Fractions and Mixed Numbers: Multiplication and Division

95

Fractions andMixed Numbers:Multiplication and Division

2CHAPTER OUTLINE

2.1 Introduction to Fractions and Mixed Numbers 96

2.2 Prime Numbers and Factorization 106

2.3 Simplifying Fractions to Lowest Terms 114

2.4 Multiplication of Fractions and Applications 123

2.5 Division of Fractions and Applications 133

Problem Recognition Exercises: Multiplication and Division of

Fractions 143

2.6 Multiplication and Division of Mixed Numbers 144

Group Activity: Cooking for Company 151

Chapter 2

In Chapter 2 we study the concept of a fraction and a mixed number. We learn how to simplifyfractions by reducing to lowest terms. Learning the terms and vocabulary relating to fractions willhelp you understand this chapter. Try the crossword puzzle shown. Refer to definitions in thechapter if you need help.

1

2

3

5

4

6

Across

2. The top number of a fraction.5. To divide two fractions, multiply the

first fraction by the of the second fraction.

6. A fraction whose numerator is greater than or equal to the denominator is called .

Down

1. The bottom number of a fraction.3. A fraction whose numerator is less than

the denominator is called .4. A whole number greater than 1 that has

only 1 and itself as factors is called anumber.

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96 Chapter 2 Fractions and Mixed Numbers: Multiplication and Division

1. Definition of a FractionIn Chapter 1, we studied operations on whole numbers. In this chapter, we work

with numbers that represent part of a whole. When a whole unit is divided into

equal parts, we call the parts fractions of a whole. For example, the pizza in Figure 2-1

is divided into 5 equal parts. One-fifth of the pizza has been eaten, and four-

fifths of the pizza remains.145 2115 2

Section 2.1 Introduction to Fractions and Mixed Numbers

Objectives

1. Definition of a Fraction

2. Proper and Improper

Fractions

3. Mixed Numbers

4. Fractions and the Number

Line

has been eaten15

remains45

Figure 2-1

A fraction is written in the form , where a and b are whole numbers and In

the fraction , the “top” number, 5, is called the numerator. The “bottom” number, 8,

is called the denominator.

numerator

denominator

5

8

58

b � 0.ab

Identifying the Numerator and Denominator of a Fraction

For each fraction, identify the numerator and denominator.

a. b. c.

Solution:

a. The numerator is 3. The denominator is 5.

b. The numerator is 1. The denominator is 8.

c. The numerator is 8. The denominator is 1.8

1

1

8

3

5

8

1

1

8

3

5

Example 1

The denominator of a fraction denotes the number of equal pieces into which a

whole unit is divided.The numerator denotes the number of pieces being considered.

Skill Practice

Identify the numerator anddenominator.

1. 2. 3.61

05

411

Answers1. Numerator: 4, denominator: 112. Numerator: 0, denominator: 53. Numerator: 6, denominator: 1

Avoiding Mistakes

The fraction can also be written asa⁄b. However, we discourage the useof the “slanted” fraction bar. In laterapplications of algebra, the slantedfraction bar can cause confusion.

ab

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Section 2.1 Introduction to Fractions and Mixed Numbers 97

tomato plants310

Figure 2-2

Writing Fractions

Write a fraction for the shaded portion and a fraction for the unshaded portion

of the figure.

Example 2

Solution:

Shaded portion:

Unshaded portion:3 pieces are not shaded.

The triangle is divided into 16 equal pieces.

3

16

13 pieces are shaded.

The triangle is divided into 16 equal pieces.

13

16

Writing Fractions

What portion of the group of doctors shown below is female?

