California Math Triumphs Vol 3B RATIOS (CALIFORNIA MATH TRIUMPHS VOL 3B)

Authors Basich Whitney • Brown • Dawson • Gonsalves • Silbey • Vielhaber

Photo Credits Cover Peter Sterling/Getty Images; iv (tl bl br) File Photo, (tc tr) The McGraw-Hill Companies, (cl c) Doug Martin, (cr) Aaron Haupt; v (1 2 3 4 6 7 8 9 11 12) The McGraw-Hill Companies, (5 10 13 14) File Photo; vii Digital Vision/PunchStock; viii CORBIS; ix Larry Brownstein/Getty Images; x CORBIS; 2–3 Michael A. Keller/ CORBIS; 27 CORBIS; 36 Rachel Epstein/PhotoEdit; 44 Getty Images; 51 Stockdisc/SuperStock; 57 Darwin Wiggett/Getty Images; 58 David Buffington/ Getty Images; 60 Darren McCollester/Getty Images; 67 Brand X Pictures/ PunchStock; 76 Ryan McVay/Getty Images; 81 Getty Images

Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240-4027 ISBN: 978-0-07-878208 MHID: 0-07-878208-2 Printed in the United States of America. 1 2 3 4 5 6 7 8 9 10 055/027 16 15 14 13 12 11 10 09 08 07

California Math Triumphs Volume 3B

California Math Triumphs Volume 1 Place Value and Basic Number Skills 1A Chapter 1 Counting 1A Chapter 2 Place Value 1A Chapter 3 Addition and Subtraction 1B Chapter 4 Multiplication 1B Chapter 5 Division 1B Chapter 6 Integers Volume 2 Fractions and Decimals 2A Chapter 1 Parts of a Whole 2A Chapter 2 Equivalence of Fractions 2B Chapter 3 Operations with Fractions 2B Chapter 4 Positive and Negative Fractions and Decimals

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Volume 3 Ratios, Rates, and Percents 3A Chapter 1 Ratios and Rates 3A Chapter 2 Percents, Fractions, and Decimals 3B Chapter 3 Using Percents 3B Chapter 4 Rates and Proportional Reasoning Volume 4 The Core Processes of Mathematics 4A Chapter 1 Operations and Equality 4A Chapter 2 Math Fundamentals 4B Chapter 3 Math Expressions 4B Chapter 4 Linear Equations 4B Chapter 5 Inequalities Volume 5 Functions and Equations 5A Chapter 1 Patterns and Relationships 5A Chapter 2 Graphing 5B Chapter 3 Proportional Relationships 5B Chapter 4 The Relationship Between Graphs and Functions Volume 6 Measurement 6A Chapter 1 How Measurements Are Made 6A Chapter 2 Length and Area in the Real World 6B Chapter 3 Exact Measures in Geometry 6B Chapter 4 Angles and Circles iii

Authors and Consultants AUTHORS

Frances Basich Whitney

Kathleen M. Brown

Dixie Dawson

Project Director, Mathematics K–12 Santa Cruz County Office of Education Capitola, California

Math Curriculum Staff Developer Washington Middle School Long Beach, California

Math Curriculum Leader Long Beach Unified Long Beach, California

Philip Gonsalves

Robyn Silbey

Kathy Vielhaber

Mathematics Coordinator Alameda County Office of Education Hayward, California

Math Specialist Montgomery County Public Schools Gaithersburg, Maryland

Mathematics Consultant St. Louis, Missouri

Viken Hovsepian Professor of Mathematics Rio Hondo College Whittier, California

Dinah Zike Educational Consultant, Dinah-Might Activities, Inc. San Antonio, Texas

CONSULTANTS Assessment Donna M. Kopenski, Ed.D. Math Coordinator K–5 City Heights Educational Collaborative San Diego, California

iv

Instructional Planning and Support

ELL Support and Vocabulary

Beatrice Luchin

ReLeah Cossett Lent

Mathematics Consultant League City, Texas

Author/Educational Consultant Alford, Florida

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CONTRIBUTING AUTHORS

California Advisory Board CALIFORNIA ADVISORY BOARD

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Glencoe wishes to thank the following professionals for their invaluable feedback during the development of the program. They reviewed the table of contents, the prototype of the Student Study Guide, the prototype of the Teacher Wraparound Edition, and the professional development plan.

Linda Anderson

Cheryl L. Avalos

Bonnie Awes

Kathleen M. Brown

4th/5th Grade Teacher Oliveira Elementary School, Fremont, California

Mathematics Consultant Retired Teacher Hacienda Heights, California

Teacher, 6th Grade Math Monroe Clark Middle School San Diego, California

Math Curriculum Staff Developer Washington Middle School Long Beach, California

Carol Cronk

Audrey M. Day

Jill Fetters

Grant A. Fraser, Ph.D.

Mathematics Program Specialist San Bernardino City Unified School District San Bernardino, California

Classroom Teacher Rosa Parks Elementary School San Diego, California

Math Teacher Tevis Jr. High School Bakersfield, California

Professor of Mathematics California State University, Los Angeles Los Angeles, California

Eric Kimmel

Donna M. Kopenski, Ed.D.

Michael A. Pease

Chuck Podhorsky, Ph.D.

Mathematics Department Chair Frontier High School Bakersfield, California

Math Coordinator K–5 City Heights Educational Collaborative San Diego, California

Instructional Math Coach Aspire Public Schools Oakland, California

Math Director City Heights Educational Collaborative San Diego, California

Arthur K. Wayman, Ph.D.

Frances Basich Whitney

Mario Borrayo

Melissa Bray

Professor Emeritus California State University, Long Beach Long Beach, California

Project Director, Mathematics K–12 Santa Cruz County Office of Education Capitola, CA

Teacher Rosa Parks Elementary San Diego, California

K–8 Math Resource Teacher Modesto City Schools Modesto, California

v

California Reviewers CALIFORNIA REVIEWERS Each California Reviewer reviewed at least two chapters of the Student Study Guides, providing feedback and suggestions for improving the effectiveness of the mathematics instruction. Melody McGuire

Math Teacher California College Preparatory Academy Oakland, California

6th and 7th Grade Math Teacher McKinleyville Middle School McKinleyville, California

Eppie Leamy Chung

Monica S. Patterson

Teacher Modesto City Schools Modesto, California

Educator Aspire Public Schools Modesto, California

Judy Descoteaux

Rechelle Pearlman

Mathematics Teacher Thornton Junior High School Fremont, California

4th Grade Teacher Wanda Hirsch Elementary School Tracy, California

Paul J. Fogarty

Armida Picon

Mathematics Lead Aspire Public Schools Modesto, California

5th Grade Teacher Mineral King School Visalia, California

Lisa Majarian

Anthony J. Solina

Classroom Teacher Cottonwood Creek Elementary Visalia, California

Lead Educator Aspire Public Schools Stockton, California

vi

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Bobbi Anne Barnowsky

Volume 3A

Ratios, Rates, and Percents

Chapter

Ratios and Rates

1

1-1 Ratios ..................................................................................4. 6NS1.2

1-2 Rates and Unit Costs ......................................................11 3NS2.7, 6AF2.2

Progress Check 1 .............................................................18 1-3 Probability as a Ratio......................................................19 6SDAP3.3

Assessment Study Guide .....................................................................26 Chapter Test .....................................................................28 Standards Practice ...................................................30

Standards Addressed in This Chapter 3NS2.7 Determine the unit cost when given the total cost and number of units. 6NS1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative size of two quantities, using appropriate notations (a/b, a to b, a:b). 6AF2.2 Demonstrate an understanding that rate is a measure of one quantity per unit value of another quantity. 6SDAP3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that the probabilities computed are reasonable; know that if P is the probability of an event, 1 - P is the probability of an event not occurring.

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Joshua Tree National Park

Chapters 1 and 2 are contained in Volume 3A. Chapters 3 and 4 are contained in Volume 3B.

vii

Contents Chapter

Percents, Fractions, and Decimals

2

Standards Addressed in This Chapter 2-1 Introduction to Percents ................................................34 5NS1.2

2-2 Percents, Fractions, and Decimals ................................41 5NS1.2

Progress Check 1 .............................................................48 2-3 Compare Data Sets of Different Sizes...........................49 5SDAP1.3, 6NS1.2

Assessment Study Guide .....................................................................56 Chapter Test .....................................................................58

5NS1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number. 5SDAP1.3 Use fractions and percentages to compare data sets of different sizes. 6NS1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative size of two quantities, using appropriate notations (a/b, a to b, a:b).

Standards Practice ...................................................60 Merced River near Yosemite National Park

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

viii

Contents Chapter

Using Percents

3

3-1 Calculate Percents ............................................................4 5NS1.2, 6NS1.4

3-2 Solve Percent Problems ..................................................11 6NS1.3, 6NS1.4, 7NS1.7

Progress Check 1 .............................................................20 3-3 Interest Problems.............................................................21 6NS1.4, 7NS1.7

3-4 Percent of Change .......................................................... 29 7NS1.6, 7NS1.7

Progress Check 2 .............................................................36 Assessment Study Guide .....................................................................37 Chapter Test .....................................................................40 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Standards Practice ...................................................42 Manhattan Beach Pier

Chapters 1 and 2 are contained in Volume 3A. Chapters 3 and 4 are contained in Volume 3B.

Standards Addressed in This Chapter 5NS1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number. 6NS1.3 Use proportions to solve problems (e.g., determine the value of N 4 N if __ = ___, find the length of a side of 7 21 a polygon simiular to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. 6NS1.4 Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips. 7NS1.6 Calculate the percentage of increases and decreases of a quantity. 7NS1.7 Solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest.

ix

Contents Chapter

Rates and Proportional Reasoning

4

Standards Addressed in This Chapter 4-1 Proportions ......................................................................46 6NS1.3

4-2 Unit Conversions ............................................................53 3AF1.4, 3MG1.4, 6AF2.1

Progress Check 1.............................................................60 4-3 Solve Rate Problems .......................................................61 3AF2.1, 3AF2.2, 6AF2.3

4-4 Solve Problems Using Proportions ............................. 69 3AF2.1, 6NS1.3, 7AF4.2

Progress Check 2.............................................................76 Assessment Study Guide ....................................................................77 Chapter Test ....................................................................80 Standards Practice...................................................82

3AF2.1 Solve simple problems involving a functional relationship between two quantities (e.g., find the total cost of multiple items given the cost per unit). 3AF2.2 Extend and recognize a linear pattern by its rules (e.g., the number of legs on a given number of horses may be calculated by counting by 4s or by multiplying the number of horses by 4). 3MG1.4 Carry out simple unit conversions within a system of measurement (e.g., centimeters and meters, hours and minutes). 6NS1.3 Use proportions to solve problems (e.g., determine the value of N 4 N if __ = ___, find the length of a side of 7 21 a polygon simiular to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. 6AF2.1 Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches). 6AF2.3 Solve problems involving rates, average speed, distance, and time. 7AF4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation.

x

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Reconstructed house in a restored Hoopa Valley Tribe village, Humboldt County

3AF1.4 Express simple unit conversions in symbolic form (e.g., ___ inches = ___ feet × 12).

R E G N E V A SC HUNT Let’s Get Started Use the Scavenger Hunt below to learn where things are located in each chapter. 1 What is the title of Chapter 4?

Rates and Proportional Reasoning 2

What is the Key Concept of Lesson 4-4?

Solve Problems Using Proportions 3

In Example 1 on page 12, how much did Theresa’s coat cost? $42

4

What are the vocabulary words for Lesson 4-3?

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

unit cost, unit rate, proportion 5

How many Examples are presented in Lesson 3-4? 3

6

What are the California Standards covered in Lesson 4-3?

3AF2.1, 3AF2.2, 6AF2.3 7

From Lesson 3-3, write the formula for simple interest. I = prt

8

How many problems are in the Standard Practice on pages 82–83? 12

9

On what pages will you find the Study Guide for Chapter 3?

pages 37–39 10

In Chapter 3, find the logo and Internet address that tells you where you can take the Online Readiness Quiz. It is

found on page 3. The URL is ca.mathtriumphs.com.

1

Chapter

3

Using Percents Suppose that you want to buy a guitar. The guitar you want is $200. You don’t have enough money to buy it. Finally, the store marks the guitar at a 60% discount. You can use percents to decide if you can buy the guitar.

Copyright © by The McGraw-Hill Companies, Inc.

2

Chapter 3 Using Percents

Michael A. Keller/Corbis

STEP

STEP

1 Quiz

Are you ready for Chapter 3? Take the Online Readiness Quiz at ca.mathtriumphs.com to find out.

2 Preview

Get ready for Chapter 3. Review these skills and compare them with what you’ll learn in this chapter.

What You Know

What You Will Learn

You know how fractions, decimals, and percents are related.

Lesson 3-1

Example: 20 students out of 100 students 20 ratio ____ 100

decimal 0.2

percent 20%

TRY IT 1

Write 0.52 as a percent.

2

What percent of the letters in test are e’s?

52%

Percents can be less than 1% or more than 100%. A percent that is less than 1% is written as a fraction or decimal. A percent more than 100% contains more than 1 whole. Examples:

_1% 4

0.5%

_

33 1 % 3

65%

118%

Copyright © by The McGraw-Hill Companies, Inc.

25%

3

Write 45% as a decimal.

4

1 as a percent. Write __ 50% 2

0.45

You know how to multiply three numbers.

Lesson 3-2

5

3 × 900 × 0.02 =

54

When you have a savings account, the bank pays you money. This payment is called interest . To calculate simple interest, follow this formula:

6

500 × 0.05 × 2 =

50

I = prt

7

0.04 × 200 × 4 =

32

8

300 × 5 × 0.03 =

45

interest = principal × interest rate × time (years)

TRY IT

or

Example: Investment of $1,000 at 5% for 6 years I = 1,000 · 0.05 · 6 I = $300

3

Lesson

3-1 Calculate Percents KEY Concept Percents can be less than 1% or more than 100%. A percent that is less than 1% is written as a fraction or decimal. A percent more than 100% is expressed with a number greater than 100. Use the equation below to solve percent problems. Express the percent as a decimal or a fraction before you multiply or divide. 5IF
UFSN
CBTF
DBO
CF VTFE
JO
QMBDF
PG
UIF XPSE
XIPMF

Percent Equation:

percent · whole = part

What is 35% of 180?

0.35

·

180

=

n

Ninety percent of what number is 45?

0.90

·

n

=

45

n

·

300

=

60

What percent of 300 is 60?

5NS1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number. 6NS1.4 Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips.

VOCABULARY percent a ratio that compares a number to 100; it uses the symbol % The word percent means hundredths or out of 100. (Lesson 2-1, p. 34)

ratio a comparison of two numbers by division Example: The ratio of 2 to 3 can be stated as 2 out 2 of 3, 2 to 3, 2:3, or __. 3 (Lesson 1-1, p. 4)

Example 1 15% of what number is 24? 1. Write the percent equation. 2. Substitute the given numbers and a variable (unknown number) into the equation. Write 15% as a decimal. 3. Solve for the variable. 4. Check your answer. 0.15 · 160 = 24✓

4

Chapter 3 Using Percents

percent · whole = part 0.15

·

= 24

n

0.15n = _ 24 _ 0.15

0.15

n = 160

Copyright © by The McGraw-Hill Companies, Inc.

You can use the percent equation to find the part, the whole, or the percent.

variable a letter or symbol used to represent an unknown quantity

YOUR TURN! 28% of what number is 182? 1. Write the percent equation. 2. Substitute the known numbers and variable into the equation.

percent

·

whole

=

part

0.28

·

n

0.28n _

=

182

0.28

=

n

=

3. Solve for the variable. 4. Check your answer.

0.28

·

650

=

182

182 _ 0.28 650

✓

Example 2 What is 35% of 1,050? percent · whole = part

1. Write the percent equation. 2. Substitute the known numbers and variable into the equation.

0.35

· 1,050 = n 367.5 = n

3. Solve for the variable.

Copyright © by The McGraw-Hill Companies, Inc.

4. Check your answer. 0.35 · 1,050 = 367.5 ✓ YOUR TURN! What is 12% of 380? 1. Write the percent equation. 2. Substitute the known numbers and variable into the equation. 3. Solve for the variable.

percent

·

whole

=

part

0.12

·

380

=

n

45.6

=

n

4. Check your answer.

0.12

·

380

=

45.6

✓

GO ON Lesson 3-1 Calculate Percents

5

Example 3 What percent of 500 is 125? percent · whole n · 500 500n 500 n

1. Write the percent equation. 2. Substitute the given numbers and a variable into the equation.

= part = 125 = 125 500 = 0.25

_ _

3. Solve for the variable.

0.25 = 25%

4. Change the decimal to a percent by multiplying by 100. 5. Check your answer. 0.25 · 500 = 125 ✓ YOUR TURN! What percent of 750 is 600? 1. Write the percent equation.

percent

·

n

·

whole

=

part

3. Solve for the variable.

750 750n 750 n

4. Change the decimal to a percent by multiplying by 100.

=

600 600 750 0.80

0.80

=

80%

2. Substitute the given numbers and a variable into the equation.

_

=

_

5. Check your answer. ·

750

=

600

✓

Who is Correct? What percent of 440 is 132?

