
10% of 45: 45.0
= 4.5
To find 10% of a number,
move the decimal point
one place to the left.
EXAMPLE [T) Estimating with Simple Percents
Use 1% or 10% to estimate the percent of each number.
3%) of 59
59 is about 60, so find 3% of 60.
I%of60 = 60.0 = 0.60
3% of 60 = 3 0.60 = 1.8
3%of59isabout 1.8.
3% equals 3 • 1%.
B 18% of 45
18% is about 20%, so find 20% of 45.
10% of 45 = 45.0 = 4.5
20% of 45 = 2 • 4.5 = 9.0
18%of45isabout9.
20% equals 2 • 10%.
EXAMPLE [4J Consumer Math Application
Eric and Selena spent S25.85 for their meals at a restaurant. About
how much money should they leave for a 15% tip?
Since $25.85 is about $26, find 15% of $26.
15% = 10% + 5% Think: 15% is 10% plus 5%.
10% of $26 = $2.60
5% of $26 = $2.60 ^ 2 = $1.30 5% is  of 10%, so divide $2.60 by 2.
$2.60 + $1.30 = $3.90 Add the 10% and 5% estimates.
Eric and Selena should leave about $3.90 for a 15% tip.
Think and Discuss
1. Describe two ways to estimate 51% of 88.
2. Explain why you might divide by 7 or mtiltiply by y to estimate a
15% tip.
3. Give an example of a situation in which an estimate of a percent
is sufficient and a situation in which an exact percent is necessary.
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63 Estimating with Percents 345
63
23j'i
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Exercises 128, 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. MultiStep 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 email
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
Email 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 33)
48. 0.8 • 96 49. 30 • 0.04
50. 1.6900
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 44)
63 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
64 Percent of a Number
349
64
i»irinTn
Homework Help Online go.hrw.com,
keyword MBibiniigw ®
Exercises 126, 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 Tshirt that was originally priced at $15.99.
44. MuitiStep Shoe Style is discounting everything in the store by 25%.
What is the sale price of a pair of flipflops 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 14karat gold is about
58.3%. A 14karat 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?
MultiStep 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 42)
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
64 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 24hour day
16.5 hours is, you can use a proportion or an equation.
t'A
Proportion method
Part ^ jt^^ 16^^ Part
100
Whole
24
Whole
... given their ratio, and
vice versa. Express one
quantity as a fraction of
another given the two
quantities Find the whole,
or one part, when a whole
IS divided into parts m
a given ration Solve
problems involving two
pairs of equivalent ratios.
» • 24 = 100 • 16.5
24«= 1,650
n = 68.75
Equation method
What percent of 24 is 16.5?
n • 24 = 16.5
" 24
n = 0.6875
Sloths spend about 69% of the day sleeping!
EXAMPLE
[ij Using Proportions to Solve Problems with Percents
! Solve.
A What percent of 90 is 45?
Write a proportion.
n _ 45
100 90
n • 90 = 100 • 45
90« = 4,500
90/1 _ 4,500
90 90
» = 50
50%of90is45.
B 12 is 8% of what number?
Set the cross products equal.
Multiply.
Divide each side by 90 to isolate the variable.
8 _ 12
100 "
Write a proportion.
8 • « = 100 •
12
Set the cross products equal.
8/1= 1,200
Multiply
8« _ 1.200
8 8
Divide each side by 8 to isolate the variable
n= 150
12 is 8% of 1
50.
352 Chapter 6 Percents
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EXAMPLE r2 J Using Equations to Solve Problems with Percents
Solve.
A What percent of 75 is 1 05?
n • 75 = 105 Write an equation.
n 75 _ 105
75 75
n= 1.4
n = 140%
140% of 75 is 105.
Divide each side by 75 to isolate tlie variable.
Write the decimal as a percent.
B 48 is 20% of what number?
48 = 20%) • n Write an equation.
48 = 0.2 • n
48 _ 0.2 • »
0.2 0.2
240 = n
48 is 20% of 240.
Write 20% as a decimal.
Divide each side by 0.2 to isolate the variable.
EXAMPLE fsj Finding Sales Tax
iJJJ^llJJJjjJi
The sales tax rate is
the percent used to
calculate sales tax.
Ravi bought a Tshirt with a retail sales price of $12 and paid SO. 99
sales tax. What is the sales tax rate where Ravi bought the Tshirt?
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 Tshirt
is 8.25%.
Think and Discuss
1. Describe two methods for solving percent problems.
2. Explain whether you prefer to use the proportion method or the
equation method when solving percent problems.
3. Tell what the first step is in solving a sales tax problem.
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65 Solving Percent Problems 353
:.'^t^%^^^i'i:ilSitM^'j:i.i^3,i.U.'tiL.A^.Jjl..v\<i/iiii
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Exercises 122, 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 12pack of cinnamonscented 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. MultiStep 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 afterschool 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 34)
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 64)
46. 45% of 26 47. 22% of 30 48. 15% of 17 49. 68% of 98
65 Solving Percent Problems 355
'■N,
CHAPTER
6
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SECTION 6A /
Quiz for Lessons 61 Through 65
Q) 61 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%
&
62 ] 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
63 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?
(^ 64 ] 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 backtoschool sale. Find the amount of
discount on fluorescent notebooks originally priced at $7.99.
(^ 65 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
61
B
':j:/im^\
7.1.8 Solue pioblenis involving percents. Find the whole given a part and the
percentage Find percentage increase or decrease.
According to the U.S. Consumer Product
Safety Commission, emergency rooms
treated more than 50,000 skateboarding
injuries in 2000. This was a 67% decrease
from the peak of 150,000 skateboarding
injuries in 1977.
\m
Vocabulary
percent of change
percent of increase
percent of decrease
EXAMPLE
jJaipjjjjEjJj
When a number is
decreased, subtract
the new amount
from the original
amount to find the
amount of change.
When a number is
increased, subtract
the original amount
from the new
amount.
A percent can be used to describe an
amount of change. The percent of change
is the amount, stated as a percent, that a
number increases or decreases. If the
amount goes up, it is a percent of increase.
If the amount goes down, it is a
percent of decrease.
You can find the percent of change by
using the following formula.
percent of change =
Finding Percent of Change
fli^,v
■*«B(iEL
amount of change
original amount
[T] Findi
Find each percent of change. Round answers to the nearest tenth
of a percent, if necessary.
Find the amount of change.
Substitute values into formula.
A 27 is decreased to 20.
27  20 = 7
percent of change = ^
« 0.259259 Divide.
«= 25.9% Write as a percent. Round.
The percent of decrease is about 25.9%.
B 32 is increased to 67.
67  32 = 35
percent of change
_ 35
32
= 1.09375
« 109.4%
Find the amount of change.
Substitute values into formula.
Divide.
Write as a percent. Round.
The percent of increase is about 109.4%.
358 Chapter 6 Percents
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EXAMPLE [2J Using Percent of Change
The regular price of an MPS player at TechSource is S79.99.
This week the MPS player is on sale for 25% off. What is the
sale price?
Step 1 Find the amount of the discount.
25 _ d
100 $79.99
25 • $79.99 = lOOrf
1999.75 _ lOOrf
100 100
$20.00 « d
Write a proportion.
Set the cross products equal.
IVIultiply. Then divide each side by 100.
The amount of the discount d is $20.00.
Step 2 Find the sale price.
regular price  amount of discount = sale price
$79.99  $20.00
The sale price is $59.99.
= $59.99
EXAMPLE
(5
The amount of
increase is also called
the markup.
Business Application
Winter Wonders buys snow globes from a manufacturer
for S9.20 each and sells them at a 95% increase in price.
What is the retail price of the snow globes?
Step I Find tlie amount ii of increase.
95% • 9.20 = n Thinl<: 95% of $9.20 is what number?
0.95 • 9.20 = n Write the percent as a decimal.
8.74 = ;;
Step 2 Find the retail price.
wholesale price + amount of increase = retail price
$9.20 + $8.74 = $17.94
The retail price of the snow globes is $17.94 each.
Think and Discuss
1. Explain what is meant by a 100% decrease.
2. Give an example in which the amount of increase or markup is
greater than the original amount. What do you know about the
percent of increase?
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66 Percent of Change 359
i3JAMB£^
GUIDED PRACTICE
jH^rlf 1111
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Exercises 112, 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 . MultiStep 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 tenyear
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 20ounce
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 45)
34. 34 mi to feet
35. 52 oz to pounds
36. 164 1b to tons
66 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, ; =
lPrt
300 = 1,000 /• 5
300 = 5,000r
300 _ 5,000r
5,000 5,000
0.06 = r
The interest rate is 6%.
, t= 5 years
Substitute.
IVIultiply.
Divide eacti side by 5,000.
362 Chapter 6 Percents
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EXAMPLE
C3
PROBLEM
SOLVING
PROBLEM SOLVING APPLICATION
Olivia deposits $7,000 in an account that earns 7% simple interest
paid annually. About how long will it take for her account balance
to reach S8,000?
n> Understand the Problem
Rewrite the question as a statement:
• Find the number of years it will take for the balance to reach $8,000.
List the important information:
• The principal is $7,000.
• The interest rate is 7%.
• Her account balance will be $8,000.
Make a Plan
Olivia's account balance i4 includes the principal plus the interest:
A — P + I. Once you solve for /, you can use I = P • r • r to find the time.