Example 3

Solution:

The group consists of 5 members. Therefore, the denominator is 5. There are

2 women being considered. Thus, of the group is female.25

In Section 1.6 we learned that fractions represent division. For example, note that

the fraction � 5 � 1 � 5. In general, a fraction of the form This implies that

any whole number can be written as a fraction by writing the whole number over 1.

n1 � n.5

1

Skill Practice

4. Write a fraction for the shaded portion and a fraction for the unshadedportion.

Skill Practice

5. Refer to Example 3. Whatportion of the group of doctorsis male?

Answers4. Shaded portion: ; unshaded portion:5.

35

58

38

BASIC—

For example, the garden in Figure 2-2 is divided into 10 equal parts. Three sections

contain tomato plants. Therefore, of the garden contains tomato plants.310

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Further recall that for and is undefined.Therefore,

and is undefined. For example:

� 0 is undefined

2. Proper and Improper FractionsIf the numerator is less than the denominator in a fraction, then the fraction is called

a proper fraction. Furthermore, a proper fraction represents a number less than 1

whole unit. The following are proper fractions.

6

7

1

3

5

0

0

5

a0

0a � 0a � 0a � 0, 0 � a � 0

1 unit

1 unit

An improper fraction is a fraction in which the numerator is greater than or equal

to the denominator. For example:

and

An improper fraction represents a quantity greater than 1 whole unit or equal to

1 whole unit.

7

7

4

3

numerator equal

to denominator7

7

4

3

numerator greater

than denominator

1 unit 1 unit

1 unit

Categorizing Fractions

Identify each fraction as proper or improper.

a. b. c.

Solution:

a. Improper fraction (numerator is greater than denominator)

b. Proper fraction (numerator is less than denominator)

c. Improper fraction (numerator is equal to denominator)12

12

5

12

12

5

12

12

5

12

12

5

Example 4

Concept Connections

Simplify if possible.

6. 7.012

40

Skill Practice

Identify each fraction as proper orimproper.

8. 9. 10.97

79

1010

Answers6. Undefined 7. 08. Improper 9. Proper

10. Improper

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3. Mixed NumbersSometimes a mixed number is used instead of an improper fraction to denote a

quantity greater than one whole. For example, suppose a typist typed pages of a

report. We would be more likely to say that the typist typed pages (read as “two

and one-fourth pages”). The number is called a mixed number and represents

2 wholes plus of a whole.

In general, a mixed number is a sum of a whole number and a fractional part of a

whole. However, by convention the plus sign is left out.

means

Suppose we want to change a mixed number to an improper fraction. From

Figure 2-3, we see that the mixed number is the same as 72.312

3 �1

23

1

2

9

4

1

4 = 2

a df l sdf dsuig. j./df jgu jldjfgklj dlkfgupe jfg gjfghl jdfj jhupt g; g;hghj gohh lh;hsous gh j;hj p5y fj djgfh jhjdgjjt ghj hj jjhp'od gjh gf;hjpo gfh jgfhj gfh 'j gh og h gjh lghj hgljk'hrtl j dfof hk h gghdmh gfm gflhj hjg mgm dfj gfg jgfugkh hjdmgj fgh fkgk fi fkj fj fruerhkfj fljd fhjdf hdkl df ghkjh The ju9j lls jfjd; fklf;fghj ssjh sdhkdg ior hg ij jkjk sjgi dj njdirjgf jfdhid ui ddf lk kdo df jogkg ofigf fofgkj kf gio gofgio gig f kbgg figjfjf dfd gfgr dfh ffk kj jfd kdfjldflewo dfl ffdl irue oufuoq mvoepe kjiure flds

a df l sdf dsuig. j./df jgu jldjfgklj dlkfgupe jfg gjfghl jdfj jhupt g; g;hghj gohh lh;hsous gh j;hj p5y fj djgfh jhjdgjjt ghj hj jjhp'od d