Dennis n ·· 132 = 440 n = 3.33 3.33 = 33%

Rena n · 440 = 132 n = 0.3 0.3 = 30%

Lester

n · 440 = 132 _ 100 440n = 132 _

100 4.4n = 132 n = 30 30 = 30% 100

_

Circle correct answer(s). Cross out incorrect answer(s). 6

Chapter 3 Using Percents

Copyright © by The McGraw-Hill Companies, Inc.

0.80

Remember that to write a percent as a decimal, you divide by 100.

Guided Practice Write each percent as a decimal. 1

76%

3

12.5%

0.76 0.125

2

1% __

4

114%

2

0.005 1.14

Step by Step Practice Solve using the percent equation. Check your answer. 5

What percent of 2,000 is 4? Step 1 Write the percent equation.

percent ·

whole

=

part

Step 2 Substitute the given numbers and a variable into the equation.

n

·

2,000

=

4

Step 3 Solve for the variable.

2,000n = 2,000n = 2,000 n = 0.002

4 4 2,000

Copyright © by The McGraw-Hill Companies, Inc.

_

_

5IJT
NFBOT
NPWF UIF
EFDJNBM
QPJOU
UP UIF
SJHIU
UXP
QMBDFT

Step 4 Change the decimal to a percent by multiplying by 100.

0.2%

Step 5 Check your answer.

0.002

2,000

·

=

4

Solve using the percent equation. Check each answer. 6

What is 140% of 60? Check: 1.4 ·

7

60

=

20% of what number is 55?

84 275

8

percent

·

whole

=

part

1.40

·

60

=

n

84

=n

40% of what number is 130?

325 GO ON

Lesson 3-1 Calculate Percents

7

Solve using the percent equation. Check each answer. 49 9 What is 175% of 28? 10 What is 45% of 80? 11

What percent of 3,000 is 12?

0.4%

What percent of 8,000 is 16?

12

Step by Step Problem-Solving Practice ASTRONOMY On Mars an object weighs 38% as much as it weighs on Earth. How much would a person who weighs 150 pounds on Earth weigh on Mars? Understand

Solve

Use a table. Look for a pattern. Guess and check. ✓ Solve a simpler problem. Act it out.

Read the problem. Write what you know. On Mars something weighs does on Earth.

Plan

38%

of what it

Pick a strategy. One strategy is to solve a simpler problem. Divide 38% into percents that are easier to compute. 38% = 30% + 5% + 3%

15 pounds 30%: What is 10% of 150 pounds? 45 Multiply this by 3 because 30 = 10 · 3. 45 So, 30% of 150 is .

3%: What is 1% of 150 pounds? 4.5 Multiply this by 3. 1.5 So, 3% of 150 is 3 ·

1.5 pounds

, or

4.5

.

38%: Add the smaller percents to find 38% of 150 pounds. 30% + 5% + 3% = 45 + 7.5 + 4.5 57 pounds =

8

You can check your answer using estimation. 1 of 150 is 50. So, the 1 . __ 38% is close to 33% or __ 3 3 answer is reasonable.

Chapter 3 Using Percents

Copyright © by The McGraw-Hill Companies, Inc.

15 5%: 10% of 150 is . Divide the amount of 10% in half to find 5%. 15 7.5 ÷2= 1 15 7.5 . So, 5% of 150 is __ · , or 2

Check

0.2%

Problem-Solving Strategies

Solve. 13

36

14

15

TENNIS In the city of Bridgeport, 75% of the parks have tennis courts. If 18 parks have tennis courts, how many parks does 24 parks Bridgeport have altogether? Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

SCHOOL There are 175 students in seventh grade at Silverado Middle School. A survey shows that 84% of them plan to volunteer during the summer. How many students plan to volunteer?

147 students Write the percent equation in three different ways: 1. when n is the part 2. when n is the whole 3. when n is the percent. Explain when to use each form.

16

Copyright © by The McGraw-Hill Companies, Inc.

See TWE margin.

Skills, Concepts, and Problem Solving Write each percent as a decimal. 17

33%

0.33

18

3 __ % 4

0.0075

19

1.8

180%

Solve using the percent equation. Check each answer. 20

2% of what number is 5?

22

What is 110% of 60?

24

What is 250% of 40?

26

250

21

6% of what number is 21?

66

23

What is 175% of 28?

49

100

25

What is 120% of 55?

66

What percent of 2,000 is 14?

27

What percent of 84 is 63?

7% 0.7% or _ 10

75%

350

GO ON

Lesson 3-1 Calculate Percents

9

Solve. 28

CHESS The chess club has 60 members. Twenty-four of the members are younger than 20 years old. What percent of the total number of members are younger than 20?

40% 29

EATING OUT Trevor and Marina’s restaurant bill came to $36. They plan to leave a 20% tip. How much should they leave?

$7.20 30

SPORTS In a recent season, the Los Angeles Angels won 95 games and lost 67 games. What percent of games played did the Angels win? Round to the nearest tenth if necessary.

58.6% Vocabulary Check Write the vocabulary word that completes each sentence.

ratio

31

A(n)

is a comparison of two quantities by division.

32

A ratio that compares a number to 100 is a(n) percent .

33

Writing in Math Write out how you would calculate a 15% tip.

Answers may vary. First calculate 10%. Then calculate 5%. Finally, add the two values together.

The letters of the words United States are placed in a bag. Find the following probabilities. (Lesson 1-3, p. 19) 34 35 36

Pulling out the letter t

_1, or 17%

6 5 , or 42% _ Pulling out a vowel 12 _5, or 83% Not pulling out the letter e 6

Solve.

(Lesson 1-2, p. 11; Lesson 2-1, p. 34)

37

FASHION Ruby can get three hats for a total cost of $45. At that $75 rate, how much would five hats cost?

38

BUSINESS The sales tax rate is 6%. Write this percent as a fraction 3 in simplest form.

10

Chapter 3 Using Percents

_ 50

Copyright © by The McGraw-Hill Companies, Inc.

Spiral Review

Lesson

3-2 Solve Percent Problems KEY Concept A proportion is an equation stating that two ratios are equal. In a proportion, a cross product is the product of the numerator of one ratio and the denominator of the other ratio. Finding both cross products is called cross multiplying. In this proportion, a common denominator of the two fractions is 4 · 5. Multiply both sides of the proportion by 4 · 5.

2.4 __ 3 ___ =

5 4 2.4 3 (4/. · 5) ___ = __ (4 · 5/) 4/ 5 /

On the left side, you can cancel the 4s. On the right side, you can cancel the 5s.

5 · 2.4 = 3 · 4

Notice that 5 · 2.4 and 3 · 4 are the same as the two cross products of the original proportion. This demonstrates how cross products of a proportion are equal. Cross multiply to find the unknown value in a proportion. 5 ____ = 100

n ___ 40

Copyright © by The McGraw-Hill Companies, Inc.

100 2

=

100n _____ 100

VOCABULARY proportion an equation stating that two ratios or rates are equivalent 28 7 Example: __ = ___ 8 32 percent a ratio that compares a number to 100; it uses the symbol % The word percent means hundredths or out of 100. (Lesson 2-1, p. 34)

ratio a comparison of two numbers by division

200 = 100n 200 ____ =

6NS1.3 Use proportions to solve problems. Use crossmultiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. 6NS1.4 Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips. 7NS1.7 Solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest.

Divide each side by 100.

(Lesson 1-1, p.4)

n

You can write the percent equation as a proportion and cross multiply to solve. percent × whole = part The percent is always the ratio of a number compared to 100.

Substitute any two of the three and solve for the unknown.

percent _ part _ = 100

commission the amount of money a salesperson earns based on a percent of sales cross product in a proportion, a cross product is the product of the numerator of one ratio and the denominator of the other ratio

whole

This is a proportion because two ratios are equal. The percent is one ratio, and the part-to-whole is the other ratio.

GO ON Lesson 3-2 Solve Percent Problems

11

Percents are used in discounts, tips, taxes, commissions, and circle graphs.

Example 1

YOUR TURN!

Theresa bought a coat that cost $42. She paid 6% sales tax. How much did she pay in tax?

Mr. Torres wants to tip 20% of the restaurant bill. If his bill came to $34, how much should he leave as a tip?

1. Write the percent proportion.

1. Write the percent proportion.

percent _ part _ =

percent _ part _ =

whole

100

2. Substitute what you know into the proportion. Let x = the amount of the tax. tax rate

_

_

⎧ 6 = x ⎨ 100 42 ⎩

amount of tax price of coat

2. Substitute the known numbers into the the tip amount . proportion. Let x =

20 = _____ x _____ 100

3. Cross multiply and solve for x. 6(42) = 100x 252 = 100x 2.52 = x

:PV
DPVME
BMTP
VTF UIF
QFSDFOU
FRVBUJPO

whole

100

34

3. Cross multiply and solve for x. 3FNFNCFS
UP
EJWJEF CZ

NPWF
UIF
EFDJNBM QPJOU
MFGU
UXP
QMBDFT

20

4. Theresa paid $2.52 in tax.

( 34 ) = 100x 680 = 100x 6.80 = x

4. Mr Torres should leave a tip.

$6.80

Tyler earns a 5% commission on sales of computer games. Last week he earned $25 in commission. How much did Tyler sell last week? 1. Write the percent equation. 2. The part is

percent · whole = part

the amount earned in commission

The whole is

the amount of sales

Substitute the known numbers into the equation. Let n = the amount of sales 3. Solve for Tyler’s amount of sales. 4. Tyler sold $500 worth of computer games last week.

12

Chapter 3 Using Percents

You could also use a percent proportion.

. . 0.05 · n = 25 0.05n = 25 0.05 0.05 n = 500

_ _

Copyright © by The McGraw-Hill Companies, Inc.

Example 2

as

YOUR TURN! Mrs. Halen earns 3.5% on sales of furniture. Last week she earned $630 in commission. How much did Mrs. Halen sell last week? 1. Write the percent equation. 2. The part is The whole is

percent · whole = part

the amount earned in commission

.

the amount of sales

.

Substitute the known numbers into the equation.

0.035 · n = 630 630 0.035n 3. Solve for Mrs. Halen’s sales. _______ = ______

0.035 n = 18,000

0.035

4. Mrs. Halen sold $18,000 worth of furniture last week.

Example 3 A skateboard that normally sells for $129 is on sale for $90.30. What is the percent of the discount? 1. Find the amount of the discount. $129 - $90.30 = $38.70 38.70 = n · 129 38.70 = 129n 129 129

Copyright © by The McGraw-Hill Companies, Inc.

2. Use the percent equation to find the percent of discount.

_ _

The percent of discount is about 30%.

0.30 ≈ n YOUR TURN! A shirt that normally sells for $18.99 is on sale for $15.19. What is the percent of the discount? 1. Find the amount of the discount.

$18.99 - $15.19 =

$3.80

2. Use the percent equation to find the percent of discount. The percent of discount is about

20%

.

3.80 = n · 3.80 = _ 18.99n _ 18.99 18.99 0.20 ≈ n

18.99

GO ON Lesson 3-2 Solve Percent Problems

13

Who is Correct? Yancy saved 33% off his purchase of $85. How much did he save?

Clara 0.33 × 85 = $28.05

Jackson

Caroline

0.033 × 85 = $2.80

$85 - 33 = $52

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice Write a proportion for each percent problem.

31.5 7 =_ _

1

7% of 450 is 31.5

3

5% of 200 is 10

5

TAXES What is the sales tax on a $299 video game system if the tax rate is 6.75%?

100 450 5 = 10 100 200

_ _

2

3.5% of 15 is 0.525

4

20% of 85 is 17

3.5 = _ 0.525 _

15 100 20 = 17 100 85

_ _

$20.18 6

TAXES is 7%?

What is the sales tax on a $69 pair of shoes if the tax rate

7

TAXES Danielle had to pay an 11% “dine in” tax. If Danielle’s food order came to $6.29, how much additional did she have to pay in tax?

$0.69 8

TIPPING Tom’s lunch is $6.83. If Tom wants to leave an 18% tip, how much tip should Tom leave?

$1.23 9

TIPPING The Evans family’s dinner is $42.23. If Mrs. Evans wants to leave a 20% tip, how much tip should she leave?

$8.45

14

Chapter 3 Using Percents



Copyright © by The McGraw-Hill Companies, Inc.

$4.83

Step by Step Practice 10

Find the amount of commission on $315 sales if the commission rate is 6%. Step 1 Write the percent proportion.

part percent _ _ = whole

100

Step 2 $315 is the

part

whole

. The amount of commission is the

.

Substitute the values that you know. Let n = the amount of commission

6 =_ n _ 100

315

Step 3 Cross multiply and solve.

6 · 315 = 100 · n 1,890 = 100n 18.90 = n

Copyright © by The McGraw-Hill Companies, Inc.

Step 4 The amount of commission is $

18.90 .

Find the amount of commission on each sales amount at the rate given. Round to the nearest cent. 11

12 z $3,055 at 12% ____ = ______ $366.60 100 3,055

13

$10,250 at 9%

15

1% $5,200 at 7__ 2

$37.95

12

$460 at 8.25%

$922.50

14

$775 at 2.5%

$19.38

$390.00

16

1% $650 at 4 __ 2

$29.25

GO ON Lesson 3-2 Solve Percent Problems

15

Solve using the percent equation or proportion. TAXES What is the sales tax on an $899 television if the tax rate is 6.5%? $58.44

18

TAXES What is the sales tax of a $14.99 book if the tax rate is 5.75%? $0.86

19

TIPPING Pearl wants to tip 15% of the restaurant bill. If her bill came to $14.67, how much should she leave as a tip?

$2.20

5IF
-

FHFO

20

TIPPING John wants to tip 18% of the restaurant bill. If his bill came to $22.34, how much should he leave as a tip? $4.02

21

DISCOUNT A pair of jeans that normally sells for $59.99 is on sale 25% for $44.99. What is the percent of discount?

22

DISCOUNT A video game that normally sells for $54.99 is on sale 15% for $46.74. What is the percent of discount?

Step by Step Problem-Solving Practice Solve. 23

COMPUTERS Andrea ordered a computer online. The computer cost $1,000 plus 7% sales tax. What was the total amount Andrea paid for her computer? Read the problem. Write what you know. The computer cost $1,000 . 7% . The sales tax rate is

Plan

Pick a strategy. One strategy is to solve a simpler problem. Find 7% of 100. By definition of percent, 7% of $100 is $7. 7% of $1,000 is 10 · $7 or $70. So, the total cost is $1,000 + $70 or $1,070. Andrea paid a total amount of $1,070.

Check

16

Does the answer seem reasonable? Is your answer about 10% greater than the cost of the computer?

Chapter 3 Using Percents

ET
P

G
$B

TUMFT

Problem-Solving Strategies Use a table. Look for a pattern. Draw a diagram. Guess and check. ✓ Solve a simpler problem. Copyright © by The McGraw-Hill Companies, Inc.

Understand

Solve

 

5IF

$BT
-FHFOET
PG
UMFT



17

24

25

26

MOVIES A video store is having a sale. DVDs are on sale for 20% off. During this sale, what is the cost of 3 DVDs that regularly cost $48 $20 each? Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

CELL PHONES Justin is buying a cell phone that has a regular price of $150. The cell phone is on sale for 15% off the regular price. What will be the sale price? $127.50 Explain the difference between solving a problem about 10% off a price and paying 10% sales tax on an item.

You calculate the 10% of the price in about the same way for both problems. For the 10%-off problem, you subtract from the price. For the sales-tax problem, you add to the price.

Skills, Concepts, and Problem Solving

Copyright © by The McGraw-Hill Companies, Inc.

Find the amount of commission on each sales amount at the rate given. Round to the nearest cent. 27

$2,018 at 8% $161.44

28

$388 at 5.75%

29

$16,500 at 11%

30

$538 at 4.5%

$22.31

$1,815

$24.21

GO ON Lesson 3-2 Solve Percent Problems

17

Use the percent equation or a proportion to solve. 31

TESTS On the written portion of her driving test, Nadina answered 84% of the questions correctly. If Nadina answered 42 questions correctly, how many questions were on the driving test?

50 32

TAXES A property tax of 2% of a home’s value is billed to residents of the city each year. How much property tax would the owner of a $115,000 home owe?

$2,300 33

COMMISSION Mr. Faccinto earns 3% on sales of cars. Last week he earned $870 in commission. How much did Mr. Faccinto sell last week?

$29,000 34

DISCOUNT Marcus is buying a DVD that normally sells for $18.99. If the DVD is on sale for 20% off, how much will Marcus pay for the DVD?