*e] Solve
A^ P+ [
8.000 = 7,000 + /
7,000 7,000
1,000= /
/= P r t
1,000 = 7,000 • 0.07 • t
1,000 = 490r
1,000 _ 4901
490 490
2.04 « t
It will take just over 2 years.
Substitute.
Subtract 7,000 from each side.
Substitute. Use 0.07 for 7%.
Multiply.
Divide each side by 490.
Q Look Back
The account earns 7% of $7,000, which is $490, per year. So after
2 years, the interest will be $980, giving a total balance of $7,980.
An answer of just over 2 years to reach $8,000 makes sense.
^^^^^^^^^^^^■^^^^^^^B
Think and Discuss
1. Write the value of t
in th
B annual
simple interest formula for a 
time period of 6 months.
2. Show how to find r
if/ =
$10
P =
$100,
and t =
2 years.
^^^
.
'Mb'j Lesson Tutorials Online mv.hiw.com
67 Simple Interest 363
67
keyword ■39EB9 W
Exercises 113, 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 2123.
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 highyield 1year
CD and the Dow lones
industrials for one year.
Investment Returns for 1 Year
Highyield 1year 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'sworkandstorage 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 2123 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 66)
31. 154 is increased to 200. 32. 95 is decreased to 75. 33. 88 is increased to 170.
67 Simple Interest 365
CHAPTER
To Go On?
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<3
2. 121 is increased to 321.
4. 45 is increased to 60.
6. 86 is increased to 95.
OntheGo Cellular Phones
Regular
Price
Price with 2year
Contract
$49
Free
$99
$39.60
$149
$47.68
$189
$52.92
$229
$57,25
GT
Quiz for Lessons 66 Through 67
66 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 OntheGo
Cellular are listed in the table. Use the
table for problems 79.
7. Find the percent discount on tlie $99
phone with a 2year contract.
8. Find the percent discount on the $149
phone with a 2year contract.
9. What happens to the percent discount that OntheGo 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?
67 ] 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 12, 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";
'' '1Hr:
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 09, list the letters of the light bulbs that are
used when that number is showing. The first few numbers have
been done for you.
n
u
g g
Once you have determined which bulbs are lit for each number,
count how many times each bulb is lit. What percent of the time is
each bulb lit? What does this tell you about which bulb will burn
out first?
Percent Bingo
Use the bingo cards with numbers and
percents provided online. The caller has a
collection of percent problems. The caller
reads a problem. Then the players solve the
problem, and the solution is a number or
a percent. If players have the solution on
their card, they mark it off. Normal bingo
rules apply. You can win with a horizontal,
vertical, or diagonal row.
A complete copy of the rules and game pieces is available online
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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
61 ] Percents (pp. 336338)
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%
62] Fractions, Decimals, and Percents (pp 340343)
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
63 ] Estimating with Percents (pp. 344347)
■ 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?
64 ) Percent of a Number (pp. 348351)
■ 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
seventhgraders currently attend
Canyon Middle School?
65] Solving Percent Problems (pp 352355)
■ Solve.
80 is 32% of what number?
80 = 32% • n
80 = 0.32 • n
80 _ 0.32 • n
0.32 0.32
250 = n
80 is 32% of 250.
Write an equation.
Write 32% as a decimal.
Divide each side by 0.32.
Solve.
33. 20% of what number is 25?
34. 4 is what percent of 50?
35. 30 is 250% of what number?
36. What percent of 96 is 36?
37. 6 is 75% of what number?
38. 200 is what percent of 720?
39. The sales tax on a $25 shirt purchased
at a store in Oak Park is $1.99. What is
the sales tax rate in Oak Park?
40. Jaclyn paid a sales tax of $10.03 on a
camera. The tax rate in her state is 8%.
About how much did the camera cost?
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Study Guide: Review 371
EXAMPLES
EXERCISES
66j Percent of Change (pp. 358361)
' 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?
67 ] Simple Interest (pp. 362365)
Find each missing value.
;l B / = ,p= $545, ;■ = 1 .5%, t = 2 years
I^ P r t
/= 545 0.0152
/= 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 seventhgraders 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+
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Applied Skills Assessment
Constructed Response
1. The graph shows the number of
boys and the number of girls who
participated in a talent show.
Talent Show Participation
a. What is the approximate percent of
increase of girls participating in the
talent show from 2002 to 2005?
b. What percent of students
participating in the talent show in
2006 were boys? Explain how you
found your answer.
2. A homemaker association has 134
members. If 31 of these members
are experts in canning vegetables,
are more or less than 25% of the
members canning experts? Explain
how you know.
Extended Response
3. Riley and Louie each have $5,000 to
invest. They both invest at a 2.5%
simple interest rate.
a. Riley keeps her money invested
for 7 years. How much interest
will she earn? How much will her
investment be worth?
b. What is the value of Louie's
investment if he invests for 3 years,
then removes and spends $1,000,
and then invests what is remaining
for 4 more years at a rate of 4%?
MultipleChoice 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 100meter 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 16 375
and
7A
Organizing and
Displaying Data
71
Frequency Tables,
StemandLeaf Plots,
and Line Plots
7.4.4
72
Mean, Median, Mode,
and Range
7.4.3
73
Bar Graphs and
Histograms
7.4.4
74
Reading and Interpreting
Circle Graphs
7.4.4
75
BoxandWhisker Plots
7.4.4
LAB
Explore BoxandWhisker
Plots
7B
Representing and
Analyzing Data
76
Line Graphs
7.4.4
LAB
Use Venn Diagrams to
Display Collected Data
77
Choosing an Appropriate
Display
7.4.1
LAB
Use Technology to
Display Data
78
Populations and Samples
7.4.4
79
Scatter Plots
7.4.4
LAB
Samples and Lines of
Best Fit
710
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.
£?.
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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?
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IBM Msioch7 TGo.
ST Vocabulary
jf Choose the best term from the Hst to complete each sentence.
1. A part of a line consisting of two endpoints and all points
between those endpoints is called a(n) ? .
2. A(n) ? is the amount of space between the marked
values on the ? of a graph.
3. The number of times an item occurs is called its ? .
circle
frequency
interval
line segment
scale
Complete these exercises to review skills you will need for this chapter.
Order Whole Numbers
Order the numbers from least to greatest.
4. 45, 23, 65, 15, 42, 18 5. 103, 105, 102, 118, 87, 104
6. 56, 65, 24, 19, 76, 33, 82 7. 8, 3, 6, 2, 5, 9, 3, 4, 2
(v) whole Number Operations
Add or subtract.
8. 18 + 26
12. 133  35
9. 23 + 17
13. 54 29
10. 75 + 37
14. 200  88
11. 98 + 64
15. 1,055  899
Locate Points on a Number Line
Copy the number line. Then graph each number.
8 10 12 14 16 18 20
19. 7
Top Speeds of Some Animals
Animal
Speed (mi/h)
Elephant
25
Lion
50
Rabbit
35
Zebra
40
I M K I 1
7
studm,idemmm.
Where You've Been
Previously, you
• used an appropriate
representation for displaying
data.
• identified mean, median, mode,
and range of a set of data.
• solved problems by collecting,
organizing, and displaying data.
In This Chapter
You will study
• selecting an appropriate
representation for displaying
relationships among data.
• choosing among mean,
median, mode, or range to
describe a set of data.
• making inferences and
convincing arguments based
on analysis of data.
Where You're Going
You can use the skills
learned in this chapter
• to analyze trends and make
business and marketing
Key
Vocabulary /Vocabulario
decisions.
to strengthen a persuasive
argument by presenting data
and trends in visual displays.
bar graph
grafica de barras
circle graph
grafica circular
frequency table
tabia de frecuencia
line graph
grafica lineal
line plot
diagrama de
acumulacion
mean
media
median
mediana
mode
moda
scatter plot
diagrama de
dispersion
stemandleaf
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 61 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, Stemai
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
stemandleaf 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
019
16
16
2039
1
17
4059
2
19
6079
1
20
380 Chapter 7 Collecting, Displaying, and Analyzing Data \ Viilaij] Lesson Tutorials Online
A stemandleaf plot uses the digits of each
number to organize and display a set of
data. Each leaf on the plot represents the
righthand digit in a data value, and each
stem represents the remaining lefthand
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
stemandleaf 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 stemandleaf 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 stemandleaf 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.
77 Frequency Tables, StennandLeaf 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 stemandleaf plot. Explain.
,^^ Learn It Online
*^ HomeworkHelpOnlinego.hrw.com,
keyword ■miawwl ®
Exercises 16, 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 stemandleaf 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 stemandleaf 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 stemandleaf plot for Exercises 79.
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 stemandleaf 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?
71 Frequency Tables, StemandLeaf 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 stemandleaf 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 stemandleaf 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
stemandleaf plot of the data in the table have?
CA) 1 CT) 3
CD 2 CS) 4
17. Extended Response Make a stemandleaf 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 310)
Find each unit rate. Round to the nearest hundredth if necessary. (Lesson 42)
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"
^MaizalloaV'
• 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: 80 = 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
72 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
glassblowing 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
72 Mean, Median, Mode, and Range 387
72
..'■i.ti^W»«KS«*
nyjdti^
liJ
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Homework Help Online go.hrw.com,
keyword MJMhWAM ®
Exercisesln,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
100mile 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 stemandleaf 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 67)
23. Make a stemandleaf plot of the following data: 48, 60, 57, 62, 43, 62, 45,
and 51. (Lesson 71)
72 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
doublebar 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 inteivals.