a df l sdf dsuig. j./df jgu jldjfgklj dlkfgupe jfg gjfghl jdfj jhupt g; g;hghj gohh lh;hsous gh j;hj p5y fj djgfh jhjdgjjt ghj hj jjhp'od gjh gf;hjpo gfh jgfhj gfh 'j gh og h gjh lghj hgljk'hrtl j dfof hk h gghdmh gfm gflhj hjg mgm dfj gfg jgfugkh hjdmgj fgh fkgk fi fkj fj fruerhkfj fljd fhjdf hdkl df ghkjh The ju9j lls jfjd; fklf;fghj ssjh sdhkdg ior hg ij jkjk sjgi dj njdirjgf jfdhid ui ddf lk kdo df jogkg ofigf fofgkj kf gio gofgio gig f kbgg figjfjf dfd gfgr dfh ffk kj jfd kdfjldflewo dfl ffdl irue oufuoq mvoepe kjiure flds

14

214

214

94

Writing Improper Fractions

Write an improper fraction to represent the fractional part of an inch for the

screw shown in the figure.

Solution:

Each 1-in. unit is divided into 8 parts, and the screw extends for 11 parts.Therefore,

the screw is in.118

Example 5

1 2

Skill Practice

11. Write an improper fractionrepresenting the shaded area.

3 whole units3 �

�

�

3 � 2 halves

1 half

1 half� �

1

2

7

2

Figure 2-3

This process to convert a mixed number to an improper fraction can be sum-

marized as follows.

PROCEDURE Changing a Mixed Number to an Improper Fraction

Step 1 Multiply the whole number by the denominator.

Step 2 Add the result to the numerator.

Step 3 Write the result from step 2 over the denominator.

Answer

11.157

Avoiding Mistakes

Each whole unit is divided into 8 pieces. Therefore the screw is

in., not in.1116

118

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Now suppose we want to convert an improper fraction to a mixed number. In

Figure 2-4, the improper fraction represents 13 slices of pizza where each slice is

of a whole pizza. If we divide the 13 pieces into groups of 5, we make 2 whole

pizzas with 3 pieces left over. Thus,

This process can be accomplished by division.

23

5

2

5�13

�10

3

13

5

13

5� 2

3

5

15

135


For example,

31

2�

3 � 2 � 1

2�

7

2

Multiply the whole

number by the

denominator.

Add the

numerator.

Write the result over

the denominator.

Converting Mixed Numbers to Improper Fractions

Convert the mixed number to an improper fraction.

a. b.

Solution:

a. b.

�42

5 �

29

4

�40 � 2

5 �

28 � 1

4

8

2

5�

8 � 5 � 2

5 7

1

4�

7 � 4 � 1

4

8

2

57

1

4

Example 6

2 groups of 5 3 left over 13 pieces � �

2 � �3

5

13

5

Figure 2-4

remainder

divisor

PROCEDURE Changing an Improper Fraction to a Mixed Number

Step 1 Divide the numerator by the denominator to obtain the quotient and

remainder.

Step 2 The mixed number is then given by

Quotient �remainder

divisor

Skill Practice

Convert the mixed number to animproper fraction.

12. 13. 1512

1058

Answers

12. 13.312

858

Avoiding Mistakes

When writing a mixed number, the� sign between the whole numberand fraction should not be written.

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The process to convert an improper fraction to a mixed number indicates that the

result of a division problem can be written as a mixed number.

divisor

remainder

divisor

remainder

Converting Improper Fractions to Mixed Numbers

Convert to a mixed number.

a. b.

Solution:

a.

b.3

39

41

3

41�162

�123

39

162

41

41

6

4

6�25

�24

1

25

6

162

41

25

6

Example 7

Writing a Quotient as a Mixed Number

Divide. Write the quotient as a mixed number.

Solution:

150

28�4217

�28

141

�140

17

�0

17

28�4217

Example 8

divisor

remainder

4. Fractions and the Number LineFractions can be visualized on a number line. For example, to graph the fraction ,

divide the distance between 0 and 1 into 4 equal parts. To plot the number , start

at 0 and count over 3 parts.