$15.19

35

TIPPING Find the total cost of Leanne’s bill with a 7.5% tax.

$11.58 36

TIPPING Leanne wants to tip 18% of the restaurant bill before tax. How much would she leave as a tip?

3IVER$AF_ %BUF

5BCMF

4FSWFS





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$HICKEN
4ANDWICH  %RINK



5OTAL



5BY 5PUBM

$1.94 37

TIPPING Suppose Leanne pays with a $20 bill. How much change should there be after tax and tip are included?

$6.48

18

Chapter 3 Using Percents

$MBJN

5IBOL
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$PNF
"HBJO

Copyright © by The McGraw-Hill Companies, Inc.

Use the receipt at the right for Exercises 35–37.

Vocabulary Check Write the vocabulary word that completes each sentence. 38

An equation stating that two ratios are equivalent is a(n)

proportion 39

.

A special kind of ratio that compares a number to 100 is a(n)

percent 40

.

Writing in Math Explain what is meant by each term in the equation: percent ______ part _______ = 100 whole

See TWE margin.

Spiral Review

Copyright © by The McGraw-Hill Companies, Inc.

Solve using the percent equation. Check your answer. 41

75 is 20% of what number?

375

42

What number is 12% of 72?

8.64

Solve. 43

(Lesson 3-1, p. 4)

(Lesson 2-3, p. 49)

FASHION In Mae’s class, 6 of the students wear earrings. If there are 24 students in her class, what percent do not wear earrings?

75%

Write each percent as a decimal and a fraction in simplest form. (Lesson 2-1, p. 34)

44

15%

45

112%

_ _

0.15; 15 = 3 100 20

_

1.12; 28 25

Lesson 3-2 Solve Percent Problems

19

Chapter

3

Progress Check 1

(Lessons 3-1 and 3-2)

Solve using the percent equation. 5NS1.2 1

13% of what number is 52? 400

3

What is 31.5% of 200?

63

11

2

What is 12.5% of 88?

4

What percent of 200 is 86? 43%

Solve using the percent proportion. 6NS1.3 5

3 is what percent of 10? 30%

6

What is 65% of 120?

78

7

81 is 54% of what number? 150

8

What is 44% of 55? 24.2

Solve. 5NS1.2, 5SDAP1.3, 6NS1.3, 6NS1.4, 7NS1.7 9

MOVIES Fourteen of the students in Rosa’s class prefer animated movies above all other types of movies. The rest of the students like action or mystery movies better. If there are 25 students in Rosa’s class, what percent do not prefer animated movies?

44% MONEY Reina earns 7% commission on her sales each day. The table shows her sales for last week. Fill in the table to show the amount of commission Reina earned each day. How much total commission did Reina earn last week?

$119 Day Monday Tuesday Wednesday Thursday Friday 11

Commission Amount

$38.50 $2.45 $17.22 $17.08 $43.75

MONEY Find the amount of commission on sales of $654 if the commission rate is 9%.

$58.86 20

Amount Sold $550 $35 $246 $244 $625

Chapter 3 Using Percents

Copyright © by The McGraw-Hill Companies, Inc.

10

Lesson

3-3 Interest Problems KEY Concept When you have a savings account, the bank pays you money. This money is called interest. The amount of money that you save is called the principal . When you invest money, you receive interest. When you borrow money, you pay interest. A common form of borrowing money is a loan. The formula I = p × r × t is used to find simple interest .

Write the interest rate as a decimal. Drop the % sign and move the decimal to the left two places. Time must be expressed in years. If the time is part of a year, write it as a decimal or fraction.

Copyright © by The McGraw-Hill Companies, Inc.

When interest is paid on both the principal and on any interest already in the account, it is called compound interest .

6NS1.4 Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips. 7NS1.7 Solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest.

VOCABULARY simple interest money that is earned on bank deposits or investments; it is also money that is paid for borrowing a specific amount principal the amount of money deposited or invested compound interest when interest is earned on both the principal and any interest already in an account

Write the interest rate as a decimal.

In the simple interest formula, you have to add the principal to the amount of interest earned to find the amount in the account. In the compound interest formula, you have to subtract the principal to find the amount of the interest earned.

GO ON Lesson 3-3 Interest Problems

21

Example 1

YOUR TURN!

Find the simple interest earned on an investment of $600 at 8.5% for 6 months. 1. Write the simple interest formula. I = prt 2. Write the decimal for 8.5%. 0.085 The time is not given in years. 6 months = 0.5 year 3. Substitute the principal, rate, and time into the formula. I = 600 · 0.085 · 0.5 = 25.50 4. The amount of interest earned is $25.50.

Find the simple interest earned on an investment of $300 at 5% for 2 years. 1. Write the simple interest formula.

I = prt

2. Write the decimal for 5%.

0.05

3. Substitute the principal, rate, and time into the formula.

I = 300 · 0.05 · 2 I = 30.00 4. How much interest is earned?

$30

Example 2

YOUR TURN!

Find the value of an investment of $2,500 for 1 year at 6% interest compounded quarterly. Round to the nearest cent.

Find the value of an investment of $3,600 for 2 years at 7% interest compounded quarterly. Use a calculator or spreadsheet to help you. Round to the nearest cent.

_

For how many years? 1 t=1 3. Substitute the principal, rate, number of times per year and number of years into the formula. ⎛ 0.06 ⎞⎥4(1) A = 2,500 ⎪⎝ 1 + 4 ⎠ = 2,500(1 + 0.015)4(1) = 2,500(1.015)4(1) = 2,500(1.015)4 = 2,500(1.061) = 2,653.408

_

4. How much money is in the account at the end of 1 year? $2,653.41 22

Chapter 3 Using Percents

_

1. Write the compound interest formula. ⎛ ⎞nt A = P ⎪1 + r⎥ n⎠ ⎝

2. How many times a year is quarterly?

4

n=4

For how many years?

t=2

2

3. Substitute the principal, rate, number of times per year, and number of years into the formula.

_

⎛ ⎞4(2) A = 3,600 ⎪⎝1 + 0.07 ⎥⎠ 4 = 3,600(1 + 0.0175)4(2) = 3,600(1.0175)4(2) = 3,600(1.0175)8 = 4,135.97

4. How much money is in the account $4,135.97 after 2 years?

Copyright © by The McGraw-Hill Companies, Inc.

1. Write the compound interest formula. ⎛ r ⎞nt A = P ⎪⎝ 1 + n ⎥⎠ 2. How many times a year is quarterly? 4 n=4

Example 3

YOUR TURN!

Charliqua borrowed $1,500 from her bank. The loan was for 5 years. The bank charged 8% interest compounded monthly. How much money will Charliqua have paid back at the end of the loan? Use a calculator or spreadsheet to help you. Round to the nearest cent. 1. Write the compound interest formula. ⎛ r ⎞nt A = P ⎪⎝1 + n ⎥⎠ 2. The principal is the amount borrowed. P = 1,500

_

1. Write the compound interest formula.

_

A = P (1 + r )nt n

2. How much is the principal?

P = 3,000

The rate is 8% or 0.08. r = 0.08

What is the rate?

Interest is compounded monthly. n = 12

What is n? (compounded monthly)

The loan is for 5 years. t=5

How long is the loan? (10 years)

3. Substitute the values into the formula. ⎛ 0.08 ⎞⎥12(5) A = 1,500 ⎪⎝1 + 12 ⎠ A = $2,234.77

_

Copyright © by The McGraw-Hill Companies, Inc.

Royale borrowed $3,000 from his bank. His loan was for 10 years. The bank charged 6% compounded monthly. How much money will Royale have paid back at the end of the loan? Use a calculator or spreadsheet to help you. Round to the nearest cent.

So, Charliqua paid $2,234.77 by the end of the loan. 4. How much interest did she pay altogether? $2,234.77 - $1,500 = $734.77

r = 6% or 0.06 n = 12 t = 10

_

3. Substitute the values into the formula.

A = 3,000(1 + 0.06 )12(10) 12 A = $5,458.19

So, Royale paid $5,458.19 by the end of the loan. 4. How much interest did Royale pay altogether?

To find the interest in a compounded formula, you need to subtract the principal.

$5,458.19 - $3,000 = $2,458.19

GO ON Lesson 3-3 Interest Problems

23

Who is Correct? Find the simple interest to the nearest cent: $500 at 4% for 9 months.

Rafael

Bobby

I = 500 · 0.04 · 9

I = 500 · 0.4 · 0.75

I = 500 · 0.04 · 0.75

I = $180.00

I = $150.00

I = $15.00

Lisa

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice Find the simple interest earned to the nearest cent.

$240

1

$1,200, 5%, 4 years

3

1 years $209.50 $955, 6.75%, 3__ 4

2

$15,000, 7.5%, 6 months $562.50

4

$565, 16%, 9 months

$67.80

Step by Step Practice 5

Semiannually means two times a year.

Find the value of an investment of $1,000 for 5 years at 8% interest compounded semiannually. Use a spreadsheet or calculator to help you. Round the answer to the nearest cent.

How many years is the investment?

t=5

Step 3 Substitute the values that you know. A = 1,000(1 +

0.08 ) _ 2

2(5)

_

Step 4 Simplify.

A = 1,000(1 + 0.08 )2(5) 2 A = 1,000(1 + 0.04) 2(5) A = 1,000(1.04) 2(5) A = 1,000(1.04) 10 A = $1,480

Step 5 What is the value of the account at the end of 5 years?

24

Chapter 3 Using Percents

$1,480

Copyright © by The McGraw-Hill Companies, Inc.

_

A = P (1 + r )nt n Step 2 How many times is interest paid per year? n = 2 Step 1 Write the compound interest formula.

Find the value of each investment using the compound interest formula. Use a spreadsheet or calculator to help you. Round each answer to the nearest cent. 6

$2,000 invested for 6 years at 12% interest compounded quarterly r )nt = A = P(1 + __ n

⎛

2,000 ⎪

4,065.59

=$ 7

0.12 ⎞ 4·6 ⎥ 4 ⎠

1 + ________

⎝

$10,000 invested for 10 years at 9% interest compounded annually

$23,673.64

Step by Step Problem-Solving Practice

Problem-Solving Strategies Use a table. Look for a pattern. Guess and check. ✓ Solve a simpler problem. Work backward.

Solve. 8

MONEY Deepak put $1,000 into a savings account. The simple interest rate is 3.5%. How much interest will Deepak earn in 1 month? in 6 months? in 12 months? Understand

Read the problem. Write what you know.

$1,000 Deepak invests 3.5% rate of . Plan Copyright © by The McGraw-Hill Companies, Inc.

Quarterly means four times a year.

at an interest Make a table to help you write the time as a ratio in simplest form.

Pick a strategy. One strategy is to solve a simpler problem.

Number of months

Use the simple interest formula to find the interest earned for each time period. Solve

Check

Use the simple interest formula to calculate the interest for 1 month, 6 months, and 1 year.

Ratio of number of months to 12 months Ratio in simplest form

1 month I =

1,000

·

0.035

·

6 months I =

1,000

·

0.035

·

12 months I =

1,000

·

0.035

·

_1 _1 12 2

1

1

6

12

12

12

12

12

2

1 _ 12 _6 _ 1 _ _1

=

$2.92

=

$17.50

=

$35

1

The amount of interest should increase as the number of months increases. GO ON Lesson 3-3 Interest Problems

25

9

FINANCE How much interest will Hannah earn in 4 years if she deposits $630 in a savings account at 6.5% simple interest?

$163.80

Check off each step.

10

✔

Understand

✔

Plan

✔

Solve

✔

Check

FINANCE Kelly’s inheritance was $220,000 after taxes. The money is invested in an account that earns $9,900 in simple interest every year. What is the interest rate on her account?

4.5% 11

Write the formulas for simple interest and compound interest. Explain what each variable stands for.

See TWE margin.

Skills, Concepts, and Problem Solving 12

$568, 16%, 8 months

$60.59

13

$1,540, 8.25%, 2 years $254.10

14

1 years $725, 4.3%, 2__ 2

$77.94

15

3 $3,500, 4.2%, 1__ years $257.25 4

16

$740, 3.25%, 2 years

$48.10

17

$350, 6.2%, 3 years

26

Chapter 3 Using Percents

$65.10

Copyright © by The McGraw-Hill Companies, Inc.

Find the amount of simple interest earned to the nearest cent.

Find the value of each investment using the compound interest formula. Use a calculator or spreadsheet to help you. Round each answer to the nearest cent. 18

$500 invested for 1 year at 5% interest compounded semiannually

19

$4,800 invested for 10 years at 9% interest compounded quarterly

20

$800 invested for 1 year at 7.5% interest compounded semiannually

$525.31

$11,688.91

$861.13

Solve. Use a calculator or spreadsheet to help you. Round to the nearest cent.

Copyright © by The McGraw-Hill Companies, Inc.

21

LOANS Refer to the photo caption at the right. The interest rate on the loan was 8% compounded monthly. How much interest did Pul’s father pay if he took 2 years to repay the loan? Remember to subtract the principal to find interest.

$207.47 LOANS Pul’s father 22

BUSINESS Raj invested $900 for 4 years at an interest rate of 8%. How much interest did he earn?

borrowed $1,200 from the bank for a riding lawn mower.

$288.00 23

FINANCE In 2002, a $5 bank note from 1886 was sold to a collector for $103,500. Suppose a person had deposited the $5 in a bank in 1886 with an interest rate of 4% compounded quarterly. After 116 years, how much money would be in the account?

$505.93 GO ON Lesson 3-3 Interest Problems Royalty Free/Corbis

27

Vocabulary Check Write the vocabulary word that completes each sentence. 24

The amount of money deposited or invested is the principal .

25

When interest is earned on both the principal and any interest already in the account, it is called compound interest .

26

Writing in Math Another form of borrowing is to use credit cards. Consider the following two credit cards. One has a compound interest rate of 18% that is compounded annually. The other has a compound interest rate of 21% that is compounded quarterly. Which interest rate is better? Explain. (If $1,000 is charged onto each card and no payments are made, what will the balance be in 2 years’ time?)

See TWE margin.

Spiral Review Solve using the percent equation. Check each answer. (Lesson 3-1, p. 4)

80%

27

What percent of 50 is 40?

28

What is 125% of 60?

29

9 is what percent of 120?

30

25.5% of 75 is what number?

75 19.13

Solve. (Lesson 3-1, p. 4) 31

SCHOOL Out of 459 students at Breckinridge Junior High, only 30% take the bus to school. How many students take the bus?

138 students 32

FASHION Becky tipped her hair stylist 15% for a haircut. If the tip was $6.30, how much was the haircut?

$42

28

Chapter 3 Using Percents

Copyright © by The McGraw-Hill Companies, Inc.

7.5%

Lesson

3-4 Percent of Change

7NS1.6 Calculate the percentage of increases and decreases of a quantity. 7NS1.7 Solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest.

KEY Concept A percent of change is a ratio that compares the change in an amount to the original amount.

VOCABULARY increase when the original amount becomes greater in value decrease when the original amount becomes smaller in value

When the change is positive, it is a percent of increase . When the change is negative, it is a percent of decrease .

percent a ratio that compares a number to 100; it uses the symbol % The word percent means hundredths or out of 100.

Convert your decimal answer to a percent. If it is positive, label it “percent of increase.” If it is negative, drop the negative sign and label it “percent of decrease.”

(Lesson 2-1, p. 34)

ratio a comparison of two numbers by division Example: The ratio of 2 to 3 can be stated as 2 out of 3, 2 to 3, 2:3, or 2/3.

Example 1

(Lesson 1-1, p. 4) Copyright © by The McGraw-Hill Companies, Inc.

Find the percent of change. At 1 year old, a tree was 3 feet tall. At 7 years old, the same tree was 7.5 feet tall. What is the percent of change in the tree’s height? 1. The original amount is 3. The new amount is 7.5. 2. Substitute the values in the formula. new - original -3 amount amount = 7.5 _______ 3 original amount 3. Simplify. 7.5 -3 _______ = 1.50 = 150% 3 4. The value is positive. 150% increase

YOUR TURN! At 4 years old, Yonnie was 40 inches tall. At 12 years old, she was 54 inches tall. What is the percent of change in her height? 1. The original amount is 40 . The new amount is 54 .

_

2. Substitute the values in the formula.

54 - 40 40

_

3. Simplify.

54 - 40 40

=

14 _

= 0.35 = 35%

40 4. The value is positive . 35% increase

GO ON

Lesson 3-4 Percent of Change

29

Example 2 Find the percent of change. A car that was worth $10,500 in 2004 is now worth $9,030. What is the percent of change in the car’s worth? 1. The original amount is $10,500. The new amount is $9,030.

YOUR TURN! Kieran’s winter coat was originally priced $135. He bought the coat on clearance and paid $74.25. What is the percent of change in the price Kieran paid?