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 doublebar graph to compare two related sets of data.
Making a DoubleBar Graph
The table shows the life
expectancies of people in three
Central American countries. Make
a doublebar 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
15
22
610
34
1115
52
1620
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 doublebar 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
73 Bar Graphs and Histograms 391
keyword HiaEBOl W
Exercises 110, 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 13.
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 doublebar 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 68.
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 doublebar 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 doublebar 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 1214 years in office?
Years in Office
Frequency
02
7
35
22
68
12
911
1214
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 72)
22. 42, 29, 49, 32, 19 23. 15, 34, 26, 15, 21, 30 24. 4, 3, 3, 3, 3, 4, 1
73 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
"spinyskinned."
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 onefourth of the circle.
Since the circle shows 100% of the data, about onefourth 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 = 0.530
= 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 74 Reading and Interpreting Circle Graphs 395
74
J^i,i^:^^!i^fS'di>iiiJ:*jifi.Zi^'lA''M>^.^^
\3Z}^JA1^^
keyword ■BiaiifcBM ®
Exercises 110, 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 13.
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 68.
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 1113.
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,
19882004
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 48)
Compare. Write <, >, or =. (Lesson 62)
21. 0.1
0.09
22. 1.71
24
23. 1,25
125%
24. 32.5
69%
74 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 deepsea fishing trip.
He chooses a fishing charter based on the
number offish caught on difl'erent charters.
A boxandwhisker plot uses a number
line to show the distribution of a set
of data.
Vocabulary
boxandwhisker 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 boxandwhisker 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 BoxandWhisker Plot
Use the data to make a boxandwhisker 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 boxandwhisker plot to analyze how data in a set are
distributed. You can also use boxandwhisker plots to help you compare
two sets of data.
EXAMPLE r2j Comparing Boxandwhisker Plots
The boxandwhisker plots below show the distribution of the
number of fish caught per trip by two fishing charters.
H — \ — \ — I — \ — h
H — I — \ — \ — \ — h
l = ReeltoReel
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 ReeltoReel 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 boxandwhisker plot indicates the
interquartile range. ReeltoReel 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
boxandwhisker 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
75 Boxandwhisker Plots 399
Homework Help Online go.hrw.com
keyword ■BHIilifl.M ®
Exercises 18,9, 11, 19
GUIDED PRACTICE
See Example 1 Use the data to make a boxandwhisker plot.
1. 46 35 46 38 37 33 49 42
35 40 37
See Example 2
Use the boxandwhisker plots of inches flown by two different paper
airplanes for Exercises 24.
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 boxandwhisker plot.
5. 81 73 88 85 81 72 86 72 79
75 76
See Example 2 Use the boxandwhisker plots of apartment rental costs in two different
cities for Exercises 68.
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 911.
12 7 15 23 10 18 39 15 20 8 13
9. Make two boxandwhisker 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 boxand
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 boxandwhisker plot of the data.
b. Describe the distribution of the number of medals won.
14. Measurement The stemandleaf plot shows the
heights in inches of a class of seventh graders.
a. Make a boxandwhisker plot of the data.
b. Threefourths of the students are taller
than what height?
c. Threefourths 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 boxandwhisker 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 boxandwhisker
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 49)
21. Mari spent $24.69 on lunch with her mom. About how much should she
leave for a 15% tip? (Lesson 63)
75 BoxandWhisker Plots 401
A LAB
Explore
BoxandWhisker Plots
Use with Lesson 75
S^.,
Learn It Online
Lab Resources Online go.hrw.com,
MSlOLab? ■Go,
You can use a graphing calculator to analyze data
in boxandwhisker 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 boxandwhisker 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 boxandwhisker 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 boxandwhisker 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 boxandwhisker plot?
Wliat values must you find before you can identify the upper and lower quartiles?
2. Wliat does the boxandwhisker 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 boxandwhisker 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 seventhgrade 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 boxandwhisker 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
boxandwhisker 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 boxandwhisker 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 Flci:^
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 boxandwhisker 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 andwhiskerplot?
3. Are there any differences in the plots? What do these differences tell you about
boys talking on the phone as compared to girls?
75 Technology Lab 403
CHAPTER
7
SECTION 7A
Ready To Go On?
^£*9Learn It Online
ResourcesOnlinego.hrw.com,
(2r
Quiz for Lessons 71 Through 75
71 ] Frequency Tables, StemandLeaf 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 stemandleaf plot of the data.
3. Make a line plot of the data.
72 ] 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.
73 ] 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 doublebar 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
74 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?
75 ) BoxandWhisker Plots
10. Make a boxandwhisker plot of the data 14, 8, 13, 20,
15, 17, 1, 12, 18, and 10.
1 1 . On the same number line, make a boxandwhisker plot
ofthedataS, 8, 5, 12,6, 18, 14,8, 15, and 11.
12. Which boxandwhisker 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
threeperson 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?
76
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
doubleline 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 2month
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 doubleline graph shows change over time for two sets of data.
EXAMPLE
9
Russia
^^' Nome
Canada
Alaska
Anchorage*
U
i^
Gulf of Alaska
Making a DoubleLine Graph
The table shows the normal daily
temperatures in degrees Fahrenheit
in two Alaskan cities. Make a
doubleline 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.
76 Line Graphs 407
76
''. Homework Help Online go.hrw.com,
keyword ■mBiiiBMiM ®
Exercises 17, 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 5year 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 doubleline 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 46.
See Example 1 4. Make a line graph of the data. Use the
graph to determine during which 5year
period the number of NBA teams
increased the most.
See Example 2
5. During which 5year 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 doubleline
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 doubleline graph rather
than two singleline 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) 19911993
CE) 19931997
CD 19972001
CE) 20012005
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 62)
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 74)
76 Line Graphs 409
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s
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Use Venn Diagrams
to Display Collected Data
Wse w/f/i Lesson 76
£?.
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
76 HandsOn Lab 411
\
&
77
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 stemandleaf 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
Gossamerwing
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 stemandleaf 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
andleaf 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
77 Choosing an Appropriate Display 413
77
<iit<*Tiiiiii
Homework Help Online qo.hrw.com,
keyword ■mmwiBiM ®
Exercises 18, 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 topselhng 42inch 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 stemandleaf 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
161170
4
171180
8
181190
12
191200
11
201210
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) Stemandleaf 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 62)
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 64)
77 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 line
<> 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
1001
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
77 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
78 Populations and Samples 419
78
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Homework Help Online qo.hrw.com,
keyword ■aWlil'AM . ®
Exercises 18, 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 randomlyselected 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 cityhall 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 randomlyselected 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 61)
19. 52% 20. 7%
Find the percent of each number. (Lesson 64)
23. 11% of 50 24. 48% of 600
21. 110%
25. 0.5% of 82
22. 0.4%
26. 210% of 16
78 Populations and Samples 421
79
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 meateating dinosaur, was about onethird 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
79 Scatter Plots 423
keyword ■mMllBigjM ®
Exercises 18, 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 911, 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 64)
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 66)
79 Scatter Plots 425
T^LAB
Samples and Lines of
Best Fit
Use after Lesson 79
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 datacollection 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.
"" =
F1:L1AH
a
K
HiS
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?
VH8iiKar^<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?
rijiiA'msaf
79 Technology Lab 427
710
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]iii<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
770 Misleading Grapiis 429
710
\L'^ttei'tfi^vX\'rau:'i«i.«jft;wo?M'»*iiL.'S[*W'5aa^^^
,ii^j'^J33^
keyword IBBiliiiaiil ®
Exercises 16, 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 2,160mile
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 50cent 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 31 1)
14. h = ^ 15
X +  — 
•*■ ^ 3 6
16. iv^f
17. x^ =
Write positive, negative, or no correlation to describe each relationship. (Lesson 79)
18. height and test scores 19. speed of a car and time required to travel a distance
770 Misleading Graphs 431
CHAPTER
7
SECTION 7B
Ready To Go On?
#^Leam it Online
»*■ ResourcesOnlinego.hrw.com,
lBBW!lMsin RTri07B^
Quiz for Lessons 76 Through 710
76 ] 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
&
77 ] 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.
(^ 78 ] 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
79 ] 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
(^ 710] 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 15, use the table.
1 . Make a stemandleaf 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 boxandwhisker plot of the data.
UTAH
K^l
/
Salt Lake
i^City
Wins by the Utah Jazz
Season
Wins
19992000
55
20002001
53
20012002
46
20022003
47
20032004
42
20042005
26
20052006
41
20062007
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
BOXANDWHISKER
PLOT
STEMANDLEAF
PLOT
CIRCLE GRAPH
iS.iiv:S.;!:zc.
It's in the Bag! 435
Vocabulary
bar graph 390
biased sample 419
boxandwhisker plot . . 398
circle graph 394
convenience sample ... 418
correlation 423
cumulative frequency . . 380
doublebar graph 390
doubleline 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
stemandleaf 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
71 ] Frequency Tables, StemandLeaf Plots, and Line Plots (pp. 380384)
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 stemandleaf plot of the data.
5. Make a line plot of the data.
72] Mean, Median, Mode, and Range (pp. 385389)
Find the mean, median, mode, and range
of the data set 3, 7, 10, 2, and 3.