10

34

34

34

Skill Practice

Convert the improper fraction to amixed number.

14. 15.9522

145

Skill Practice

Divide and write the quotient asa mixed number.

16. 5967 � 41

Answers

14. 15. 16. 1452241

4722

245

BASIC—

15017

28

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Plotting Fractions on a Number Line

Plot the point on the number line corresponding to each fraction.

a. b. c.

Solution:

a. Divide the distance between 0 and 1 into 2 equal parts.

b. Divide the distance between 0 and 1 into 6 equal parts.

c. Write as a mixed number.

Thus, is located one-fifth of the way between 4 and 5 on the number line.

543210

415

215 � 41

5

215

21

5� 4

1

5

10

56

5

6

10

12

1

2

21

5

5

6

1

2

Example 9Skill Practice

Plot the numbers on a numberline.

17. 18. 19.134

13

45

Answers17.

18.

19.

60 1 2 3 4 5

314

10

13

10

45

Study Skills Exercises

1. After doing a section of homework, check the odd-numbered answers in the back of the text. Choose a

method to identify the exercises that gave you trouble (i.e., circle the number or put a star by the number).

List some reasons why it is important to label these problems.

2. Define the key terms.

a. Fraction b. Numerator c. Denominator

d. Proper fraction e. Improper fraction f. Mixed number

Boost your GRADE atALEKS.com!

• Practice Problems • e-Professors

• Self-Tests • Videos

• NetTutor

Section 2.1 Practice Exercises

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Objective 1: Definition of a Fraction

For Exercises 3–6, identify the numerator and the denominator for each fraction.

(See Example 1.)

3. 4. 5. 6.

For Exercises 7–14, write the fraction as a division problem and simplify, if possible.

7. 8. 9. 10.

11. 12. 13. 14.

For Exercises 15–22, write a fraction that represents the shaded area. (See Example 2.)

15. 16. 17. 18.

19. 20. 21. 22.

23. Write a fraction to represent the portion of gas

in a gas tank represented by the gauge.

10

10

11

0

2

0

0

7

0

3

8

8

2

2

9

1

6

1

1

2

12

11

8

9

2

3

E

F

1

2

24. Write a fraction that represents the portion of

medicine left in the bottle.

25. The scoreboard for a recent men’s championship

swim meet in Melbourne, Australia, shows the

final standings in the men’s 100-m freestyle

event. What portion of the finalists are from

the USA? (See Example 3.)

26. Refer to the scoreboard from Exercise 25. What

portion of the finalists are from the Republic of

South Africa (RSA)?

27. The graph categorizes a sample of people by

blood type. What portion of the sample represents

people with type O blood?

28. Refer to the graph from Exercise 27. What

portion of the sample represents people with

type A blood?

Name Country Time

Maginni, Filippo ITA 48.43

Hayden, Brent CAN 48.43

Sullivan, Eamon AUS 48.47

Cielo Filho, Cesar BRA 48.51

Lezak, Jason USA 48.52

Van Den Hoogenband, Pieter NED 48.63

Schoeman, Roland Mark RSA 48.72

Neethling, Ryk RSA 48.81

Type O 41

Sample by Blood Type

Type A43

Type AB7

Type B12

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Objective 2: Proper and Improper Fractions

For Exercises 31–38, label the fraction as proper or improper. (See Example 4.)

31. 32. 33. 34.

35. 36. 37. 38.

For Exercises 39–42, write an improper fraction for the shaded portion of each group of figures. (See Example 5.)

39. 40.

41. 42.

Objective 3: Mixed Numbers

For Exercises 43–44, write an improper fraction and a mixed number for the shaded portion of each group of figures.

43. 44.

45. Write an improper fraction and a mixed number to represent the length of the nail.

13

21

15

17

21

20

7

2

3

3

10

10

2

3

7

8

1 in. 2 in.

29. A class has 21 children—11 girls and

10 boys. What fraction of the class is

made up of boys?