$135 1. The original amount was $74.25 The new amount is . 2. Substitute the values in the formula.

__

74.25 - 135 135

2. Substitute the values in the formula. new original - 10,500 amount - amount = 9,030 _____________ 10,500 original amount 3. Simplify. -1,470 9,030 - 10,500 _______ _____________ = = -0.14 = -14% 10,500 10,500

.

3. Simplify.

-60.75 _ 135

=

-0.45

=

-45%

4. The value is negative .

45% decrease

4. The value is negative. 14% decrease

Example 3

1. The original amount is 60. The new amount is 63.5. 2. Substitute the values in the formula. new - original - 60 amount amount = 63.5 _________ 60 original amount 3. Simplify. − 63.5 - 60 ___ 3.5 _________ = = 0.0583 ≈ 5.8% 60 60 Round to the nearest whole percent. 6% 4. The value is positive. 6% increase

30

Chapter 3 Using Percents

Copyright © by The McGraw-Hill Companies, Inc.

Warren got a score of 60 on his science exam. He was able to retake the test and scored a 63.5. Find the percent of change in the test scores. Round to the nearest whole percent.

YOUR TURN! Ms. Vetten runs 5 miles every day. During the winter, she finishes her run in 56 minutes. During the summer, she finishes her run in 49 minutes. Find the percent of change in Ms. Vetten’s time from winter to summer. Round to the nearest whole percent. 1. The original amount is The new amount is

56 .

49 .

2. Substitute the values in the formula.

-7 _

_

49 - 56 56

3. Simplify.

56

= -0.125 = -12.5%

4. Round to the nearest whole percent.

-13%

Remember to drop the negative sign when writing as a decrease.

5. The value is negative .

13% decrease

Who is Correct? On Elan’s birthday he will receive a $2 increase in his allowance. If his current allowance is $8, what is the percent increase?

Copyright © by The McGraw-Hill Companies, Inc.

Trent

_ _

8-2 =6 8 8 = 0.75 = 75% increase

Ted

Dora

_ _

_ _

10 - 8 = 2 8 8 = 0.25 = 25% increase

10 - 8 = 2 10 10 = 0.20 = 20% increase

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice Find each percent of change. Round to the nearest whole percent. 1

original price: $30; new price: $110

267% increase 2

original price: $91; new price: $77

15% decrease

GO ON Lesson 3-4 Percent of Change

31

Step by Step Practice 3

The original price of a DVD player was $105. The new price of the player is $45. Step 1 The original amount is 105 . The new amount is 45 . Step 2 Substitute the values in the formula.

_

45 - 105 105

Step 3 Simplify.

-60 _ 105

= -0.5714 =

-57%

Step 4 The value is negative .

57% decrease

Find each percent of change. Round to the nearest whole percent. 4

Vincent had an average test score of 80%. After studying hard during a grading period, his average changed to 90%. new - original 90 - 80 10 amount amount = = = 0.125 80 80 original amount

_

_

13% increase 5

50% increase 6

A bush is 1.5 feet tall. It is pruned until it is 0.55 feet tall.

63% decrease 7

The ribbon was 3.75 inches long, but was cut to two inches.

47% decrease 8

The price was $2.50. Now it is $1.99.

20% decrease 9

Linda put $100 in the bank. Now she has $140.

40% increase

32

Chapter 3 Using Percents


JO


JO

Copyright © by The McGraw-Hill Companies, Inc.

Amy receives an allowance of $15. She used to receive $10.

Step by Step Problem-Solving Practice

Problem-Solving Strategies Use a table.

Solve. 10

✓ Use logical reasoning.

BUSINESS A store buys a pair of running shoes for $68. What is the selling price of the shoes if the store has a markup of 75%? Understand

Guess and check. Solve a simpler problem. Act it out.

Read the problem. Write what you know. A pair of shoes cost percent markup.

$68

before a

A markup is a percent increase in the price.

75%

Pick a strategy. One strategy is to use logical reasoning. The store buys the shoes and then marks them up by 75%.

Plan

To find the selling price, calculate 75% of the store’s price. Add this to $68 for the selling price of the shoes. To find 75% of $68, multiply.

Solve

$68

×

original price

0.75

=

markup percent as a decimal

$51 markup amount

To find the selling price, add.

$68

Copyright © by The McGraw-Hill Companies, Inc.

original price

$51.00 = markup amount

$119 selling price

A markup of 75% means the price of the shoes is almost double what the store paid. Does your answer make sense?

Check

11

+

SPORTS A sports bag costs a sporting supply store $18. What is the selling price if the store has a markup of 55%? How much $27.90; $9.90 profit is made if the bag sells at that price? Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

Profit is the difference between the selling price of an item and the cost to the store.

GO ON Lesson 3-4 Percent of Change

33

12

FASHION Lupe’s clothing store buys a dress for $48 and sells it for 45% $69.60. What is her percent of profit?

13

A shirt that the store buys for $15 is marked up 60%. During a sale, the shirt is discounted 20% from its selling price. Explain how to find the price of the shirt.

See TWE margin.

Skills, Concepts, and Problem Solving Find each percent of change. Round to the nearest whole percent. 14

original: 1.5; new: 2.5

15

67% increase 16

original: 63; new: 45

29% decrease

original: 84; new: 111

32% increase 17

original: 6.8; new: 8.2

21% increase

Find each percent of change. Round to the nearest whole percent. 18

Gloves that cost $14.99 are on sale for $12.64.

16% decrease Sunglasses are on sale for the price marked on the glasses shown at the right.

  

21% decrease 20

In 7th grade, Rachel read 15 books. In 8th grade, she read 18 books.

20% increase 21

In 2005, the yearbook had 236 photos. In 2006 it had 214 photos.

9% decrease 22

The population of U.S. immigrants was 4,298,000 in 1990 and 7,858,000 in 2000.

83% increase

34

Chapter 3 Using Percents

Copyright © by The McGraw-Hill Companies, Inc.

19

Solve. 23

BOOKS Ray chose a book at a store that sold for $14.95. The book was on sale and cost $12.71 before tax. What percent discount was applied to the nearest whole percent?

15%

24

SALES A bookstore buys a textbook for $18 and marks the price up 60%. How much will the bookstore sell the book for? What is the profit made if the book sells at that price?

$28.80; $10.80

Vocabulary Check Write the vocabulary word that completes each sentence.

decrease

is when an original quantity goes

increase

is when an original quantity goes up.

25

A(n) down.

26

A(n)

27

Writing in Math In a percent of change, what two quantities are being compared?

The amount of change is compared to the original amount.

Spiral Review Copyright © by The McGraw-Hill Companies, Inc.

Find the amount of simple interest earned to the nearest cent. 28

$96

$400, 12%, 2 years

19 0.095; _

29

(Lesson 3-3, p. 21)

$3,500, 6.5%, 5 years

Write each percent as a decimal and a fraction in simplest form. 30

9.5%

32

75%

Solve. 34

_

200 0.75; 3 4

31

125%

33

25%

$1,137.50

_

(Lesson 2-1, p. 34)

1.25; 1 1 4 1 0.25; 4

_

(Lesson 1-3, p. 19)

CONTACT LENSES The probability of buying a single vial that 3 contains 2 instead of 1 contact lens is ____. Find the probability of 100 97 not receiving an extra contact.

_ 100

Lesson 3-4 Percent of Change

35

Chapter

3

Progress Check 2

(Lessons 3-3 and 3-4)

Find each amount of simple interest earned to the nearest cent. 6NS1.4, 7NS1.7 1

$200, 9.5%, 2 years

$38

2

$10,600, 8%, 6 months

$424

Find the value of each investment using the compound interest formula. Round each answer to the nearest cent. 6NS1.4, 7NS1.7 3

$10,000 invested for 5 years at 5% interest compounded $12,800.85 semiannually

Find each percent of change. Round to the nearest whole percent. 7NS1.6, 7NS1.7 4

A can of soup is on sale as marked.

14% decrease 5

Hats that cost $9.99 each are sold for $5 each.

50% decrease In fifth grade, Andres was 56 inches tall. In sixth grade, he was 7% increase 60 inches tall.

7

Last month it rained 50 centimeters. This month it rained 22% decrease 39 centimeters.

Copyright © by The McGraw-Hill Companies, Inc.

6

Solve. 6NS1.4, 7NS1.6, 7NS1.7 8

MOVIES Movie tickets cost $7.50 each. With a student discount card, Alvin pays $5. What is the percent discount?

_

9

33 1 % 3

BUSINESS A shop buys shoes for $15 and marks them up 45%. What is the selling price of the shoes?

4LQBOP

$21.75

ELECTIONS The bar graph at the right shows the number of people who voted in a local election. To the nearest tenth, what is the percent of change?

16.7% decrease 36

Chapter 3 Using Percents

Rachel Epstein/PhotoEdit

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Chapter

Study Guide

3

Vocabulary and Concept Check commission, p. 11

Write the vocabulary word that completes each sentence.

compound interest, p. 21

1

principal The money deposited or invested.

2

decrease A(n) occurs when the original quantity becomes smaller in value.

cross products, p. 11 decrease, p. 29 increase, p. 29 principal, p. 21 3

proportion, p. 11 simple interest, p. 21

When interest is earned on both the principal and any interest already earned, it is known as

compound interest 4

is the amount of

.

Commission

is the amount of money a salesperson earns based on the amount of sales.

Write the correct vocabulary term in the blank that corresponds to the example.

proportion

5

6

simple interest

8 2 = ___ __

Copyright © by The McGraw-Hill Companies, Inc.

3

principal

7

I = prt

12

Lesson Review

3-1

Calculate Percents

(pp. 4–10)

Solve using the percent equation. Check each answer. 8

25% of what number is 90? 360

9

15% of what number is 33? 220

Example 1 20% of what number is 32? Write the percent equation.

percent · whole = part

Substitute the known numbers and variables.

0.20 ·

Solve for the variable. Check your answer.

n = 32 0.20n = 32 0.20 0.20 n = 160

0.20 · 160 = 32 ✓

Chapter 3 Study Guide

37

Solve using the percent equation. Check each answer. 10

What is 22% of 740? 162.8

11

What is 7% of 810? 56.7

Example 2 What is 12% of 400?

12

What percent is 140 of 200? 70%

Write the percent equation. Substitute the known numbers and variables. Solve for the variable.

13

What percent is 26 of 520? 5%

Check your answer. 0.12 · 400 = 48 ✓

3-2

Solve Percent Problems

SHOPPING Derrick bought an MP3 player that cost $199. He paid 7.25% sales tax. How much did he pay in tax?

$14.43 15

SALES Debra earns 3% commission on the sale of a home she lists. Last week she sold a home and earned $10,140 in commission. For what amount did the home sell?

Example 3 Colin bought a baseball glove that cost $45. He paid 5.75% sales tax. How much did he pay in tax? percent ______ part _______ =

Write the proportion.

100 whole Substitute what you know into the proportion. Let x = the amount of tax. Cross multiply and solve for x. 5.75 ____ ____ = x 100

45

5.75(45) = 100x 258.75 = 100x 2.5875 = x Round the amount of tax to the nearest cent. Colin paid $2.59 in tax.

$338,000 17

SALES Kathleen earns 2% commission for every dress she sells. Last week she earned $17 in commission. What was the total amount of her sales last week?

$850

38

Chapter 3 Study Guide

Copyright © by The McGraw-Hill Companies, Inc.

SHOPPING Denise bought a magic set that cost $29.97. She paid 6.5% sales tax. How much did she pay in tax?

$1.95 16

0.12 · 400 = n 48 = n

(pp. 11–19)

Solve using the percent proportion. 14

percent · whole = part

3-3

Interest Problems

(pp. 21–28)

Find each amount of simple interest earned to the nearest cent.

Example 4

18

$900, 3.5%, 4 years $126

Find the simple interest earned on an investment of $1,500 at 5% for 2 years. Round to the nearest cent.

19

$500, 4%, 6 months $10

Write the simple interest formula. I = prt Write the decimal for 5% 0.05

Find the value of each investment using the compound interest formula. Use a calculator or spreadsheet to help you. Round each answer to the nearest cent. 20

I = 1,500 · 0.05 · 2 I = 150 How much interest is earned? $150

$2,000 invested for 3 years at 7% interest compounded annually

$2,450.09

3-4

Copyright © by The McGraw-Hill Companies, Inc.

Substitute the principal, rate, and time into the formula.

Percent of Change

(pp. 29–35)

Find each percent of change. Round to the nearest whole percent. 21

A share of stock in the Peabody Alarm Company that was worth $200 in 2001 is now worth $50. What is the percent of change in the stock’s worth?

75% decrease 22

Albert brought his average grade in science up from a 74 to a 92. What is the percent of change in Albert’s average grade in science?

24% increase

Example 5 Find the percent of change. A house that was worth $129,000 in 2004 was worth $148,350 in 2006. What was the percent of change in the value of the house in those two years? Substitute the values of the original and new amounts into the formula. new - original amount amount 148,350 - 129,000 = ________________ 129,000 original amount Simplify. 148,350 - 129,000 _______ 19,350 ________________ = = 0.15 = 15% 129,000 129,000 The value is positive, so it is an increase. 15% increase Chapter 3 Study Guide

39

Chapter

3

Chapter Test

Solve using the percent equation. Check each answer. 5NS1.2 1

90 is 15% of what number?

600

2

What number is 6.5% of 3,600?

234

3

125 is what percent of 625?

20%

4

What number is 12% of 600?

5

8 is 50% of what number?

16

6

23 is what percent of 69?

8

What number is 24% of 120?

28.8

10

What number is 15% of 800?

120

12

6 is 37.5% of what number?

16

72

33%

Solve using the percent proportion. 6NS1.3

14%

7

7 is what percent of 50?

9

30 is 24% of what number?

11

12 is what percent of 60?

125 20%

$1,960

14

$12,014, 7.5%, 18 months $1,351.58

16

$5,500, 3%, 2 years

$330

18

$3,400, 4%, 6 years

$816

13

$2,800, 3.5%, 20 years

15

$250, 8%, 1 year

17

$1,000, 6%, 5 years

19

Find the value of $400 invested for 2 years at 5% interest compounded annually. Use a spreadsheet or calculator to help you. $441 7NS1.7 Round to the nearest cent.

$20 $300

GO ON 40

Chapter 3 Test

Copyright © by The McGraw-Hill Companies, Inc.

Find each amount of simple interest earned to the nearest cent. 7NS1.7

Find each percent of change. Round to the nearest whole percent. 7NS1.6 20

Purses that normally cost $49.99 each are on sale for $37.49.

25% decrease 21

Last December it snowed a total of 4 inches. This December it snowed 12 inches.

200% increase 22

The value of the antique table went from $1,300 to $1,500.

15% increase Solve. 5NS1.2, 6NS1.3, 7NS1.6, 7NS1.7 23

SPORTS In the 2006 season, the Lakeside Lions won 52 of their 70 total games, and they lost all the rest. What percent of their games did they lose? Round to the nearest tenth.

25.7% 24

PACKAGING The width of a box made by the Triple H packing company was recently changed from 18 inches to the new length shown at the right. What is the percent of change of the length? Round to the nearest whole percent.

Copyright © by The McGraw-Hill Companies, Inc.

11% increase 25

FINANCES Rico’s income tax refund was $5,200. If he invests the entire amount for 3 years at 7% interest compounded annually, what is the value of his investment after three years? Round your answer to the nearest cent.

$6,370.22

Correct the mistakes made by the students taking the quiz. 26

The following question was on the quiz: The price of a school-play ticket rose from $5.75 to $6.25. What is the percent of increase for the price of a ticket? Who is correct? Explain your reasoning.

Kata: the change must be compared to the original price, $5.75, not the new price, $6.25.


JO

Kata =

0.50 _

5.75 = 0.087 or 8.7%

Felipe =

0.50 _

6.25 = 0.08 or 8%

Chapter 3 Test

41

Chapter

3

Standards Practice

Choose the best answer and fill in the corresponding circle on the sheet at right. 1

2

3

A store buys its bracelets for $1.50 each. It marks up the price 150%. At what price does this store sell these bracelets? A $2.25

C $3.75

B $3.00

D $4.50

7NS1.7

Shiva makes a 3% commission on every new car she sells. This month, she sold 4 new cars at $12,500 each. How much did she make in commission? F $375

H $1,500

G $750

J $37,500

B $32 to $3,849 C $212 to $4,093 D $244 to $4,061

6

7

Which is correct? 5NS1.2 1 = 0.33 = 3.3% A __ 3 3 B __ = 0.75 = 7.5% 4 2 = 0.4 = 40% C __ 5 3 D __ = 0.6 = 60% 6

Identify the percent modeled. 5NS1.2

F 53%

H 63%

G 57%

J 73%

The Johnson family went out to lunch. If the bill was $48 and Mr. Johnson gave a 20% tip, how much money did he spend on lunch? 6NS1.4 A $9.60

C $56.60

B $38.40

D $57.60

What is the simple interest on a $9,500 loan at 5.5% interest over 5 years? 7NS1.7 F $522.50 G $2,090 H $2,612.50 J $52,250

42

Chapter 3 Standards Practice

GO ON

Copyright © by The McGraw-Hill Companies, Inc.