Mean: 3 + 7+ 10 1213 = 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
73] Bar Graphs and Histograms (pp. 390393)
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 doublebar graph of the data.
Favorite Pet
Girls
Boys
Cat
42
31
Dog
36
52
Fish
3
10
Other
19
7
74 J Reading and Interpreting Circle Graphs (pp 394397)
About what
percent of
people said
yellow was
their favorite
color?
about 25%
Favorite Colors
75l BoxandWhisker Plots (pp 398401)
I Use the data to make a boxandwhisker
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 1314:
33, 38, 43, 30, 29, 40, 51, 27, 42, 23, and 31.
13. Make a boxandwhisker plot.
14. What is the interquartile range?
\^6} Line Graphs (pp. 406409)
I Make a line graph of the rainfall data:
Apr, 5 in.; May, 3 in.; Jun, 4 in.; Jul, 1 in.
15. Make a doubleline 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
77 j Choosing an Appropriate Display (pp. 412415)
Choose the type of graph that would best
represent the population of a town over a
10year 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
78] Populations and Samples (pp. 418421)
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'
79] Scatter Plots (pp. 422425)
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
710] Misleading Graphs (pp. 428431)
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 18.
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 stemandleaf plot of the data.
5. Make a line plot of the data. 6. Make a histogram of the data.
7. Make a boxandwhisker 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 doublebar 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
seventhgraders?
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
Shortresponse 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 2point 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 2point rubric.
2point 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 of central tendency that best represer
ts
the data is
ih
e mean.
because if shows the
averacje
number of hours
thai
Leann
studied fcefore
lisr 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.
0point 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 boxandwhisker
■ plot shows the height in inches of
seventhgrade 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
013
H
H
1HZ6
1
11
11 HO
1
11
H15H
1
13
5568
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 5acre lots, another part into
two 10acre 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 twelfthgraders 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 doublebar 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.
MultipleChoice 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 uvzuuic 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 stemandleaf 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 20012007, soccer
participation increased by 100%.
C. From 20022006, 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 boxand
whisker plot below?
H — \ — I — \ — \ — \ — \ — \ — I — \^
8 10 12 14 16 18 20 22 24 26
16. The key in a stemandleaf plot states
that 2I5 means 2.5. What value is
represented by l8 ?
Cumulative Assessment, Chapters 17 443
CHAPTER
8A Lines and Angles
81 Building Blocks of
Geometry
LAB Explore Complementary
and Supplementary
Angles
82 Classifying Angles
LAB Explore Parallel Lines and
Transversals
83 Line and Angle
Relationships
LAB Construct Bisectors and
Congruent Angles
8B Circles and Polygons
84 Properties of Circles
LAB Construct Circle Graphs
85 Classifying Polygons
86 Classifying Triangles
87 Classifying Quadrilaterals
88 Angles in Polygons
7.3.1
r: If
Use facts about distance
and angles to analyze
figures.
• Find unknown measures of
angles.
mti
5l!liil^M
IL 1 , _
L.
w
A ^ 1 "■
111' L
'// If Liu:»a
R'
^,
.\ ^
I
I ' ! /I ■ I <
A
./J I
\
8C
Transformations
89
Congruent Figures
7.3.4
810
Translations, Reflections,
and Rotations
7.3.2
LAB
Explore Transformations
EXT
Dilations
811
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 Mfo
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. I3*'
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
81 Building Blocks of Geometry 449
li
81
<iii<iriiiii[
Homework Help Online go.hrw.com,
keyword ■QgUOggH (J)
Exercises 112, 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 24)
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 66)
27. 85 is decreased to 60. 28. 35 is increased to 120. 29. 6 is decreased to 1.
81 Building Blocks of Geometry 451
naM<,bv\
Explore Complementary and
Supplementary Angles
Use with Lesson 82
£?.
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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.
82 HandsOn Lab 453
82
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
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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°.
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82 Classifying Angles 455
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Exercises 118, 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 72)
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 81) • —
K
82 Classifying Angles 457
'mA
LAB
Explore Parallel Lines
and Transversals
Use with Lesson 83
REMEMBER
• Two angles are supplementary if the sum of their measures is 180°.
• Angles with measures greater than 0° but less than 90° are acute.
• Angles with measures greater than 90° but less than 180° are obtuse.
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Parallel lines are lines in the same plane that never cross. When two
parallel lines are intersected by a third line, the angles formed have
special relationships. This third line is called a transversal.
In San Francisco, California, many streets are parallel such as Lombard St.
and Broadway.
Lombard St.
Broadway
Columbus Ave. is a transversal that runs diagonally across them. The eight
angles that are formed are labeled on the diagram below.
Columbus Ave.
Activity
O Copy the table below. Then use a protractor to measure angles 18 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
83A HandsOn Lab 459
83
B
Line and Angle
Relationships
7.3.1 Identify and use basic properties of angles formed by transversals
intersecting pairs of parallel lines.
Wlien lines, line segments, or rays
intersect, they form angles. If the angles
formed by two intersecting lines measure
90°, the lines are perpendicular lines .
Some lines in the same plane do
not intersect at all. These lines are
parallel lines . Segments and rays that are
parts of parallel lines are also parallel. The
blue lines in the photograph are parallel.
Vocabulary
perpendicular lines
parallel lines
skew lines
adjacent angles
vertical angles
transversal
Skew lines do not intersect, and yet
they are also not parallel. They lie in
different planes. The yellow lines in the
photograph are skew.
EXAMPLE 1
Interactivities Online ►
The symbol _L means
"is perpendicular to."
The symbol  means
"is parallel to."
Identifying IParailel, Perpendicular, and
Skew Lines
Tell whether the lines in the figure appear
parallel, perpendicular, or skew.
A AB and AC
ABLAC
B CE and BD
CE and BD are skew.
C AC and BD
acWbd
t
jf
1
f/i f/
A 1 j/
1
1
C
B
^
^
•
D
y
"
The lines appear to intersect to form
right angles.
The lines are in different planes and do
not intersect.
The lines are in the same plane and do
not intersect.
Adjacent angles have a common vertex and a
common side, but no common interior points.
Angles 2 and 3 in the diagram are adjacent.
Adjacent angles formed by two intersecting lines
are supplementary.
460 Chapter 8 Geometric Figures
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Reading Math
Angles with the
same number of tick
marks are congruent
/^' • / Vertical angles are the opposite angles formed
^ by two intersecting lines. Angles 1 and 3 in the
diagram are vertical angles. Vertical angles have
the same meastire, so they are congruent.
A transversal is a line that intersects two or more lines that lie in the
same plane. Transversals to parallel lines form special angle pairs.
Alternate interior
angles
Alternate exterior
angles
Corresponding
angles
PROPERTIES OF TRANSVERSALS TO PARALLEL LINES
If two parallel lines are intersected by a transversal,
• corresponding angles are congRient,
• alternate interior angles are congaient,
• and alternate exterior angles are congruent.
EXAMPLE [2] Using Angle Relationships to Find Angle Measures
Line n \\ line p. Find the measure of each angle.
A Z6 7/6
mZB = 55° Vertical angles are congruent. P 55/5
^ 4/3
B ^1 *n m
mZl = 55° Corresponding angles are congruent
C Z7
mZ7 + 55° = 180° Adjacent angles formed by two
— 55° — 55° intersecting lines are supplementary.
mZ7 = 125°
D Z3
mZ3 = 55° Alternate interior angles are congruent.
Think and Discuss
1. Draw a pair of parallel lines intersected by a transversal. Use tick
marks to indicate the congruent angles.
2. Give some examples in which parallel, perpendicular, and skew
relationships can be seen in the real world.
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83 Line and Angle Relationships 461
;i(^2B333
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Exercises 112,13,15,17,19,21,23
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 1316, 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. MultiStep 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 32)
30. 3.58312.759) 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 82)
33.
34.
35.
148"^
83 Line and Angle Relationships 463
,(\v\6<>ov\
Construct Bisectors and
Congruent Angles
Use with Lesson 83
REMEMBER
• Congruent angles have the same measure, and
congruent segments are the same length.
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To bisect a segment or an angle is to divide it into two congruent parts.
You can bisect segments and angles, and construct congruent angles
without using a protractor or ruler. Instead, you can use a compass
and a straightedge.
Activity
O Construct a perpendicular bisector of a line segment.
a. Draw a line segment JS on a piece of paper. ^*®P i^
b. Place your compass on endpoint / and, using an
opening that is greater than half the length of •
JS, draw an arc that intersects JS.
c. Place your compass on endpoint S and draw an arc
using the same opening as you did in Step b. The arc
should intersect the first arc at both ends.
d. Draw a line to connect the intersections of the arcs.
Label the intersection of /S and the line point K.
Measure JS, JK, and KS. What do you notice?
The bisector of/S is a perpendicular bisector because all of the
angles it forms with JS measure 90°.
Q Bisect an angle.
a. Draw an acute angle GHE on a piece of paper. Label the vertex H.
b. Place the point of your compass on H and draw an
arc through both sides of the angle. Label points G
and £■ where the arc crosses each side of the angle.
c. Without changing your compass opening, draw
intersecting arcs from point G and point E. Label
the point of intersection D.
d. Draw HD.
Use your protractor to measure angles GHE, GHD, and DHE.
What do vou notice?
Y
Step b
K
A
Stepd
464 Chapter 8 Geometric Figures
Construct congruent angles.
a. Draw /LABM on your paper.
b. To construct an angle congruent to /_ABM,
begin by drawing a ray, and label its endpoint C.
c. With your compass point on B,
draw an arc through Z^45M.
d. With the same compass opening, place the
compass point on C and draw an arc through
the ray. Label point D where the arc crosses the ray.
e. With your compass, measure the arc in /^BM.
f. With the same opening, place your compass point
on D, and draw another arc intersecting the first one.
Label the intersection F. Draw CF .
Use your protractor to measure /LABM and /LFCD.
What do you find?
Think and Discuss
1 . How many bisectors would you use to di\dde an angle into four
equal parts?
2. An 88° angle is bisected, and then each of the two angles formed are
bisected. What is the measure of each of the smaller angles formed?
Try This
Use a compass and a straightedge to perform each construction.
1. Draw and bisect a line segment.
2. Trace and then bisect /.GOB.
3. Draw an angle congruent to AGOB.
83B HandsOn Lab 465
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SECTION 8A
Quiz for Lessons 81 Through 83
^ 81 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) 82 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?
&
83 ] 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
84
Vocabulary
circle
center of a circle
arc
radius
diameter
chord
central angle
sector
of Circles
^m
Completed in 1893 for the Chicago World's
Fair, the first Ferris wheel could carry up to
2,160 people. George Ferris relied on the
idea of a circle when he modeled his design
on a bicycle wheel.
A circle is the set of all points in a plane
that are the same distance from a given
point, called the center of a circle .
A circle is named by its center. For
example, if point A is the center of a circle,
then the name of the circle is circle A.
There are special names for the different
parts of a circle.
Navy Pier Ferris Wheel, Chicago
Arc
Part of a circle
named by its
endpoints
Radius
Line segment whose
endpoints are the
center of a circle
and any point on
the circle
Diameter
Line segment that
passes through
the center of a circle,
and whose endpoints
lie on the circle
Chord
Line segment
whose endpoints
are any two points
on a circle
EXAMPLE
Reading Math
Radii is the plural
form of radius.
[lj Identifying Parts of Circles
Name the parts of circle P.
A
radii
PA. PB, PC, PD
B
diameter
BD
C
chords
AD, DC, AB.
BC.BD
468 Chapter 8 Geometric Figures
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A central angle of a circle is an
angle formed by two radii. A
sector of a circle is the part of the
circle enclosed by two radii and an
arc connecting them.
The sum of the measures of all of
the nonoverlapping central angles
in a circle is 360°. We say that there
are 360° in a circle.
EXAMPLE i2l PROBLEM SOLVING APPLICATION
Sector
Central angle
Keep the Penny?
PROBLEM
SOLVING
The circle graph shows the results
of a survey to determine how
people feel about keeping the
penny. Find the central angle
measure of the sector that shows
the percent of people who are
against keeping the penny.
P!f* Understand the Problem
List the important information: source usa Today
• The percent of people who are against keeping the penny is 32%.
Make a Plan
The central angle measure of the sector that represents those people
against keeping the penny is 32% of the angle measure of the whole
circle. The angle measure of a circle is 360°. Since the sector is 32% of
the circle graph, the central angle measure is 32% of 360°.
32% of 360° = 0.32 360°
•1] Solve
0.32 • 360°= 115.2° Multiply.
The central angle of the sector measures 115.2°.
Q Look Back
The 32% sector is about onethird of the graph, and 120° is
onethird of 360°. Since 1 15.2° is close to 120°, the answer is
reasonable.
Tfiink and Discuss
1. Explain why a diameter is a chord but a radius is not.
2. Draw a circle with a central angle of 90°.
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84 Properties of Circles 469
84
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Exercises 18, 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 63)
19. 28% of 150 20. 21% of 90 21. 2% of 55
Use the alphabet at right. (Lesson 83)
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
84 Properties of Circles 471
UVBI^ Construct Circle Graphs
Use with Lesson 84
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.
84 HandsOn Lab 473
85
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.
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85 Classifying Polygons 475
■:y«V ' ■:•.. Sfjll
85
keyword ■BHIilAM ®
Exercises 118, 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 14gon. What is the name of
the large starshaped 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 54)
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 65)
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?
85 Classifying Polygons 477
86
Classifying Triangles
Vocabulary
scalene triangle
isosceles triangle
equilateral triangle
acute triangle
obtuse triangle
right triangle
A harnessed rider uses the
triangleshaped 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.
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86 Classifying Triangles 479
[•Til 1 1 II
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Exercises 18, 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. Multistep 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 211)
Name each polygon, and tell whether it is a regular polygon. If it is not, explain
why not. (Lesson 85)
30. rn h
"•/^^
86 Classifying Triangles 481
87
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 foursided
enclosure, or a quadrilateral.
Some quadrilaterals have
properties that classify them
as special quadrilaterals.
Parallelogram /s ^
A — ^ —
The Liberal Arts Quadrangle at
the University of Washington, Seattle
Opposite sides are parallel and
congruent. Opposite angles are
congruent.
Rectangle
J " L
"> 11 ■"
Parallelogram with four right angles.
Rhombus
/ /
Parallelogram with four congruent
sides.
Square
n I n
Parallelogram with four congruent
sides and four right angles.
Trapezoid
Exactly one pair of opposite
sides is parallel.
Quadrilaterals can have more than one name because the special
quadrilaterals sometimes share properties.
EXAMPLE [Tj Classifying Quadrilaterals
Give all of the names that apply to each quadrilateral. Then give
the name that best describes it.
The figure lias opposite sides thiat are
parallel, so it is a parallelogram. It has
four right angles, so it is also a rectangle.
Rectangle best describes this quadrilateral.
482 Chapter 8 Geometric Figures
[71ilbu Lesson Tutorials Online inv.hrw.com
Give all of the names that apply to each quadrilateral. Then give
the name that best describes it.
J
The figure has exactly one pair of opposite
sides that is parallel, so it is a trapezoid.
Trapezoid best describes this quadrilateral.
5 cm
5 cm
5 cm
5 cm
The figure has two pairs of opposite sides
that are parallel, so it is a parallelogram.
It has four right angles, so it is also a
rectangle. It has four congruent sides,
so it is also a rhombus and a square.
Square best describes this quadrilateral.
The figure has two pairs of opposite sides
that are parallel, so it is a parallelogram. It
has four congruent sides, so it is a rhombus.
It does not have four right angles, so it is
not a rectangle or a square.
RJioiubiis best describes this quadrilateral.
EXAMPLE
[ 2 J Drawing Quadrilaterals
Draw each figure. If it is not possible to draw, explain why.
A a parallelogram that is not a rhombus
The figure has two pairs of parallel sides,
but all sides are not congruent.
B a trapezoid that is also a rectangle
A trapezoid has exactly one pair of opposite sides that is parallel,
but a rectangle has two pairs of opposite sides that are parallel.
It is not possible to draw this figure.
Think and Discuss
1. Describe how you can decide whether a rhombus is also a square.
Use drawings to justify your answer.
2. Draw a Venn diagram to show how the properties of the five
quadrilaterals relate.
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87 Classifying Quadrilaterals 483
■ ■tOLlMlI
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keyword ■QSEB9I W
Exercises 113, 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 71)
33. Make a stemandleaf plot of the data.
34. Make a cumulative frequency table of the data.
Classify each triangle according to the measures of its angles. (Lesson 86)
35. 50°, 50°, 80° 36. 40°, 50°, 90° 37. 20°, 30°, 130° 38. 20°, 60°, 100°
87 Classifying Quadrilaterals 485
88
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 foursided
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 nonadjacent vertices of a polygon. Since the
sum of the angle measures in each triangle is 180°,
the sum of the angle measures in a foursided figure
is 2 180°, or 360°.
Diagonal
486 Chapter 8 Geometric Figures
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SUM OF THE ANGLES OF A QUADRILATERAL
The sum of the measures of
the angles in a quadrilateral
is 360°.
m/11 + mzl2 + m^3 + mZ.4 = 360°
EXAMPLE [2] Finding an Angle Measure in a Quadrilateral
Find the unknown angle measure
in the quadrilateral.
98° + 137° + 52° + .V = 360°
287° + x = 360°
 287°  287°
A = 73°
The sum of the
angle measures is 3b0 .
Combine like terms.
Subtract 287° from both sides.
The imknowTi angle measure is 73°.
In a convex polygon, all diagonals can be drawn within the interior of
the figure. By dividing any convex polygon into triangles, you can find
the sum of its interior angle measures.
EXAMPLE [3] Drawing Triangles to Find the Sum of Interior Angles
Divide the polygon into triangles to find the sum of its
angle measures.
There are 5 triangles.
5 • 180° = 900°
The sum of the angle measures
of a heptagon is 900°.
TftiHk and Discuss
1. Explain how to find the measure of an angle in a triangle when
the measures of the two other angles are known.
2. Determine for which polygon the sum of the angle measures is
greater, a pentagon or an octagon.
3. Explain how the measure of each angle in a regular polygon
changes as the number of sides increases.
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88 Angles in Polygons 487
88
p^
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Exercises 118, 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°
MultiStep 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 44)
8 _ 24
P 27
32. ^ = 30
3 18
33.
34.
35.
0.5
Name the types of quadrilaterals that have each property. ( Lesson 87)
36. two pairs of opposite, congruent sides 37. four congruent sides
88 Angles in Polygons 489
To Go On?