The student council held a fund-raiser for an upcoming field trip. They spent $212 on supplies and $32 on advertising. The fund-raiser earned $4,061. What was the ratio of earnings to expenses from this fund-raiser? 6NS1.2 A $4,061 to $244

4

5

8

9

10

If 25% of a number is 4, what is 75% of the number? 6NS1.3 F 8

H 16

G 12

J 20

The Activities Committee sponsored a middle-school dance. They spent $90 on supplies, which included $18 on advertising. What percent of the supplies was spent on advertising? 6NS1.3 A 2%

C 18%

B 5%

D 20%

Selena reads 189 pages every 3 days. At this rate, how many pages does she read in 8 days? 6NS1.3

Copyright © by The McGraw-Hill Companies, Inc.

F 54 pages G 63 pages H 189 pages

ANSWER SHEET Directions: Fill in the circle of each correct answer. 1

A

B

C

D

2

F

G

H

J

3

A

B

C

D

4

F

G

H

J

5

A

B

C

D

6

F

G

H

J

7

A

B

C

D

8

F

G

H

J

9

A

B

C

D

10

F

G

H

J

Success Strategy For multiple-choice answers, read each choice as a true-false question. This might help to eliminate some answer choices right away. Draw a clear picture or diagram if you are stuck. Sometimes seeing it drawn out makes the problem clearer.

J 504 pages

Chapter 3 Standards Practice

43

Chapter

4

Rates and Proportional Reasoning We often compare things to understand them. Can you run a faster mile than I can? How much of this medicine should I take (based on my age and weight)? You can use proportional reasoning to make these comparisons.

Copyright © by The McGraw-Hill Companies, Inc.

44

Chapter 4 Rates and Proportional Reasoning

Getty Images/StockByte

STEP

STEP

1 Quiz

Are you ready for Chapter 4? Take the Online Readiness Quiz at ca.mathtriumphs.com to find out.

2 Preview

Get ready for Chapter 4. Review these skills and compare them with what you’ll learn in this chapter.

What You Know

What You Will Learn

You know how to multiply.

Lessons 4-1 and 4-2

Example: 4 × 3 = 12

Proportions are two equivalent ratios. 28 7 = ___ __ 8 32 To solve a proportion, you cross multiply .

TRY IT 1

12 × 5 =

60

2

2,000 × 6 =

12,000

n 12 in. = ___ _____

3

16 × 20 =

320

4

5,280 × 5 =

26,400

1

1 · n = 6 · 12 n = 72 in. 2

Copyright © by The McGraw-Hill Companies, Inc.

You know how to translate sentences into math problems. Example: Arturo had fifteen pennies. He found some more. Now he has thirty-three. Which number sentence could be used to find how many pennies he found? 15 + x = 33 x = 18

Write the solution. 72 in. is equal to 6 ft.

Lessons 4-2, 4-3, and 4-4 You can use proportions to solve word problems. The rain was falling at a rate of 2 inches per hour. If the rain fell at that rate for a full 24 hours, how much rain would fall? x in. 2 in. = ____ ____ 1h 24 h 2 · 24 = x · 1 48 = x

TRY IT 5

1 ft 6 ft Find the cross products. Solve.

Brittany saw 24 birds sitting on a wire. Some birds flew away. There were 15 left. How many birds flew away?

There would be 48 inches of rain in 24 hours.

24 - x = 15 x=9

45

Lesson

4-1 Proportions KEY Concept An equation stating that two ratios are equivalent is a proportion . For two ratios to form a proportion, their cross products must be equal.

3 1 = __ __

1 · 6 is one cross product.

2

6

1·6=2·3 6=6

2 · 3 is the other cross product. The cross products are equal.

Why does this work? A common denominator of the two fractions is 6 · 2 or 12. Multiply both sides of the proportion by 6 · 2. On the left side we can cancel the 2s and 1 3 (2 6) / · 6) __ = __ (2 · / on the right side we can cancel the 6s. 2 / 6 /

6NS1.3 Use proportions to solve problems. Use crossmultiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse.

VOCABULARY proportion an equation stating that two ratios are equivalent

28 7 Example: __ = ___ 8 32 (Lesson 3-2, p. 11)

cross product in a proportion, a cross product is the product of the numerator of one ratio and the denominator of the other ratio

This gives us 1 · 6 = 2 · 3. Notice that 1 · 6 and 2 · 3 are the same as the two cross products of the original proportion. Cross multiply to solve proportions when one value in the proportion is not known. A rate is a ratio with different units. If two bagels cost $0.50, you can use a proportion to find the cost of 12 bagels. Copyright © by The McGraw-Hill Companies, Inc.

0.50 ___ ____ = y

2 12 12 · 0.50 = 2 · y 2y 6 ___ __ = 2 2 3=y

Cross multiply. Divide by 2. 12 bagels would cost $3.

Example 1 Determine whether the ratios are proportional. 7 _3 = _ 4

21

1. Find the cross products. 2. The cross products are not equal. Therefore, the ratios are not proportional. 46

Chapter 4 Rates and Proportional Reasoning

3 · 21 = 7 · 4 63 ≠ 28

YOUR TURN! Determine whether the ratios are proportional. 4 _2 = _ 5

10

2 · 10 = 4 · 5

1. Find the cross products.

20 = 20

2. The cross products are equal. Therefore, the ratios are proportional.

Example 2

YOUR TURN! Solve for w.

Solve for t.

2.8 _w = _

10 14 _ _ =

7 6 1. Find the cross products.

t 11 1. Find the cross products.

2. Solve. w·7= 7w = w=

2. Solve. 14 · 11 = t · 10 154 = 10t t = 15.4

6 16.8 2.4

·

2.8

Copyright © by The McGraw-Hill Companies, Inc.

Example 3 Wyley Auto Sales orders new vehicles at a ratio of 8 cars to 3 trucks. How many cars are ordered when 15 trucks are ordered? 1. Write the ratio of cars to trucks.

_8 3

Do you remember how to write a ratio? Example: There are 3 girls in every 5 students. The ratio of 3 girls to students is . 5

_

2. Write another ratio of cars to trucks c using 15 for the number of trucks. 15 3. Write these two ratios as a proportion.

_

4. Cross multiply and solve.

_ _

cars → 8 c ← cars = trucks → 3 15 ← trucks

_8 = _c

3 15 8 · 15 = 3c 120 = 3c 3 3 40 = c

_ _

Forty cars are ordered when 15 trucks are ordered.

GO ON GO ON Lesson 4-1 Proportions

47

YOUR TURN! A recipe that makes 3 dozen cookies calls for 7 cups of flour. How many dozens of cookies can be made with 28 cups of flour? 1. Write the ratio of dozens of cookies to cups of flour.

_3

7 dozens of cookies cups of flour 2. Write another ratio of to f 28 using for the number of cups of flour. 28

_

is the number of dozens made with 28 cups of flour.

4. Cross multiply and solve.

_3

3

·

7 28

84 _ 7

= = = f=

12

cookies → 3 cups → 7

=

28 ← cups

_f 28 7f

_7f

7 12

dozen cookies can be made with 28 cups of flour.

Irina uses 4 inches of wire for every 3.6 feet of ribbon to make big bows. If she has 48 inches of wire, how many feet of ribbon feet does she need to use all the wire?

Kraig

48 4 =_ _ z

3.6 4z = 48 · 3.6 4z = 1,728 z = 432

Irina needs 432 feet of ribbon.

Uma

z 4 =_ _

48 3.6 3.6z = 192 53.3 = z

Irina needs 53.3 feet of ribbon.

Circle correct answer(s). Cross out incorrect answer(s).

Chapter 4 Rates and Proportional Reasoning

Frances

48 4 =_ _ z

3.6 4z = 48 · 3.6 4z = 172.8 z = 43.2

Irina needs 43.2 feet of ribbon.

Copyright © by The McGraw-Hill Companies, Inc.

Who is Correct?

48

f

_f ← cookies

_

3. Write these two ratios as a proportion.

, where

Guided Practice Determine whether each pair of ratios is proportional. Write = or ≠ in each circle. 5 10 18 2 1 __ = ___ 2 __ ≠ ___ 9 7 18 42 3

36 ___ 12

12 = ___

4

4

350 _____

2 = ___

1,750

10

Step by Step Practice 5

a 2 = ___ Solve the proportion. ___ 12 36

2 × 36 = 12 × a 72 = 12a 72 = 12a 12 12 a=6

Step 1 Cross multiply.

_ _

Step 2 Solve. Step 3 The solution is

6

Solve each proportion. n 3 6 __ = __ 8 4

n 3 6

Copyright © by The McGraw-Hill Companies, Inc.

= n=

.

7

4 8

· ·

34 w = ___ __ 3

3 ___ 0.2 __ =

_1 or 0.−3

11

3 ___ __ = 27

2

10

7.5 2.5 _____ = ___

12

Two apples cost $0.58. How many apples cost $2.32?

13

Refer to the price sticker on the books to the right. 3 How many books cost $17.85?

14

Twenty bows make 8 centerpieces. How 75 many bows make 30 centerpieces?

15

Thirty-two yards of cloth make 6 blankets. 48 How many yards make 9 blankets?

14.52

10

x

43.56

· ·

9

9 b ___ = ___ 15

w 3 2

= w=

8

6

51

5

n

51 34

3

d

18

8

 

GO ON Lesson 4-1 Proportions

49

Step by Step Problem-Solving Practice

Problem-Solving Strategies ✓ Make a table.

Solve. 16

Look for a pattern. Guess and check. Solve a simpler problem. Work backward.

PETS Cats drink about 2 milliliters of water for each gram of food they eat. If a cat eats about 15 grams of food, how much water will it drink? Understand

Read the problem. Write what you know. Cats drink

1

2

milliliters of water for every gram of food.

Pick a strategy. One strategy is to make a table.

Plan

water (mL)

2

4

6

food (g)

1

2

3

8 4

10 5

12 6

14 7

16 8

18 9

What pattern do you observe? Write the pattern as 2 mL water a ratio.

Solve

_ 1 g food

Use the table or the ratio to find the solution.

30 mL

Check

BIKING A group of bicyclers went on a bike ride. After 3 hours, they had traveled 48 miles. How far did they travel in one hour? 16 miles How long will it take for them to complete a 60-mile round trip? Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

_

3 3 hours 4

18

SCHOOL On a playground there are 3 pieces of play equipment for every 24 students. If there are 12 pieces of equipment, how many students can be on the playground? 96

50

Chapter 4 Rates and Proportional Reasoning

Copyright © by The McGraw-Hill Companies, Inc.

17

Use your answer to write a ratio. Is your ratio equivalent to the ratio in the table?

Describe a situation in which you might use a proportion.

19

Sample answers may include descriptions of maps, scale models, unit rates, or recipes (doubling or tripling ingredients).

Skills, Concepts, and Problem Solving Determine whether each pair of ratios is proportional. Write = or ≠ in each circle. 5 2 4 4 20 __ ≠ ___ 21 ___ = __ 9 8 16 10 22

3 __ 4

12 ≠ ___ 3

Copyright © by The McGraw-Hill Companies, Inc.

Solve each proportion. 2 8 20 x= 24 __ = __ 5 x

23

72 ___

25

2 = __ 4 __

12

7

12 = ___ 2

y=

y

26

3 ___ __ = b

b=

18

27

c 2 = ___ __

28

d 4 = ___ __

d=

20

29

20 ___ 10 ___ =

30

3 d = __ ___

d=

6

31

c 1.2 = ___ ___

5

5

30

25

8

16

9

4

9

36

f

c=

f=

1.5

c=

14 8 2 0.2

Solve. 32

READING James read 4 pages in 6 minutes. At this rate, how long would it take him to read 6 pages?

9 minutes 33

FOOD How many Calories would you expect to find in 5 slices of the same kind of pizza? (See the photo at right.)

1437.5 Calories 34

BUSINESS Jermaine can type 180 words in 3 minutes. How many words would you expect him to type in 10 minutes?

600 words

2 slices = 575 Calories

GO ON Lesson 4-1 Proportions

Stockdisc/SuperStock

51

Vocabulary Check Write the vocabulary word that completes each sentence. 35

A(n)

proportion

is an equation stating that two ratios or

rates are equivalent. 36

37

unit rate A(n) is a ratio of two measurements made with different units in which the second amount is 1. Writing in Math Describe the relationship between the two ratios in a proportion.

The two ratios in a proportion are equal. They have a constant rate or ratio.

Spiral Review Use the percent equation or proportion to solve.

(Lesson 3-2, p. 11)

38

Find the sale price of a $10 shirt at a 25% discount.

39

Find the total cost of a $19.50 book with 7.8% sales tax.

40

Find the sale price of a $179.99 bike marked off 15%.

Solve.

$7.50 $21.02 $152.99

(Lesson 3-1, p. 4)

#VTJOFTT
4DIPPM
.BKPST

BUSINESS Review the graph at right. How many business 9 school majors have not found jobs?

42

FASHION When ordering colors for the new fall line, a dressmaker chose yellow and orange for 13 of her selections. If the dressmaker selected 65 items, what percent are not 80% yellow or orange?

Order each set of numbers from least to greatest. 75 0.7, 75%, 75 43 75%, ___, 0.7 50 50

_

_

44

54 55%, 0.055, ____ 100

0.055, 54 , 55% 100

45

19 0.019, ___, 19% 50

0.019, 19%, 19 50

52

Chapter 4 Rates and Proportional Reasoning

_

(Lesson 2-3, p. 57)


CVTJOFTT
TDIPPM
NBKPST
GPVOE
FNQMPZNFOU
BGUFS
HSBEVBUJPO



Copyright © by The McGraw-Hill Companies, Inc.

41

Lesson

4-2 Unit Conversions KEY Concept Convert units within the same measurement system or between the two different measurement systems using proportions . 1 kilogram equals about 2.2 pounds. How many kilograms does an 8-pound cat weigh? Let x = weight of cat in kilograms. 2.2 lbs _____ 8 lbs ______ = x kg

1 kg

2.2 · x = 8 · 1 8 2.2x = ___ ____ 2.2

2.2

x ≈ 3.6

Make certain that the numerators and the denominators of the two ratios have the same units. Find the cross products. Simplify. So, an 8-pound cat weighs about 3.6 kilograms.

Copyright © by The McGraw-Hill Companies, Inc.

This symbol means approximately equal to or about.

1 pint equals 16 fluid ounces. How many ounces does a 2.5 pint pitcher contain? Let n = the number of ounces in 2.5 pints. 16 fl oz n fl oz _______ = ______ 1 pt 2.5 pt 16 · 2.5 = n · 1 n = 40

Find the cross products.

3AF1.4 Express simple unit conversions in symbolic form. 3MG1.4 Carry out simple unit conversions within a system of measurement. 6AF2.1 Convert one unit of measurement to another.

VOCABULARY customary system of measurement a measurement system that includes units such as foot, pound, and quart metric system of measurement a measurement system that includes units such as meter, kilogram, and liter proportion an equation stating that two ratios or rates are equivalent 28 7 Example: __ = ___ 8 32 (Lesson 3-2, p. 11)

unit rate a rate that describes how many units of the first type of quantity are equal to 1 unit of the other type of quantity (Lesson 1-2, p. 11)

Simplify.

So, a 2.5-pint pitcher contains 40 fluid ounces.

Example 1

YOUR TURN!

_

1 There are 3 feet in 1 yard. Convert 5 yards 2 to feet, f, using a proportion. n ft _ _ _1 5 yd _ n2 _1 16

3 ft 1. Write a proportion. 1 yd = Use the ratio of 1 feet to yards. 3·5 = 2 2. Cross multiply n= and solve.

2

There are 1,000 milliliters in a liter. Convert 3,200 milliliters to liters, z, using a proportion.

_ _

1. Write a proportion. 1,000 mL = 32 mL Use the ratio of 1L zL mL to L . =

1,000z

2. Cross multiply and solve.

32

z = 0.032 Lesson 4-2 Unit Conversions

53

Example 2 A piece of ribbon is 4.5 inches long. One inch equals about 2.54 centimeters. How many centimeters, x, long is the ribbon? The units are the same in the numerators and in the denominators.

4.5 in. 1 in. = ______ ________ 2.54 cm

x

1. Find the cross products. Solve. 1 · x = 4.5 · 2.54 x ≈ 11.43 cm 2. Write the solution. The ribbon is about 11.43 cm long. YOUR TURN!

1 lb = _ 4.8 lb _

A bag of apples weighs 4.8 pounds. One pound equals 16 ounces. How many ounces, z, does the bag weigh?