.^pLearn It Online
t* RP<
Resources Online go.hrw.com,
IBBIWIIm^i RTGosB^ Go;
&
Quiz for Lessons 84 Through 88
84 ] 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 85 j Classifying Polygons
Name each polygon, and tell whether it is a regular polygon. If it is not,
explain why not.
6.
8.
(^ 86 ] Classifying Triangles
Classify each triangle according to its sides and angles.
9. P
10.
11.
(vj 87 j Classifying Quadrilaterals
Give all of the names that apply to each quadrilateral. Then give the name
that best describes it.
15.
16.
(v) 88 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 rhombusshaped 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
89
7.3.4
Recognize, describe, or exten
words, or symbols
Vocabulary
SideSideSide 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
81 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 SideSideSide 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 SideSideSide 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
89 Congruent Figures 493
i3.^^M3^
HonieworkHelpOnlinego.hrw.com, "
keyword BBbiWKflgM <S)
Exercises 114, 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 51 )
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°
89 Congruent Figures 495
81
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 Xaxis.
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 .vaxis, (.v, y) — » U'. .v)
In a reflection across the yaxis, (.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.
.vaxis
*y
f(3, 3)
G(1,4)
B yaxls
♦ y
xcoordinates —*■ same
xcoordinates — » opposites
ycoordinates — » opposites
ycoordinates — » 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 810 Translations, Reflections, and Rotations 497
EXAMPLE [4j Graphing Rotations on a Coordinate Plane
The point that a
figure rotates
around may be on
the figure or away
from the figure.
Triangle JKL has vertices J(0, 0), K{0, 3), and L(4, 3). Rotate
AJKL 90° counterclocl<wise about the origin. Write the
coordinates of the vertices of the image.
— , — ft~ "^ — [
The corresponding sides, JK and JK',
make a 90° angle.
Notice that vertex K Is 3 units below
the origin, and vertex K' is 3 units to
the right of the origin.
The coordinates of the vertices ofAjK'L' are/(0, 0), A" (3, 0),
and L' (3, 4).
Think and Discuss
1. Explain how a figure skater might perform a translation and
a rotation at the same time.
See Example 1 Identify each type of transformation.
** Hni
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Homework Help Online go.hrw.com,
keyword lAHMabl ®
Exercises 114, 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. yaxis
Ay
ky
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. yaxis 13. .vaxis
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
810 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 boxandwhisker plot for Exercises 19 and 20. (Lesson 75)
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 89)
21. S E 22. N
6 ir
4m \ — ■ ■•■/ 4m
A 3rr\ C
■"— ~"^"
500 Chapter 8 Geometric Figures
tfedjlaJSfaw
?^LAB7\ Explore Transformations
Use with Lesson 810
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 5sided 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°.
810 Technology Lab 501
LESSON 810
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'(42, 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 810 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 810 have line
symmetiy.
Vocabulary
line symmetry
line of symmetry
asymmetry
rotational symmetry
center of rotation
Many architects and artists use symmetry in their buildings. The structure
of the Puerta de Europa towers in Madrid, Spain, is symmetrical. You
can draw a line of symmetry between the towers.
When a figure is not symmetrical, it has asymmetry , or is asymmetrical.
EXAMPLE
Identifying Line Symmetry
Decide whether each figure has line symmetry. If it does, draw al
the lines of symmetry.
3 lines of symmetry
4 lines of symmetry
, EXAMPLE [2j Social Studies Application
Find all the lines of symmetry in each flag
A ^ ^ 1 B
There is 1 line of symmetry.
There are no lines of symmetry.
504 Chapter 8 Geometric Figures
Vjdaii Lesson Tutorials Online my.hrw.com
EXAMPLE
When you rotate a figure, you can create
a figure with rotational symmetiy.
A figure has rotational symmetry if,
when it is rotated less than 360° around
a central point, it coincides with itself.
The central point is called the
center of rotation.
If the stained glass window at right is
rotated 90°, as shown, the image looks
the same as the original stained glass
window. Therefore the window has
rotational symmetry.
[ 3 J Identifying Rotational Symmetry
i Tell how many times each figure will show rotational symmetry
within one full rotation.
Center of
rotation
The starfish will show
rotational symmetn,' 5 times
within a 360° rotation.
m
The pinwheel will show
rotational symmetry' 4 times
within a 360° rotation.
Draw lines from thie center of
tfie figure out tfiroughi identical
places in the figure.
Count the number
of lines drawn.
Draw lines from the center of
the figure out through identical
places in the figure.
Count the number
of lines drawn.
ThiHk and Discuss
1. Draw a figure that does not have rotational symmetry.
2. Determine whether an equilateral triangle has rotational
symmetry. If so, tell how many times it shows rotational symmetry
within one full rotation.
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811 Symmetry 505
•ruiiii
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Exercises 118, 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 410)
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 810)
29. Translate the triangle 4 units right and 2 units down.
30. Reflect the triangle across theyaxis.
811 Symmetry 507
Bl^ Create Tessellations
Use with Lessons 810 and 8 1 1
£?.,
Tessellations are patterns of identical shapes that completely
cover a plane with no gaps or overlaps. The artist M. C. Escher
created many fascinating tessellations.
Activity
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Q Create a translation tessellation.
The tessellation by M. C. Escher shown
at right is an example of a translation
tessellation. To create your own
translation tessellation, follow the
steps below.
a. Start by drawing a square, rectangle, or
other parallelogram. Replace one side
of the parallelogram with a curve,
as shown.
b. Translate the curve to the opposite side
of the parallelogram.
c. Repeat steps a and b for the other two
sides of your parallelogram.
The figure can be translated to create an
interlocking design, or tessellation. You can
add details to your figure or divide it into
two or more parts, as shown below.
508 Chapter 8 Geometric Figures
Q Create a rotation tessellation.
The tessellation by M. C. Escher shown
at right is an example of a rotation
tessellation. To create your own rotation
tessellation, follow the steps below.
a. Start with a regular hexagon. Replace one
side of the hexagon with a curve. Rotate the
curve about point B so that the endpoint at
point A is moved to point C.
b. Replace side CD with a new curve, and rotate
it about point D to replace side DE.
Replace side fFwith a new curve, and rotate it
about point f to replace side FA.
The figure can be rotated and fitted together with
copies of itself to create an interlocking design, or
tessellation. You can add details to your figure, if desired.
Think and Discuss
1. Explain why the two t\'pes of tessellations in this activity' are known as
translation and rotation tessellations.
Try Til is
1. Create your own design for a translation or rotation tessellation.
2. Cut out copies of your design from 1 and fit them together to fill
a space with your pattern.
8n HandsOn Lab 509
CHAPTER \
Ready To Go On?
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Quiz for Lessons 89 Through 811
(^ 89 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) 810] 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 xaxis.
6. Rotate triangle JKL 90°
clockwise about die origin.
*y
Q) 811] 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 27, 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 rhombusshaped tiles to build
a variety of polygons. Each side of a tile
is a different color. Build each design by
matching the samecolored 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
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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 righthand side. Turn the top lefthand corner
back and under to form a pocket. Figure B
^ Turn the whole thing over and fold the top righthand
corner back and under to form a pocket. Repeat steps
13 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
SideSideSide 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
81] Building Blocks of Geometry (pp. 448451)
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
82J Classifying Angles (pp. 454457)
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.
83] Line and Angle Relationships (pp. 460463)
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 1316, use the figure at left.
Find the measure of each angle.
14. A3
16. Z6
84J Properties of Circles (pp. 468471)
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
85J Classifying Polygons (pp. 474477)
■ 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
86] Classifying Triangles (pp 478481)
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
87 ] Classifying Quadrilaterals (pp. 482485)
■ 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.
88 j Angles in Polygons (pp. 485489)
■ 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.
89] Congruent Figures (pp. 492495)
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
810] Translations, Reflections, and Rotations (pp 496500)
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.
811] Symmetry (pp. 504507)
■ 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
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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.
MultipleChoice 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
extendedresponse 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 xcoordinate of point A' be after
the figure is reflected across the yaxis?
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 18 519
CHAPTER
TififoDimensional
Figures
9A Perimeter,
Circumference,
and Area
91 Accuracy and Precision
LAB Explore Perimeter and
Circumference
92 Perimeter and
Circumference
LAB Explore Area of
Polygons
93 Area of Parallelograms
94 Area of Triangles and
Trapezoids
LAB Compare Perimeter and
Area of Similar Figures
95 Area of Circles
96 Area of Irregular Figures
9B Using Squares and
Square Roots
LAB Explore Square Roots and
Perfect Squares
97 Squares and Square
Roots
EXT Identifying and Graphing
Irrational Numbers
LAB Explore the Pythagorean
Theorem
98 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
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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 foursided figure with opposite sides
that are congruent and parallel.
3. The ? of a circle is onehalf 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. 0.8221 17. 2.746.6 18. 409.54
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 +34.3
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: TwoDimensional 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
33 = 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: TwoDimensional 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
DoubleBar Graph
udent Enrollme
2,000
2004 2005 2006 2007
I Seventhgraders Year
I Eighthgraders
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
mjJiA' •
Read the title of the graph and any
special notes.
Blue indicates seventhgraders.
Purple indicates eighthgraders.
Read each axis label and note the
intervals of each scale.
.vaxis — year increases by 1.
yaxis — enrollment increases by
400 students.
Determine what information is presented,
student enrollment for seventh
and eighthgraders per year
"ny This
Look up each graphic in your textbook and answer the following questions.
1. Lesson 48 Exercise 1; Which side of the smaller triangle corresponds to BC?
Which angle corresponds to /LEDFl
2. Lesson 73 Example 1: By what interval does the .vaxis scale increase? About
how many people speak Hindi?
Measurement: TwoDimensional Figures 523
u
91
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[rIl]in 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: TwoDimensional 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.
42 = 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
91 Accuracy and Precision 525
91
iiii<«riiiiii
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Exercises 113, 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: TwoDimensional 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 2930, 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 85)
31.
33
<^
57 Accuracy and Precision 527
k^
Explore Perimeter &
Circumference
Use with Lesson 92
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
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Activity 1
O Cut a piece of string that is slightly longer than
18 inches. Tie the ends together to form an
18inch 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: TwoDimensional 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.
92 HandsOn Lab 529
92
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: TwoDimensional Figures
y'lddpl 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.
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92 Perimeter and Circumference 531
92
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Exercises 120, 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: TwoDimensional 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 KailuaKona to
Waimea to Hilo and back to KailuaKona.
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 65)
34. 18 is 20% of what number?
35. 78% of 65 is what number?
Choose the more precise measure in each pair. (Lesson 91
36. 4 ft, 1 yd 37. 2 cm, 21 mm
38. 5^ in., 5 in.
39. 37 g, 37.0 g
92 Perimeter and Circumference 533
Explore Area of Polygons
Use with Lessons 93, 94 and 95
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
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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_
„, — ^
 i0,^,^^tH~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: TwoDimensional 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.
93 HandsOn 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: TwoDimensional 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"^.
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93 Area of Parallelograms 537
93
H'
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Exercises 116, 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.
2cm
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: TwoDimensional 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 6footby8foot 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 82)
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
93 Area of Parallelograms 539
94
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: TwoDimensional 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 "bone"
or "b subone."
A = ^hib^ + bj
Use the formula.
5 in.
A^\4{\0 + 6)
.4 = i4{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
94 Area of Triangles and Trapezoids 541
94
[733333
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Exercises 114, 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: TwoDimensional 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. MultiStep 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 88)
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 93)
94 Area of Triangles and Trapezoids 543
Compare Perimeter and
Area of Similar Figures
Use with Lesson 94
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
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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: TwoDimensional 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.
94 HandsOn Lab 545
d
A circle can be cut into equalsized
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
onehalf 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 = nr
The order of
operations calls for
evaluating the
exponents before
multiplying.
EXAMPLE 1
]l3jjjaijjija,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: TwoDimensional 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
95 Area of Circles 547
Homework Help Online go.hrw.com,
keyword ■mianitJiM ®
Exercises 112, 13, 15, 17, 19, 21
G011IEKPRIWI£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 14inch 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 75mile 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: TwoDimensional 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 20story building. Each
turbine can produce 24 megawatthours 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 freethrow 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 12inch diameter.
It also offers a "mega" pizza with a 24inch 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 83)
31. mZl 32. mZ2 33. mZ3
135°
Graph the polygon with the given vertices. Identify the polygon and 3/
then find its area. (Lesson 94) '^
34. (1,1), (0,4), (4,1) 35. (3,3), (2,3), (1,1), (1,1)
a
b
95 Area of Circles 549
rea or irregular iigures
A composite figure is made up of simple geometric shapes, such
as triangles and rectangles. You can find the area of composite and
other irregular figures by separating them into nonoverlapping
familiar figures. The sum of the areas of these figures is the area of
the entire figure. You can also estimate the area of an irregular figure
by using graph paper.
EXAMPLE [lj Estimating the Area of an Irregular Figure
Vocabulary
composite figure
EXAMPLE
S3
The area of a
semicircle is 1 the
area of a circle.
A = l(7rr')
Estimate the area of the figure. Each square represents 1 ft^.
Count the number of filled or almost
filled squares: 35 yellow squares.
Count the number of squares that
are about halffilled: 6 blue squares.
Add the number of filled squares plus
I the number of halffilled squares:
35 + (^ 6)^ 35 + 3 = 38.
The area of the figure is about 38 ft".
12 m
Finding the Area of a Composite Figure
Find the area of the figure. Use 3.14 for x
Step 1: Separate the figure into
smaller, familiar figures.
Step 2: Find the area of each
smaller figure.
Area of the square:
A = s
A= 12" = 144
tL
12 m
Area of the semicircle:
.4«i(3.146)
^« ^(113.04) « 56.52
Step 3: Add the areas to find the total area.
>1 « 144 + 56.52 = 200.52
The area of the figure is about 200.52 m".
550 Chapter 9 Measurennent: TwoDimensional Figures
l/jdau Lesson Tutorials Online
EXAMPLE
C3
PROBLEM
SOLVING
Helpful Hint
There are often
several different
ways to separate an
Irregular figure into
familiar figures.
PROBLEM SOLVING APPLICATION
Chandra wants to carpet the floor of her
closet. A floor plan of the closet is shown at ^^
right. How much carpet does she need?
12ft
3ft
^* Understand the Problem
Rewrite the question as a statement:
• Find the amount of carpet needed to cover the floor of the closet.
List the important information:
• The floor of the closet is a composite figure.
• The amount of carpet needed is equal to the area of the floor.
Make a Plan
Find the area of the floor by separating the
figure into familiar figures: a rectangle and a 4 ^^
triangle. Then add the areas of the rectangle
and triangle to find the total area.
*ll Solve
Find the area of each smaller figure.
Area of the triangle:
12 ft
4 ft
3ft
Area of the rectangle:
A = ew
A =12 4:
A = 48 ft
A^^bh
A = U5){3 + 4)
A = U35) = 17.5 ft'
Add the areas to find the total area, yl = 48 + 17.5 = 65.5
Chandra needs 65.5 ft" of carpet.
EZ Look Back
The area of the closet floor must be greater than the area of the
rectangle (48 ft), so the answer is reasonable.
Thmk and Discuss
1. Describe two different ways to find the
area of the irregular figure at right.
2. Explain how dividing the figure into
two rectangles with a horizontal line
would affect its area and perimeter.
4 in.
J
1
2in^
2 in
L
r
2 in.
in.
/yJb'j Lesson Tutorials Online mv.hrw.com
96 Area of Irregular Figures 551
96
Ci3j'3J33:3
•LlMlI
Homework Help Online go.hrw.com,
keyword BBbiniig^M ®
Exercises 112, 13, 15, 17
GUIDED PRACTICE
See Example 1 Estimate the area of each figure. Each square represents 1 ft^.
1. '—r—.^ , 2.
See Example 2 Find the area of each figure. Use 3.14 for ;r.
3. \T
18 ft
10ft
10ft
n_
18 ft
12 m
5.
10ft
See Example 3
6. Luis has a model train set. The layout of the
track is shown at right. How much artificial
grass does Luis need in order to fill the
interior of the layout? Use 3.14 for n:.
1
1
f2ft
1
1
f
1
1
. n
1
n
4.5 ft
INDEPENDENT PRACTICE
See Example 1 Estimate the area of each figure. Each square represents 1 ft^
7. ^1 I I I i I ' I I 8.
See Example 2 Find the area of each figure. Use 3.14 for ;r.
10.
4m/ \ 4 m
4 m
8 ft
4 m
■\Z7
6 m
J L
3ft
4ft
n
L
4ft
3 ft
L
r
11.
3 ft
2 cm
3 cm
4 cm
4 cm
CJ"
5 cm
4 cm
5 cm
See Example 3 12. The figure shows the floor plan for a gallery of a
museum. The ceiling of the gallen>' is to be covered
with soundproofing material. How much material
is needed? Use 3.14 for /r.
552 Chapter 9 Measurement: TwoDimensional Figures
Extra Practice
See page EP24.
PRACTICE AND PROBLEM SOLVING
Find the area and perimeter of each ilgure. Use 3.14 tor k.
13. 3 ft 14. 5 m 4m 15.
3ft
4 ft
3 ft
2 ft
12 m
lOm,
8 m
16. Critical Thinking Will the area and perimeter change for the figure in
Exercise 14 if the triangle part is reflected to the left side? Explain.
17. Critical Thinking The figure at right is made up of
an isosceles triangle and a square. The perimeter of
the figure is 44 feet. What is the value of .v?
xft
@
18. MultiStep Afigurehasvertices/l(8, 5),B(4, 5),C(4, 2),D(3,2),
£(3, 2), F(6, 2), G(6, 4), and H(Q, 4). Graph the figure on a
coordinate plane. Then find the area and perimeter of the figure.
19. Choose a Strategy A figure is formed by combining a square and a triangle.
Its total area is 32.5 m". The area of the triangle is 7.5 m". What is the length of
each side of the square?
20. Write About It Describe how to find the area of the
composite figure at right.
10 cm
21. Challenge Find the area and
perimeter of the figure at right. ^ \ ( 1 8 cm
Use 3.14 for ;r.
12 in.
5 in.
m
Test Prep and Spiral Review
22. Multiple Choice A rectangle is formed by two congruent right triangles.
The area of each triangle is 6 in". Each side of the rectangle is a whole number
of inches. Which of these CANNOT be the perimeter of the rectangle?
CA) 26 in.
(X 24 in.
Cc:) 16 in.
CE) 14 in.