16 oz

z

1. Find the cross products. Solve.

16 · 4.8 = z · 1 z=

76.8 4.8 lbs is equal to 76.8 oz.

2. Write the solution.

Example 3 If Ernesto can run 100 meters in 10 seconds, how long, in minutes, does it take him to run 4,200 meters? x seconds 10 seconds x = 420 seconds

2. Cross multiply and solve.

Now convert the 420 seconds to minutes. Set up the proportion so that the units cancel. The seconds in the numerator cancel the 1 minute seconds in the denominator. 420 seconds · = 7 minutes 60 seconds 3. Write the solution. It takes Ernesto 7 minutes to run 4,200 meters.

__

YOUR TURN! The width of Melinda’s bedroom measures 9 yards. How many feet does it measure?

3 feet = _ n _ 1 yard

9 yards

1. Write a proportion. 2. Cross multiply and solve. 3. Write the solution. 54

3·9=1·n

27 feet is equal to 9 yards.

Chapter 4 Rates and Proportional Reasoning

,n=

27

Copyright © by The McGraw-Hill Companies, Inc.

4,200 meters 100 meters = __ __

1. Write a proportion.

Who is Correct? There are 4 quarts in a gallon. 9 qt = ? gal

Shawanda

Tito

Kendrick

_ _

_ _

x 4 qt = qt 9 1 gal x = 2.5 9 qt = 36 gallons

9 qt 4 qt = x 1 gal 4 · x = 1.9 4x = 9 4 4 x = 2.25 gal

9 × 4 = 36 9 qt = 36 gallons

_ _

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice Convert. 1 1 3__ mi = 4 2 3

11 pt = 6c=

17,160 ft 5.5 3

qt

Length

Weight

Capacity

1 ft = 12 in.

1 lb = 16 oz

1 c = 8 fluid oz

1 yd = 3 ft

1 ton (T) = 2,000 lb

1 pt = 2 c

1 mi = 5,280 ft

1 qt = 2 pints

pt

1 gal = 4 qts

Copyright © by The McGraw-Hill Companies, Inc.

Step by Step Practice 4

Convert 3.5 miles to kilometers. There is about 0.621 mile in 1 kilometer. Step 1 Write a ratio with the number of miles, 3.5, over y,the 3.5 miles unknown number of kilometers.

_ y

Write the ratio with 0.621 mile to 1 kilometer.

0.621 mile _ 1 km

Step 2 Set up a proportion. Step 3 Find the cross products. Simplify.

0.621 mile 3.5 miles = _ _ y

1 km

3.5 · 1 = y · 0.621 3.5 = _ 0.621y _ 0.621 0.621 y = 5.6

Step 4 3.5 miles is about is

5.6 kilometers

.

GO ON

Lesson 4-2 Unit Conversions

55

Convert each unit using a proportion. 5

One meter is equal to about 39.37 inches. How many meters are in 158 inches? Let x = the number of meters. Find the cross products: 158 · 158 inches are equal to about

4

4

1

= 39.37 ·

158 = _ 39.37 _ x 1

x

meters.

32

6

48 in. =

ft

7

4c=

8

40 oz = 2.5 lb

9

6 yd =

10

5.6 km ≈ 3.48 mi (1 mi ≈ 1.609 km)

11

5 qt ≈ 4.73 L (1 L ≈ 1.057 qt)

fl oz

18

Step by Step Problem-Solving Practice

Problem-Solving Strategies Use a table. ✓ Look for a pattern. Guess and check. Act it out. Solve a simpler problem.

Solve. 12

COOKING A recipe for ice cream calls for 66 fluid ounces of milk. Marylynn has a 1-cup measuring tool. How many times will she have to fill it with milk? Understand

Read the problem. Write what you know.

66

The recipe calls for 1 cup equals Plan

ft

8

fluid ounces.

fluid ounces.

Pick a strategy. One strategy is to make a table to look for a pattern. 1

2

3

Fluid ounces

8

16

24

4 32

5 40

6 48

7 56

8 64

9 72

Start with what you know: 1 cup = 8 fluid ounces Continue filling in the table. Solve.

Two cups equal

16

fluid ounces.

8 Notice that cups are equal to 64 ounces. How many cups are in 66 ounces then? 8 2 cups with ounces left over. Write the leftover ounces as a fraction and as a decimal. 1 or 0.25

_

Solve Check

56

_1

4 8.25 cups or 8 cups How many cups are in 66 ounces? 1 times. 4 Therefore, Marylynn will have to fill the cup 8__ 4 Does the answer make sense? Look over your solution. Did you answer the question?

Chapter 4 Rates and Proportional Reasoning

10 80

Copyright © by The McGraw-Hill Companies, Inc.

Cups

13

BUSINESS The United States exports over 200 billion pounds of coal. How many tons are exported? Check off each step.

over 100 million

14

✔

Understand

✔

Plan

✔

Solve

✔

Check

_

60 2 3

NATURE How many yards high is Niagara Falls?

NATURE The height of

When you set up and solve a proportion, what operation(s) do you use? Explain.

15

Niagara Falls is 182 feet.

Multiplication and sometimes division are used; the first step is to multiply. If the variable has a coefficient, then you divide.

Skills, Concepts, and Problem Solving

Copyright © by The McGraw-Hill Companies, Inc.

Convert. 16

720 cm =

18

57,000 mL =

7.2

m

57

L

17

64 kg =

19

3.2 cm =

64,000 g 32

mm

Metric Units Length

Weight

Capacity

1 km = 1,000 m

1 kg = 1,000 g

1 kL = 1,000 L

1 m = 100 cm

1 g = 1,000 mg

1 L = 1,000 mL

1 cm = 10 mm

Convert each unit using a proportion. Round to the nearest tenth.

88.6

20

1 m ≈ 39.37 in.; 2.25 m ≈

21

1 kg ≈ 2.2 lb; 15 lb ≈

22

1 in ≈ 2.54 cm; 19 in. ≈

48.26

23

1 gal ≈ 3.79 L; 11 L ≈

2.9

6.8

in.

kg cm gal GO ON Lesson 4-2 Unit Conversions

Darwin Wiggett/Getty Images

57

Simplify. 24

1 minute 540 seconds · __________ = 60 seconds

25

1 yard 180 inches · _________ = 36 inches

5 yards

26

1 pound 352 ounces · _________ = 16 ounces

22 pounds

27

1 cup 920 fl ounces · __________ = 8 fl ounces

9 minutes

115 cups

Solve. 28

If Sofia can run 200 meters in 15 seconds, how long, in minutes, does it take her to run 6,000 meters? 7.5 minutes

29

If Karen can read 25 pages of her book in 50 minutes, how long, in 19 hours hours, will it take her to read a 570-page book?

30

If Harold uses 8 lemons to make 6 quarts of lemonade, how much, 5.25 gallons in gallons, will 28 lemons make?

Solve. 31

BABIES The weight of a baby girl at birth was 7.5 pounds. The weight of a baby boy was 125 ounces. Which baby weighed more and by how much?

32

CONSTRUCTION How many tons does the Statue of Liberty weigh?

225 T 33

BAKING Natalie needs 85 grams of chopped walnuts for a recipe. If she has a 2-ounce package, does she have enough? A gram equals about 0.035 of an ounce.

No, she needs almost a full ounce more.

CONSTRUCTION The Statue of Liberty weighs 450,000 pounds.

58

Chapter 4 Rates and Proportional Reasoning

David Buffington/Getty Images

Copyright © by The McGraw-Hill Companies, Inc.

The baby boy weighed 5 ounces more.

Vocabulary Check Write the vocabulary word that completes each sentence. 34

Meters, grams, and liters are measurements in the metric system.

35

Miles, pounds, and ounces are measurements in the customary system.

36

Writing in Math Explain how to set up a proportion to convert 26 ounces to pounds. (There are 16 ounces in 1 pound.)

Sample answer: Set up a proportion, making sure that the numerators are of one unit and the denominators are of the other unit of measurement. Then cross multiply to solve for the unknown unit. 16 oz = 26 oz 26 oz = 1.625 lbs 1 lb x lbs

_ _

Spiral Review Determine whether each pair of ratios is proportional. Write = or ≠ in each circle. (Lesson 4-1, p. 46) 9 3 15 35 27 14 37 ___ = ___ 38 __ ≠ ___ 39 ___ ≠ ___ 33 25 50 11 4 21

Copyright © by The McGraw-Hill Companies, Inc.

Solve using the percent equation. Check your answer. 40

What percent of 225 is 18?

41

What is 105% of 60?

42

45 is what percent of 180?

Solve.

(Lesson 3-1, p. 4)

8%

63 25%

(Lesson 1-1, p. 4)

43

BASEBALL The batting average of a baseball player is a ratio of number of hits to the number of batting attempts, or at-bats. What is the batting average of a player who has had 18 hits with 25 0.72 at-bats? Write in decimal form.

44

EDUCATION There are 13 girls in a class of 28 students. Write a ratio in simplest form to show the number of boys to the total number of students in this class. 15

_ 28

Lesson 4-2 Unit Conversions

59

Chapter

4

Progress Check 1

(Lessons 4-1 and 4-2)

Determine whether each pair of ratios is proportional. Write = or ≠ in each circle. 5SDAP1.3, 6AF1.2 5 10 15 16 1 __ ≠ ___ 2 ___ ≠ ___ 2 5 5 2 3

26 ___ 8

= 3__1 4

Solve each proportion. 5SDAP1.3, 6AF1.2 34 n 11 1 5 ___ = __ n= 15 5 3

_

7

0.75 ___ ____ = n 5

25

n=

3.75

4

45 ___

6

72 2 = ___ __

d=

108

8

90 27 = ___ ___

c=

10

5

10 ≠ ___ 42

3

d

c

3

Convert each unit using a proportion. 3AF1.4, 3MG1.4, 6AF1.4 9

8 fluid ounces = 1 cup; 24 fluid ounces =

10

1,000 mm = 1 m; 125 mm =

11

1 in. ≈ 2.54 cm; 10 in. ≈

25.4

cm

12

1 km ≈ 0.62 mi; 5 km ≈

3.1

mi

13

1 qt = 2 pt; 13 pt =

14

1 mi = 5,280 ft; 7,920 ft =

1.5

mi

15

1 m = 100 cm; 11.75 m =

1,175

cm

16

1 L = 1,000 mL; 5.6 L =

6.5

0.125

3

c

m

qt Copyright © by The McGraw-Hill Companies, Inc.

5,600

mL

Solve. 3AF1.4, 3AF2.2, 3MG1.4, 6AF1.4 17

TRAVEL The gasoline tank of Roxanna’s car holds 60 liters of gas. A gallon of gas costs $2.10. If Roxanna has $35, does she have enough money to fill the tank? Exactly how much will she spend if the tank is completely empty? (1 L = 1.057 qt and 1 gallon = 4 qt)

Yes, she would need $33.30 to fill the tank. GEOGRAPHY The Ted 18

GEOGRAPHY How many yards is the Ted Williams Tunnel?

2,816 yards 60

Chapter 4 Rates and Proportional Reasoning

Darren McCollester/Getty Images

Williams Tunnel under Boston Harbor is 8,448 feet long.

Lesson

4-3 Solve Rate Problems

3AF2.1 Solve simple problems involving a functional relationship between two quantities. 3AF2.2 Extend and recognize a linear pattern by its rules. 6AF2.3 Solve problems involving rates, average speed, distance, and time.

KEY Concept A rate is a ratio that involves different units. A unit rate has a denominator of 1 unit.

VOCABULARY

Average speed is a type of unit rate that expresses the distance traveled per unit of time. For example, suppose a car travels 330 miles in 6 hours. Its average speed is 55 mi/hr. 330 miles ____ 330 _________ = 55 mi/hr = 6 6 hours The distance formula shows the relationship between distance, rate, and time. d is distance

d=r·t

t is time

unit cost the cost of a single piece or item (Lesson 1-2, p. 11) unit rate a rate that describes how many units of the first type of quantity are equal to 1 unit of the other type of quantity (Lesson 1-2, p. 11) proportion an equation stating that two ratios or rates are equivalent 28 7 Example: __ = ___ 8 32

r is rate

(Lesson 3-2, p. 11)

Copyright © by The McGraw-Hill Companies, Inc.

To solve rate problems, you can extend linear patterns or solve a proportion.

Example 1 A semitruck has 18 wheels. How many wheels are on 6 semitrucks? 1. Extend the pattern by using repeated addition. 18 + 18 + 18 + 18 + 18 + 18 = 108 2. You can also extend the pattern by using ratios. Write a ratio for wheels to trucks. Then use a proportion to find the number of wheels, x, on 6 trucks. 18 __ ___ =x 6 1 x = 108

YOUR TURN! A horse has 4 legs. How many legs are on 14 horses? 1. Extend the pattern by using repeated addition.

4+4+4+4+4+4+4+4+ 4 + 4 + 4 + 4 + 4 + 4 = 56 2. Write a ratio for legs to horses. Then use a proportion to find the number of legs, x, on 14 horses.

_4 1

= x = 56

_x 14

GO ON Lesson 4-3 Solve Rate Problems

61

Example 2

YOUR TURN!

Peaches cost $4.18 for 2 pounds. How much will 5 pounds of peaches cost?

Raisins cost $1.12 for 8.2 ounces. How much will one pound cost? Round to the nearest cent.

1. Write a ratio of the cost to the number of pounds. $4.18 2 lb

_

2. Set up a proportion to find the cost, c, of 5 pounds. $4.18 c = 5 lb 2 lb

_ _

3. Cross multiply and solve. $4.18 · 5 = 2c $20.90 = 2c $10.45 = c Five pounds of peaches will cost $10.45.

1. Write a ratio of the cost to the number of ounces.

$1.12 _ 8.2 oz

2. Set up a proportion to find the cost, c, of 1 pound.

$1.12 = _ c _ 8.2 oz

16 oz

3. Cross multiply and solve.

$1.12 · 16 = 8.2c $17.92 = 8.2c $2.185 = c One pound of raisins will cost

$2.19 .

Example 3

1. Write a ratio for the number of miles to the number of hours. 405 mi 3h

_

2. Simplify to a unit rate. 135 mi ÷3 _ 405 mi = 405 _ _ = 3h

1h

3÷3

Substitute the distance and rate into the distance formula. Solve for t. 550 = 135 · t 550 =t 135 4.07 = t

_

3. It will take the helicopter 4.07 hours to fly 550 miles. 62

Chapter 4 Rates and Proportional Reasoning

d=r·t

Copyright © by The McGraw-Hill Companies, Inc.

A helicopter flew 405 miles in 3 hours. How long will it take the helicopter to fly 550 miles?

YOUR TURN! Toshi took a boat ride traveling 67 miles. The boat took 50 minutes for the trip. How long will it take the boat to travel 100 miles?

67 mi _

1. Write a ratio for number of miles to number of minutes. 2. Simplify to a unit rate.

50 min

67 mi = __ 1.34 mi 67 ÷ 50 = __ __ 50 min 50 ÷ 50 1 min

The average speed of the boat was per minute.

1.34

miles

d=r·t

100 = 1.34 · 74.6 = t

Substitute the distance and rate into the distance formula.

t

3. It will take the boat 74.6 minutes to travel 100 miles.

Who is Correct? A high-speed train travels at 160 miles per hour. How many hours would it take for the train to travel 800 miles?

Copyright © by The McGraw-Hill Companies, Inc.

Julia d=r∙t 800 = 160 ∙ t 800 = 160t 160 160 t=5

_ _

urs. The train will take 5 ho

Amos d=r∙t 160 = 800 ∙ t 160 = 800t 800 800 .2 t=

_ _

urs. The train will take .2 ho

Tracee

800 miles 160 miles = _ _ t hours 1 hour 160t = 800 160t = 800 160 160 t=5

_ _

urs. The train will take 5 ho

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice Extend each pattern. Use repeated addition. 1

Each package has 8 hot dogs. How many hot dogs are in 9 packages? 72 hot dogs

2

Each music CD has 16 songs. How many songs are on 7 CDs?

112 songs

GO ON

Lesson 4-3 Solve Rate Problems

63

Solve. Use a proportion. 3

Each bicycle has 2 wheels. How many wheels are on 12 bicycles?

24 wheels 4

Each book has 108 pages. How many pages are in 9 books?

972 pages Find each total cost. 5

One orange is $0.15. How much do 12 oranges cost?

$1.80 6

Three pounds of cheese cost $9.93. How much do 5 pounds cost?

$16.55 7

Five pounds of potatoes cost $2.98. How much do 2 pounds cost?

$1.19 8

Refer to the art at the right. How much would 5 pounds of bananas cost?


POUNDS FOR


$2.00 9

Jelly costs $0.16 an ounce. How much does 12 ounces cost?

Step by Step Practice 10

Robert drove 360 miles in 6 hours. How long will it take him to drive 990 miles? Step 1 Write a ratio with the number of miles to the number of hours. Step 2 Find the unit rate. 360 ÷ 6 = 60 Substitute the rate into the distance formula. Solve for t. Step 3 It will take Robert 16.5 hours to drive 990 miles.

64

Chapter 4 Rates and Proportional Reasoning

360 mi _ 6h d=r·t

990 = 60 · t 990 = t 60 16.5 = t

_

Copyright © by The McGraw-Hill Companies, Inc.

$1.92

Solve. 11

Two slices of Dan’s Famous Pizza have 460 Calories. How many Calories would you expect to be in 5 slices of the same pizza?

460 2 460 = ________ x ________ 2 5 ________

Write a ratio.

Set up a proportion.

Cross multiply and solve. 12

x

2x

= 2,300


FOR 

1,150 calories

=

Refer to the art at the right. How many grapefruits can be 6 purchased for $2.10?

Step by Step Problem-Solving Practice

Problem-Solving Strategies ✓ Write an equation.

Solve. 13

Look for a pattern. Draw a diagram. Solve a simpler problem. Work backward.

MOVIES The cost for 4 tickets to the movies is $30. There are 76 students in the seventh grade. If they all go on a field trip to see a movie, how much will it cost? Understand

Read the problem. Write what you know.

4

$30

movie tickets cost

.

There are 76 students. Copyright © by The McGraw-Hill Companies, Inc.

Plan

Pick a strategy. One strategy is to write an equation. Let t = the cost of 1 ticket. Set up a proportion.

Solve

$30 t ________ = _______ 4 tickets

1 ticket

Cross multiply and solve.

30

·

1 = 30 =

t · 4t t = 7.50

4

One ticket costs $7.50. Set up another equation. $570 $7.50 × 76 = 76 tickets would cost Check

$570

.

Compare the ratios. In simplest form, they all should be the same. $30 $7.50 $570 ________ = _______ = _________ 4 tickets 1 ticket 76 tickets

GO ON

Lesson 4-3 Solve Rate Problems

65

14

HIKING A group of people went on a weekend hike. After 9 hours, they had traveled 27 miles. On average, how far did they travel in 3 miles one hour? . About how long will it take for them to complete a 40-mile round 13.3 hours trip? Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

15

TRAVEL Duane traveled a distance of 300 miles in 5 hours. How 120 miles far had he traveled after 2 hours?

16

Describe a time in life when knowing how to calculate a unit cost will be useful. Explain how to find a unit cost.

See TWE margin.

Skills, Concepts, and Problem Solving Extend each pattern. Use repeated addition. 17

Each shirt has 8 buttons. How many buttons on 8 shirts?

18

Each guitar has 6 strings. How many strings on 11 guitars?

19

One package has 9 markers. How many markers in 12 packages?

20

One can has 15 raviolis. How many raviolis in 4 cans?

64 buttons

108 markers

60 raviolis

Solve. Use a proportion. 21

Each puzzle has 750 pieces. How many pieces in 10 puzzles?

22

Each necklace has 36 pearls. How many pearls on 7 necklaces?

23

Each jar has 48 pickles. How many pickles in 5 jars?

66

Chapter 4 Rates and Proportional Reasoning

7,500 pieces

252 pearls

240 pickles

Copyright © by The McGraw-Hill Companies, Inc.

66 strings

24

A school cabinet can display 26 trophies. How many trophies can 156 trophies be displayed in 6 cabinets?

Find each total cost. Round to the nearest cent. 25

One shirt costs $16.99. How much do 3 shirts cost?

26

Twelve cans of dog food cost $9.99. How much is 1 can?

27

Refer to the art at the right. How much do 4 tires cost?

28

Sixteen ounces of beef cost $4.99. How much does 1 ounce cost?

29

Jelly costs $0.16 for 1 ounce. How much does a 12-ounce jar cost?

30

Peanut butter costs $0.22 for 1 ounce. How much does a 16-ounce $3.52 jar cost?

31

Gravel costs $4.98 for 1 pound. How much does a 40-pound bag $199.20 cost?

$50.97

$0.83

$267.96 $0.31

 
TIRE

$1.92

Solve.

Copyright © by The McGraw-Hill Companies, Inc.

32

TRAVEL Cole rides at a rate of 18 miles per hour. How far will he travel in 2.5 hours?

45 miles

33

PAINTING The line-painting machine covers 255 feet in 10 minutes. At what rate does the machine work?

25.5 feet per minute

34

SNOW The snow was falling at a rate of 3 inches per hour. If the snow fell at that rate for 24 hours, how much snow would be on the ground?

72 inches

35

PAINTING Maya can paint 200 meters of a wall border per hour. What length of border will she paint in a 7.5-hour workday?

1,500 meters

36

EARTH SCIENCE Surface waves from an earthquake travel about 6 kilometers per second through Earth’s crust. How long would it take for a surface wave to travel 810 kilometers?

135 seconds

GO ON

Lesson 4-3 Solve Rate Problems © Brand X Pictures/PunchStock

67

37

COMMUNITY SERVICE Ivan participated in a fund-raiser for 18 hours. He raised $90. Sal participated in a different fund-raiser and raised $120. If Sal worked for 25 hours, which boy collected money at a higher rate?

Ivan

38

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LIFE SCIENCE Refer to the art shown at the right. If both earthworms start at the same place, how far apart will they be in 5 hours?

350 feet

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Vocabulary Check Write the vocabulary word that completes each sentence. 39 40

41

The cost of a single item is the

unit cost

.

rate A(n) is a ratio of two measurements or amounts made with different units. 5 miles _______ x miles _______ =

Writing in Math Teal and Ian are solving the proportion shown at the right. Teal says that the unit rate is 10 miles per hour. Ian says it is 2.5 miles per hour. Who is correct? Explain.

2 miles

1 hour

Ian is correct. To find the value of x, which is the unit rate, divide 5 by 2.

Spiral Review

42

8 2 = __ __

Solve. 44

x=

x

7

(Lesson 4-1, p. 46)

28

43

15 45 ___ ___ = 27

z

z=

(Lesson 2-3, p. 57)

BUSINESS Eighteen out of 72 customers prefer express service, while 24 out of 50 prefer carryout. Write the ratios as fractions in simplest form. Which service do more customers prefer?

12 ; more customers prefer carryout _1 < _ 4

45

25

FASHION Rina noticed that 0.75 of the boys in her class wore 9 jeans. She noticed that ___ of the girls wore jeans. Which is greater, 15 the portion of the boys or the portion of the girls who were not wearing jeans?

See TWE margin. 68

Chapter 4 Rates and Proportional Reasoning

9

Copyright © by The McGraw-Hill Companies, Inc.

Solve each proportion.

Lesson

3AF2.1 Solve simple problems involving a functional relationship between two quantities. 6NS1.3 Use proportions to solve problems. Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of KEY Concept both sides of an equation by a multiplicative inverse. 7AF4.2 Solve multistep problems You can use proportions to solve problems about similar involving rate, average speed, figures. Similar figures have the same shape but may have distance, and time or a direct variation. different sizes. The corresponding angles of similar figures are

4-4 Solve Problems Using Proportions

congruent, or the same. The corresponding sides of similar figures are proportional. &

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VOCABULARY


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similar figures figures that have the same shape but may have different sizes

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proportion an equation stating that two ratios or rates are equivalent 28 7 Example: __ = ___ 8 32

The corresponding angles of triangles ABC and DEF are congruent. The ratios of their corresponding sides are 15 3 18 12 = = = =3 equivalent. 5 6 4 1

_ _ _ _

(Lesson 3-2, p. 11)

Copyright © by The McGraw-Hill Companies, Inc.

If two figures are similar, the ratio of the two sides of one of the figures is equal to the ratio of the corresponding two sides AB , of the other figure. In triangles ABC and DEF, the ratio ____ AC 15 5 DE , which is ___ . which is __, is equal to the ratio ____ DF 4 12

Example 1 Find the value of x. The two rectangles are similar. 1. The ratio of the corresponding sides 9 TU and XY is ____. 31.5 2. The ratio of the corresponding sides x. TS and XW is ___ 14 3. Since the two rectangles are similar, these two ratios are equal. Write the proportion and solve for x. 4. The length of side TS is 4 meters.

5


N

6


N

:


N

Y
N 4

9 _ _x =

31.5 14 31.5x = 14(9) 126 31.5x = _ _ 31.5

9

31.5

x=4

7

8

;

Be careful to write the ratios in the same order.

Find the cross products. Simplify. Divide by 31.5. GO ON

Lesson 4-4 Solve Problems Using Proportions

69

YOUR TURN! Find the value of x. The two figures are similar. 1. The ratio of side AB to side AD is 2. The ratio of side EF to side EH is 3. Since the two figures are similar, these two ratios are equal. Write the proportion and solve for x.

_5

_x

4

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.

3


DN

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.

_5 = _x

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4 3 5(3) = 4(x) 15 = 4x

4x 15 = _ _

4 4 x = 3.75

The length of side EF is 3.75 cm .

Proportions can be used to find unknown measures in similar figures, in percent proportions, and for unit conversions.

Example 2

1. Write a ratio for feet to seconds.

28 ft _____

2. Set up a proportion to find the time it would take to travel 70 feet.

28 ft = _____ 70 ft _____

3. Cross multiply and solve.

4s 4s

t

28t = 4 · 70 28t = 280 t = 10

It will take 10 seconds for the car to travel 70 feet.

70

Chapter 4 Rates and Proportional Reasoning

Copyright © by The McGraw-Hill Companies, Inc.

A battery-powered toy car traveled 28 feet in 4 seconds. How long would it take the car to travel a total of 70 feet?

YOUR TURN! Shane’s grandmother lives 198 miles from his house. He left his home and drove 90 miles in 1.5 hours before he stopped for gas. How long will it take Shane to make the trip to his grandmother’s house if he travels at the same rate for the whole trip? 1. Write a ratio for miles to hours. 2. Set up a proportion to find the time it would take to travel 198 miles.

90 mi _

1.5h

90 mi = _ 198 mi _

t

1.5h

90t = 1.5 · 198

3. Cross multiply and solve.

90t = 297 t = 3.3 It will take 3.3 hours for Shane to get to his grandmother’s house.

Who is Correct? Jerry ran 5 miles in 35 minutes. Use a proportion to find how long it would take him to run 7.5 miles at that rate.

Janet

Copyright © by The McGraw-Hill Companies, Inc.

_ _

5 mi = x 75 35 min 5 · 75 = 35x x = 11 miles

Aiden

x 5 mi = _ _

35 7.5 mi 5 · 35 = 7.5x x = 23 minutes

Sergio

7.5 mi 5 mi = _ _ x

35 min 5x = 35 · 7.5 x = 52.5 minutes

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice Find the value of x in each pair of similar figures. 1

x = 3.6 

2

x = 5.2

 Y

Y









GO ON Lesson 4-4 Solve Problems Using Proportions

71

Step by Step Practice 3

Raul traveled 150 miles in 3 hours. At that point, he reduced his rate of speed by 20%. How far did he travel in 2 hours? Step 1 Find Raul’s rate of speed at the beginning of the trip. Step 2 Reduce that amount by 20%.

40 mph

His new rate of speed is

50

20% of

50 miles per hour =

10 .

.

Step 3 Use the distance formula, d = r · t, to find his new distance. d= Raul could travel

80

40

·

2

.

miles in 2 hours.

Solve. 4

Levi earns $35 for mowing 5 lawns. At that rate, how many lawns would he need to mow in order to earn $140? He earns

$7

Levi needs to mow 5

5

=

$7

for mowing one lawn.

= number of lawns he needs to mow to

20

lawns.

Savoia wants to attend a class trip to an amusement park. The trip will cost $65. She has already saved $25. She can earn $4 an hour baby-sitting. How many hours will she need to baby-sit to pay for the trip?

10 hours 6

A bird flew for 2 hours at 30 miles per hour, then slowed down and flew 3 more hours at 25 miles per hour. How far did the bird fly?

135 miles

72

.

Chapter 4 Rates and Proportional Reasoning

Copyright © by The McGraw-Hill Companies, Inc.

$ 140 ÷ earn $140.

$35 ÷

Step by Step Problem-Solving Practice

Problem-Solving Strategies ✓ Use a table.

Solve. 7

GEOMETRY

1 times as Each side of polygon ABCD is 3__

Understand

Read the problem. Write what you know.

Look for a pattern. Guess and check. Solve a simpler problem. Act it out.

4 long as the corresponding side of polygon FGHI. Find the perimeter of polygon ABCD.

_

31 4

Each side of polygon ABCD is times as long as the corresponding side of polygon FGHI. "


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JO

' 
JO

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*

First, fill in the corresponding sides of polygons FGHI and ABCD. Then, fill in the measurements of the sides of polygon FGHI. All measurements are in inches. 1 or 3.25 to Multiply the length of each side of polygon FGHI by 3__ 4 find the lengths of the sides of polygon ABCD. Complete the table.

Solve

side

FGHI ABCD

Copyright © by The McGraw-Hill Companies, Inc.

(

Pick a strategy. One strategy is to make a table showing the corresponding sides.

Plan

$

#

−− FG −− AB

length

side

3

−− GH

9.75

−− BC

length

2 6.5

side

−− HI

−− CD

length

side

length

−− IF

5 16.25

−− DA

3 9.75

The perimeter is the sum of the lengths of the sides. What is 42.25 inches the perimeter of polygon ABCD? Does the answer make sense? Look over your solution. Did you answer the question?

Check

8

PROPORTIONS At the same time of day the height of an object and its shadow are proportional to the height of another object and its shadow. If a 6-ft-high doghouse casts a shadow 5 feet long, how tall is a flagpole that casts a 40-ft shadow? Check off each step.

Y 
GU


GU 
GU

48 feet ✔

Understand

✔

Plan

✔

Solve

✔

Check

GO ON Lesson 4-4 Solve Problems Using Proportions

73

9

EVENTS Monica decided to participate in a walk-a-thon for charity. She walked 18 miles and then took a 5-minute break. She then walked another 6 miles to complete the event. It took her 6 hours to complete the walk-a-thon from start to finish. About how many miles did she travel each hour? about 4 miles per hour List the types of problems that proportions can be used to solve.

10

Sample answer: Proportions can be used to find missing lengths in similar figures, indirect measurement, percents, unit costs, and unit rates.

Skills, Concepts, and Problem Solving Find the value of x in each pair of similar figures. 11

10.2 ft

x=

12

12 mm

x=

Y 
GU 
GU


NN


NN 
GU Y

13

11 cm

x= "

Y
DN


DN

& #


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x=

20 in. 5

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4


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3 
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Y
JO 7

Use a proportion to solve each problem. 15

A jet can travel at about 225 miles in 30 minutes. At this speed, how long will it take to travel 800 miles?

about 107 minutes or 1 hour and 47 minutes

16

Enola ran 2 miles in 29 minutes. At this rate, how long would it take Enola to run 3 miles?

43.5 minutes

17

Light travels 720,000 kilometers in 2.4 seconds. Find the speed of light per second.

300,000 kilometers per second

18

A bird flies 30 miles in 18 minutes. How far would the bird fly if it continued at the same rate for 45 minutes?

75 miles

74

$

" (

Chapter 4 Rates and Proportional Reasoning


JO &


JO

%

Copyright © by The McGraw-Hill Companies, Inc.

$

14


NN

Solve. 19

FITNESS Harry jogged 5 miles on Saturday morning at a rate of 0.71 hours 7 miles per hour. How long did he jog? He jogged 3 miles on Monday at a rate of 6 miles per hour. Did he jog for a longer amount of time on Saturday or Monday?

He jogged for a longer amount of time on Saturday. 20

SCUBA DIVING A company produces 39 wetsuits every 2 weeks. How long will it take the company to produce 429 wetsuits?

22 weeks

Vocabulary Check Write the vocabulary word that completes each sentence. 21

Figures whose shapes are the same but may have different sizes are similar figures .

22

A(n) division.

23

Writing in Math Candace says that if two figures are similar, their corresponding sides are equal and their corresponding angles are proportional. Is she correct? Explain.

ratio

is a comparison of two numbers by

Sample answer: No, Candace is not correct. If two figures are similar, their

Copyright © by The McGraw-Hill Companies, Inc.

corrresponding angles are equal and their corresponding sides are proportional.

Spiral Review Solve.

(Lesson 4-3, p. 61)

24

If a car travels 30 miles per hour, what is the distance it travels in 165 miles 5.5 hours?

25

If a turtle travels 7.5 feet in 15 minutes, what is its rate per minute?

26

ENTERTAINMENT Elisa sold 330 tickets to the museum in 5 hours while working at the ticket booth. Later, Ron sold 480 tickets while working an 8-hour shift. Who sold tickets at a higher rate? Explain.