23. Extended Response The shaded area of the garden
represents a patch of carrots. Veronica estimates that she
will get about 12 carrots from this patch. Veronica is going to
plant the rest of her garden with carrots. Estimate the total
number of carrots she can expect to grow. Show your work.
Z.1 and Z.2 are complementary angles. Find mZZ. (Lesson 82)
24. mZl = 33° 25. mZl = 46° 26. mZl = 60°
Given the diameter, find the area of each circle to the nearest tenth.
Use 3.14 for X (Lesson 95)
28. <^ = 30m 29. rf = 5.5cm 30. r/=18in.
>
27. mZl = 25.5°
31. f/= lift
96 Area of Irregular Figures 553
CHAPTER
To Go On? „<Cf
Learn It Online
Resources Online go.hrw.com,
l^ !y) HTTrsiORTG09A IgI
Quiz for Lessons 91 Through 96
Q) 91 j Accuracy and Precision
Choose the more precise measurement in each pair.
1. 5 in. 2. 6c
56 ft
8floz
&
92 j Perimeter and Circumference
3. Find the perimeter of the figure at right.
4. If the circumference of a wheel is 94 cm, what is its
approximate diameter?
0.2 m
3.0 m
Qj 93 ] Area of Parallelograms
5. The area of a rectangular courtyard is 1,508 m'^, and the length is 52 m.
What is the width of the courtyard?
6. lackson's kitchen is 8 yd by 3 yd. Wliat is the area of his kitchen in square
feet?
^J 94 ] Area of Triangles and Trapezoids
7. Find the area of the trapezoid at right.
8. A triangle has an area of 45 cm' and a base of 12.5 cm.
What is the height of the triangle?
3 in.
8 in.
n
12 in.
(2f 95 ] Area of Circles
9. Find the area of the circle to the nearest tenth. Use 3.14 or ^ for tt.
10. The radius of a clock face is 8t in. Wliat is its area to the nearest
4
whole number?
Q) 96 I Area of Irregular Figures
Find the area of each figure to the nearest tenth if necessary. Use 3.14 for jr.
11. 21cm 12. /i:::^s^3ft 13. 9 yd
21 cm
rr
EL
_d
_xi
6 cm
13 ft
6 cm
7ft
■q
10 ft
11 yd
4ft
554 Chapter 9 Measurement: TwoDimensional Figures
X
Focus on Problem Solving
,^^^
"^
Understand the Problem
• Identify too much or too little information
Problems involving realworld situations sometimes give too much
or too little information. Before solving these types of problems,
you must decide what information is necessary' and whether you
have all the necessary information.
If the problem gives too much information, identify which of the
facts are really needed to solve the problem. If the problem gives
too little information, determine what additional information is
required to solve the problem.
Copy each problem and underline the information you need to solve
it. If necessary information is missing, write down what additional
information is required.
Q Mrs. Wong wants to put a fence around
her garden. One side of her garden
measures 8 feet. Another side measures
5 feet. What length of fencing does Mrs.
Wong need to enclose her garden?
Q Two sides of a triangle measure 17 inches
and 13 inches. The perimeter of the triangle
is 45 inches. What is the length in feet of the
third side of the triangle? (There are
12 inches in 1 foot.)
O During swim practice, Peggy swims 2
laps each of freests'le and backstroke. The
dimensions of the pool are 25 meters by
50 meters. What is the area of the pool?
O Each afternoon, Molly walks her dog
two times around the park. The park is a
rectangle that is 315 yards long. How far
does Molly walk her dog each afternoon?
Q A trapezoid has bases that measure
12 meters and 18 meters and one side that
measures 9 meters. The trapezoid has no
right angles. What is the area of the
trapezoid?
Focus on Problem Solving 555
Explore Square Roots and
Perfect Squares
Use with Lesson 97
tan lat
Learn It Online
Lab Resources Online go.hrw.com,
lUBiffifMSl Lab9 KGoJI
You can use geometric models such as tiles or graph paper to represent
squares and square roots.
Activity 1
O Copy the three square arrangements below on graph paper.
Continue the pattern until you have drawn 10 square
arrangements.
D
Q Copy and complete the table below. In the first column, write the
number of small squares in each figure you drew. To complete the
second column, use a calculator to find the square root.
(To find the square root of 4, press
SI 4m B
Total Number of Small Squares
Square Root
1
1
4
2
9
3
Q Shade in one column of each square arrangement that you drew in O
D
556 Chapter 9 Measurement: TwoDimensional Figures
Think and Discuss
1. How does the square root relate to the total number of small squares
in a figure?
2. How does the square root in the table relate to the shaded portion of
each figure?
Try This
Use graph paper to find each square root.
1. 121 2. 144
3. 196
Activity 2
Follow the steps below to estimate V 14.
O On graph paper, use one color to draw the smallest possible
square arrangement using at least 14 small squares.
O On the same arrangement, draw the largest possible square
arrangement using less than 14 small squares.
Q Count the number of squares in each arrangement. Notice that
14 is between these numbers.
Number in small arrangement Number in large arrangement
9 < 14 < 16
O Use a calculator to find \ 14 to the nearest tenth. \ 14 = 3.7. Use
inequality symbols to compare the square roots of 9, 14, and 16.
V9 <V14 <
3<3.7<4
'16
The square root of 9 is less than the square root
of 14, which is less than the sauare root of 16.
Q Use dashed lines on the figure to sketch a square that is 3.7 units
on each side.
M
t !
H ^BCjISill
i

L
:
l"^3.7r
Thinic and Discuss
1. Describe how to use two numbers to estimate the square roots of
nonperfect squares without using a calculator.
2. Explain how you can use graph paper to estimate \ 19.
3. Name three numbers that have square roots between 5 and 6.
Try This
Use graph paper to estimate each square root. Then use a calculator to
find the square root to the nearest tenth.
1.
'19
2. VIO
^^28
^35
97 HandsOn Lab 557
97
7.1.5 Recognize and use the inverse relationship between squaring and finding the
square root of a perfect square integer
A square with sides that measure
3 units each has an area of 3 • 3,
or 3". Notice that the area of the square
is represented by a power in which the
base is the side length and the
exponent is 2. A power in whicli the
exponent is 2 is called a square.
Base
Exponent
EXAMPLE [lj Finding Squares of Numbers
Vocabulary ^'"'^ ^^'^ ^^"^''^
perfect square
square root
radical sign
A
Method 1
: Use a model.
,
.
4 = (w
\ = 6 • 6
A = 36
The square of
6 is 36.
B 14^
Method 2: Use a calculator.
Press 14
14^= 196
The square of 14 is 196.
k!"S;]
V 16 = 4 is read as
"The square root of
16 is4."
A perfect square is the square of a whole number. The number 36 is a
perfect square because 36 = 6" and 6 is a whole number.
The square root of a number is one of the two
equal factors of the number. Four is a square
root of 16 because 4 • 4 = 16. The symbol for a
square root is V , which is called a radical sign
EXAMPLE r 2 J Finding Square Roots of Perfect Squares
Find each square root.
A V 64
Method 1: Use a model.
The square root of 64 is 8.
558 Chapter 9 Measurement: TwoDimensional Figures [VJii^J^i Lesson Tutorials Online
Find each square root.
B V324
Method 2: Use a calculator. Press
324
V324 = 18
The square root of 324 is 18.
You can use perfect squares to estimate the square roots of
nonperfect squares.
EXAMPLE r3j Estimating Square Roots
Estimate \ 30 to the nearest whole number. Use a calculator to
check your answer.
List some perfect squares.
Find the perfect squares nearest 30.
1,4,9, 16,25,36, .. .
25 < 30 < 36
\ 25 < \ 30 < \ 36
5< \ 30< 6
\ 30 « 5
Check
\ 30 «= S.'l I I lc'SSIS Use a calculator to approximate \ 30
5 is a reasonable estimate.
Find ttie square roots of 25 and 36.
30 is closer to 25 than to 36.
EXAMPLE MlJ Recreation Application
While searching for a lost hiker,
a helicopter pilot covers a square
area of 150 mi^ What is the
approximate length of each side
of the square area? Round your
answer to the nearest mile.
The length of each side of the square is vTSO.
144 < 150 < 169 Find the perfect squares nearest 150.
\ 144 < \ T50 < \ T69
12 < \ T50 < 13 Find the square roots of 144 and 169.
\/T50 « 12 150 is closer to 144 than to 169.
Each side of the search area is about 12 miles long.
Think and Discuss
1. Explain how to estimate \/75.
2. Explain how you might find the square root of 3'.
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97 Squares and Square Roots 559
97
^il£)3£l
GUIDED PRACTICE
See Example 1 Find each square.
1. 4 2. 17'
See Example 2 Find each square root.
L 5. V400 6. \ 9
3. 9^
'144
iiiorniiiiii
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Exercises 126, 29, 31, 37, 41,
47,49,51
4. 15
8. \'529
See Example 3 Estimate each square root to the nearest whole number. Use a calculator to
check your answer.
9. V20 10. V45 11. Vm 12. V58
See Example 4 13. A Coast Guard ship patrols an area of 125 square miles. The area the ship
I patrols is a square. About how long is each side of the area? Round your
L answer to the nearest mile.
INDEPENDENT PRACTICE
See Example 1 Find each square.
L 14. 3 15. 16
See Example 2 Find each square root.
L 18. VSel 19. \ 16
16. 8
20. V1