0.5 feet per minute

(Lesson 1-2, p. 11)

_

_

Elisa’s rate was 330 = 66 tickets/h. Ron’s rate was 480 = 60 tickets/h. 5 8 Elisa; 66 > 60 Lesson 4-4 Solve Problems Using Proportions

75

Chapter

Progress Check 2

4

(Lessons 4-3 and 4-4)

Find each total cost. Round to the nearest cent. 3AF2.1, 3AF2.2

$51.96

1

How much are 4 hats?

2

8 tickets cost $60. How much is 1 ticket?

3

One pound of nuts costs $1.99. How much do 5 pounds cost?

$7.50 $9.95

Write the ratio. Then find each unit rate. Round to the nearest tenth. 6AF2.3, 7AF4.2 4

904 people passed through the gate in 8 hours

5

336 meters in 400 minutes

336 _ 400

;

904 _

8 0.8 m/min

;

113 people/h

Find the value of x in each pair of similar figures. 3AF2.1, 6NS1.3 6

6 in.

x=

7

x=

16 mm


JO

Y


NN


NN


JO


JO

Y
NN


NN

Use a proportion to solve each problem. 3AF2.1, 6NS1.3, 8

6AF2.3, 7AF4.2

about 44 pounds

9

Ricardo runs 50 meters in 9 seconds. What is his average speed? 5.6 meters per second Round to the nearest tenth.

10

Two dozen shrimp cost $26. How much do 3.5 dozen cost?

$45.50

Solve. 3AF2.1, 6NS1.3, 6AF2.3, 7AF4.2 11

TRAVEL Paloma drives at a rate of 60 miles per hour for 3 hours. Then she decreases her speed to 55 miles per hour and drives another 1.5 hours. What distance will she travel altogether?

262.5 miles

12

EARTH SCIENCE Surface waves from an earthquake travel about 6 kilometers per second through Earth’s crust. How long would 250 seconds it take for a surface wave to travel 1,500 kilometers?

76

Chapter 4 Rates and Proportional Reasoning

Ryan McVay/Getty Images

Copyright © by The McGraw-Hill Companies, Inc.

A child weighs about 20 kilograms. If 1 kilogram is about 2.2 pounds, how much does the child weigh in pounds?

Chapter

Study Guide

4

Vocabulary and Concept Check proportion, p. 46

Write the vocabulary word that completes each similar figures, p. 69 sentence. unit cost, p. 61

1

unit rate, p. 46 2

3 4 5

An equation stating that two ratios are equivalent is a proportion .

unit rate A describes how many units of one type of quantity are equal to 1 unit of another type of quantity. The

unit cost

is the cost of a single piece or item.

Similar figures have the same shape but may have different sizes. Which of the three sets of figures below are not similar figures? "

#

B

$

Lesson Review

Copyright © by The McGraw-Hill Companies, Inc.

4-1

Proportions

(pp. 46–52)

Determine whether each pair of ratios is proportional. 3 1 yes 6 __ = ___ 6 18 7

1 = ___ 4 __ 4

no

20

Solve each proportion. x 7 8 __ = ___ x = 21 8 24 9

p ___ 10 __ = 3

7.5

p=4

Example 1 Determine whether the ratios are proportional. 9 3 __ = ___ 4 12 Find the cross products. 3 · 12 = 9 · 4 36 = 36 The cross products are equal. The ratios form a proportion.

Example 2 Solve for n. Find the cross products

Solve.

5 __ n =9 10 · 9 = 5n 5n 90 ___ ___ = 5 5 18 = n 10 ___

Chapter 4 Study Guide

77

4-2

Unit Conversions

(pp. 53–59)

Convert each unit using a proportion.

282

Example 3

10

28,200 cm =

11

5 quarts = 1.25 or 1 gallons

_1

meters

4

Solve. 12

A bag weighs 48 ounces. There are 16 ounces in one pound. How many pounds does the bag weigh? 3 lb

13

A piece of wood is 11 inches long. One inch is about 2.54 centimeters. About how long is the piece of wood in centimeters? 27.94 cm

4-3

Solve Rate Problems

(pp. 61–68)

Solve. 14

Example 4

Julian can run 3 miles in 27 minutes. What is Julian’s average time per mile?

9 minutes per mile 16

The Minton’s Turkey Farm tracks the number of turkeys they sell each fall. They sold 444 turkeys during the last 12 falls. What is the average number of turkeys sold each fall? 444 turkeys Write a ratio for turkeys to falls. ___________ 12 falls Simplify to a unit rate. 444 turkeys 444 ÷ 12 37 turkeys ___________ = ________ = __________ 12 falls 12 ÷ 12 1 fall The Minton Turkey Farm sold an average of 37 turkeys per fall.

Derek has a 286-mile drive home from college. He drove 136 miles in 2 hours. Based upon this fact, about how long will it take Derek to make it home?

4.2 hours 17

It was raining at a rate of 0.8 inches per hour. If it continues over the next 8 hours, how much rain will have fallen?

6.4 inches 78

Chapter 4 Study Guide

Copyright © by The McGraw-Hill Companies, Inc.

An ice-cream store tracks the number of customers who come into the store. After 6 months, the count is 6,372 customers. What is the average number of customers per month?

1,062 per month 15

1 feet to inches using a proportion. Convert 3__ 2 There are 12 inches in 1 foot. n in. 12 in. = _____ _____ Write a proportion. Use 1 ft 1 ft 3__ the ratio of feet to 2 inches. 1=n 12 · 3__ 2 Cross multiply. Solve. 42 = n 1 feet = 42 inches 3__ 2

4-4

Solve Problems Using Proportions

Find the value of x in the pair of similar figures. 18


GU

9

6


GU

4

"

:

9 
DN


DN

7 8

x=

Find the value of x in the pair of similar figures.


GU

Y

19

Example 5

x = 9 ft 5

(pp. 69–75)

;

20 

 Y



$


DN

#

; Y
DN :

The ratio of sides AC to CB in 12 . triangle ABC is ___ 10 The ratio of the corresponding side measures 8 in triangle XYZ is __ x. Set up a proportion and solve for x. 8 12 = __ ___

10 x 12x = 8(10) 80 12x = ___ ____ 12 12

Find the cross products. Simplify. Divide by 12.

Copyright © by The McGraw-Hill Companies, Inc.

−− 2 centimeters. The length of YZ is 6 __ 3 Use a proportion to solve the problem. 20

21

A 12-oz package of chocolate chunks contains 21 pieces. How many pieces would you expect to find in a 16-oz 28 package? Todd ran 6 miles in 36 minutes. Use a proportion to find how long it would take him to run 8 miles.

48 minutes

Example 6 Use a proportion to solve the problem. About 17 out of every 25 customers at the bakery purchase some sort of homemade bread or rolls. On a day when there are 75 customers, how many customers would be expected to purchase some bread or rolls? Write a proportion. Use n to represent the number of customers to buy bread or rolls.

n 17 = ___ ___ 25

75

75(17) = 25n

Solve for n. On a day with 75 customers, 51 would be expected to purchase bread or rolls.

1,275 = 25n 25n 1275 = ____ _____ 25 25 51 = n

Chapter 4 Study Guide

79

Chapter

4

Chapter Test

Determine whether the ratios are proportional. Write = or ≠ in each circle. 3AF2.2, 5SDAP1.3 3 5 15 2 9 ≠ 10 1 __ ≠ __ 2 __ = ___ 5 3 9 27

Solve each proportion. 3AF2.2, 5SDAP1.3 p 3 3 ___ = __ p = 27 18 2

4

36 9 ___ = ___ 25

f

135 = 135

f = 100

Convert each unit using a proportion. Round to the nearest tenth if necessary. 3AF1.4, 3MG1.4, 6AF2.1 5

16 ounces = 1 pint; 48 ounces =

7

1 in. ≈ 2.54 cm; 8 in. ≈ 20.3 cm

9

1 mi = 5,280 ft; 15,840 ft =

3

3

mi

pints

6

1,000 m = 1 km; 2,050 m = 2.1 km

8

1 lb = 16 oz; 56 oz = 3.5 lbs

10

1 T = 2,000 lb; 7,000 lb = 3.5 T

Find each total cost. 3AF2.1, 3AF2.2 One ticket costs $7.75. How much are 12 tickets?

$93 12

$98.85 buys 3 admissions to the theme park. How much is 1 admission?

$32.95

Write a ratio. Then find each unit rate. Round to the nearest tenth if necessary. 3AF2.1, 3AF2.26AF2.3 13

$75 earned in 6 hours

14

296 miles in 6 hours

80

Chapter 4 Test

75 , $12.50/h _ 6

296 , 49.3 mi/h _ 6

Copyright © by The McGraw-Hill Companies, Inc.

11

15

Find the value of x in each pair of similar figures. 6NS1.3 
JO

"

#


JO %

$

&

16

Y


JO )

Find the value of x in each pair of similar figures. 6NS1.3

' 
DN


DN

(

x = 9 in.

Y
DN 
DN

x = 7.2

Solve. 3AF2.1, 5SDAP1.3, 3AF2.2, 6AF2.3, 3AF1.4, 6NS1.3, 7AF4.2 17

SPORTS A group of friends jogged during lunch time. After 60 minutes, they had jogged 6 miles. If they kept a fairly steady pace, then how far had they jogged after 45 minutes?

4.5 miles

18

ART The two frames shown are proportional. What is the width of the second frame?

_

Copyright © by The McGraw-Hill Companies, Inc.

13 1 inches 2 Correct the mistakes. 19

1 pounds of walnuts for a recipe. She Jeremy’s mother needed 1__ 2 looked at the store’s ad and saw that walnuts were on sale for $6.99 per pound. She gave Jeremy a 10-dollar bill to buy the walnuts. What mistake did she make?

He would need more than $10 in order to purchase the amount his mother requested. He would need about $0.50 more.

20

Roberta wanted to make lemonade. The recipe called for 2 gallons of water. Roberta had only a 2-cup measuring cup for measuring. She filled the measuring cup 8 times with water to make the lemonade. What mistake did she make?

Roberta should have filled the measuring cup 16 times.

Chapter 4 Test Getty Images

81

Chapter

4

Standards Practice

Choose the best answer and fill in the corresponding circle on the sheet at right. 1

A group of students wants to ride on the newest roller coaster at the amusement park. The height requirement is 5 feet. Which students will be allowed on the ride? 3MG1.4 Student

Height (inches)

Isabelle Brady Omar Frankie

52 61 58 63

4

5

A family’s meal totals $43.80, including tax. After a 15% tip is added, what is the total cost of the meal? 7NS1.7 F $6.57

H $50.37

G $43.80

J $52.56

A store buys DVDs of new movies for $6 each. It marks up the price 350%. At what price does this store sell these DVDs? 7NS1.7

A Isabelle and Omar

A $7.17

C $21

B Brady and Frankie

B $18

D $27

C Brady and Omar D Isabelle and Frankie 6

2

F 200 feet

H 1,200 feet

F $25,017

H $50,035

G 900 feet

J 1,800 feet

G $38,473

J $5,003,500

Hector ran 116 laps during track practice this week. LaShawn ran 75% of Hector’s total laps. How many laps did LaShawn run this week? 6NS1.4 A 58 laps

C 143 laps

B 87 laps

D 203 laps

7

How many grams equal 7 milligrams? 3MG1.4 A 0.007 g

C 700 g

B 0.7 g

D 7,000 g

GO ON 82

Chapter 4 Standards Practice

Copyright © by The McGraw-Hill Companies, Inc.

3

Naomi is running the 600-yard medley. How many feet is this? 3MG1.4

Mr. Alvarez earns a 4% commission on each home he sells. Last year, his home sales totaled $1,250,875. How much did Mr. Alvarez make in commission? 7NS1.7

8

9

Copyright © by The McGraw-Hill Companies, Inc.

10

Juanita uses 12.5 cups of mashed bananas to make 5 loaves of banana bread. How many cups of mashed bananas does she need to make 3 loaves? 6NS1.3 F 2.5 cups

H 5 cups

G 4.17 cups

J 7.5 cups

Antonio needs 5 feet of rope. Which package of rope should he purchase? 3MG1.4 A 36 inches

C 55 inches

B 48 inches

D 60 inches

Marissa has a picture measuring 100 mm by 150 mm. Which frame will best fit this picture? 3MG1.4 F 1 cm by 1.5 cm G 100 cm by 150 cm H 10 cm by 15 cm

12

If 1 inch equals 2.54 centimeters, how many inches equal 12.7 centimeters? F 3 inches

H 8 inches

G 5 inches

J 32.258 inches

ANSWER SHEET Directions: Fill in the circle of each correct answer. 1

A

B

C

D

2

F

G

H

J

3

A

B

C

D

4

F

G

H

J

5

A

B

C

D

6

F

G

H

J

7

A

B

C

D

8

F

G

H

J

9

A

B

C

D

10

F

G

H

J

11

A

B

C

D

12

F

G

H

J

J 0.10 cm by 0.15 cm Success Strategy 11

A bird flew at 13 miles per hour for 2 hours. Then the bird flew at 15 miles per hour for 3 hours. How far did the bird fly in all? 7AF4.2 A 26 miles

C 71 miles

B 45 miles

D 195 miles

After you select your answer, reread the question to check that the answer is reasonable. Check for careless mistakes, like skipping over words that change the meaning of the question.

Chapter 4 Standards Practice

83

Index A Algebra and Functions, 46, 53, 61, 69

K Key Concept, 4, 11, 21, 29, 46, 53, 61, 69

Answer sheet, 43, 83 Assessment, 40–41, 80–81

C California Mathematics Content Standards, 4, 11, 21, 29, 46, 53, 61, 69 Chapter Preview, 3, 45

M Mathematical Reasoning, see Step-by-Step Problem Solving Measurement and Geometry, 53 metric system of measurement, 53–59

Chapter Test, 40–41, 80–81 commission, 11–19 compound interest, 21–28 Correct the Mistakes, 41, 81

N Number Sense, 4, 11, 21, 29, 69

cross multiply, 11–19 cross products, 11–19, 46–52 customary system of measurement, 53–59

D

P percent, 4–10, 11–19, 21–28, 29–35 percent of change, 29–35

distance, 61–68

Problem-Solving, see Step-byStep Problem Solving Progress Check, 20, 36, 60, 76

E equation, 11–19, 21–28, 29–35, 46–52, 53–59, 61–68, 69–75

I increase, 29–35 interest compound, 21–28 simple, 21–28

84

Index

proportion, 11–19, 46–52, 53– 59, 61–68, 69–75

R rate, 21–28, 46–52 ratio, 4–10, 11–19, 21–28, 29–35 Real-World Applications art, 81 astronomy, 8 babies, 58 baking, 58 baseball, 59 biking, 50 books, 35

Reflect, 9, 17, 26, 34, 51, 57, 67, 73

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

principal, 21–28 decrease, 29–35

business, 10, 27, 33, 36, 51, 52, 57, 68 cell phones, 17 chess, 10 commission, 18 community service, 68 computers, 16 construction, 58 contact lenses, 35 cooking, 56 discount, 16, 18 earth science, 68 eating out, 10 education, 59 elections, 36 entertainment, 75 events, 74 fashion, 10, 19, 28, 34, 52, 68 finance, 26, 27 finances, 26, 41 fitness, 75 food, 51 geography, 60 geometry, 73 hiking, 66 life science, 68 loans, 27 money, 20, 25 movies, 17, 20, 36, 66 nature, 57 packaging, 41 painting, 67 pets, 50 proportions, 73 reading, 51 sales, 35, 38 school, 9, 28, 50 scuba diving, 75 shopping, 38 snow, 67 sports, 10, 33, 41, 81 studying, 19 surveys, 18 taxes, 14, 16, 18 tennis, 9 tests, 18 tipping, 14, 16, 18 travel, 60, 67, 76

S scale, 69–75 similar figures, 69–75 simple interest, 21–28 Spiral Review, 10, 19, 28, 35, 52, 59, 68, 74 Standards Practice, 42–43, 82–83 Statistics, Data Analysis, and Probability, 4, 46 Step-by-Step Practice, 7, 15, 24, 32, 49, 55, 65, 72

Step-by-Step Problem Solving Practice, 8–9, 16–17, 25–26, 33, 50, 56–57, 66, 73 Look for a pattern, 56 Make a table, 50 Solve a simpler problem, 8, 16, 25 Use a table, 73 Use logical reasoning, 33 Write an equation, 66

V Vocabulary, 4, 11, 21, 29, 46, 53, 61, 69 Vocabulary and Concept Check, 37, 77 Vocabulary Check, 10, 19, 28, 35, 52, 59, 68, 75

Study Guide, 37–39, 77–79 Success Strategy, 43, 83

U unit cost, 61–68

W Who is Correct?, 6, 14, 24, 31, 48, 55, 64, 71 Writing in Math, 10, 19, 28, 35, 52, 59, 68, 74

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

unit rate, 46–52, 53–59, 61–68

Index

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