30. In a neighborhood in Ft. Lauderdale, 10 houses are

for sale and 53 are not for sale. Write a fraction

representing the portion of houses that are for sale.

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46. Write an improper fraction and a mixed number that represent

the fraction of the amount of sugar needed for a batch of cookies,

as indicated in the figure.

For Exercises 47–58, convert the mixed number to an improper fraction. (See Example 6.)

47. 48. 49. 50.

51. 52. 53. 54.

55. 56. 57. 58.

59. How many eighths are in ? 60. How many fifths are in ?

61. How many fourths are in ? 62. How many thirds are in ?

For Exercises 63–74, convert the improper fraction to a mixed number. (See Example 7.)

63. 64. 65. 66.

67. 68. 69. 70.

71. 72. 73. 74.

For Exercises 75–82, divide. Write the quotient as a mixed number. (See Example 8.)

75. 76. 77. 78.

79. 80. 81. 82.

Objective 4: Fractions and the Number Line

For Exercises 83–92, plot the fraction on the number line. (See Example 9.)

83. 84.

85. 86.10

1

510

1

3

10

1

210

3

4

34�69515�1874257 � 238913 � 11

7213 � 85281 � 54�9217�309

115

7

23

6

51

10

133

11

67

12

52

9

43

18

27

10

19

4

39

5

13

7

37

8

52

31

3

4

23

52

3

8

151

221

3

812

1

611

5

12

103

57

1

48

2

33

3

7

31

54

2

96

1

31

3

4

12

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1. Factors and FactorizationsRecall from Section 1.5 that two numbers multiplied to form a product are called

factors. For example, indicates that 2 and 3 are factors of 6. Likewise,

because , the numbers 1 and 6 are factors of 6. In general, a factor of a

number n is a nonzero whole number that divides evenly into n.

The products and are called factorizations of 6. In general, a

factorization of a number n is a product of factors that equals n.

1 � 62 � 3

1 � 6 � 6

2 � 3 � 6

Section 2.2 Prime Numbers and Factorization

Objectives

1. Factors and Factorizations

2. Divisibility Rules

3. Prime and Composite

Numbers

4. Prime Factorization

5. Identifying All Factors of a

Whole Number

Finding Factorizations of a Number

Find four different factorizations of 12.

Solution:

12 � μ

1 � 12

2 � 6

3 � 4

2 � 2 � 3


1. Find four different factorizations of 18.

TIP: Notice that afactorization may includemore than two factors.

BASIC—

Answer1. For example. 1 18

2 93 62 3 3��

��

87. 88.

89. 90.

91. 92.

Expanding Your Skills

93. True or false? Whole numbers can be written both as proper and improper fractions.

94. True or false? Suppose m and n are nonzero numbers, where m n. Then is an improper fraction.

95. True or false? Suppose m and n are nonzero numbers, where m n. Then is a proper fraction.

96. True or false? Suppose m and n are nonzero numbers, where m n. Then is a proper fraction.n3m

nm

mn

20 1

3

220 1

5

3

20 1

7

520 1

7

6

10

5

610

2

3

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Section 2.2 Prime Numbers and Factorization 107

Answers2. Divisible by 23. Divisible by 3 and 54. Divisible by 2, 3, 5, and 10

PROCEDURE Divisibility Rules for 2, 3, 5, and 10

• Divisibility by 2. A whole number is divisible by 2 if it is an even number.

That is, the ones-place digit is 0, 2, 4, 6, or 8.Examples: 26 and 384

• Divisibility by 5. A whole number is divisible by 5 if its ones-place digit is

5 or 0.Examples: 45 and 260

• Divisibility by 10. A whole number is divisible by 10 if its ones-place digit

is 0.Examples: 30 and 170

• Divisibility by 3. A whole number is divisible by 3 if the sum of its digits is

divisible by 3.Example: 312 (sum of digits is which is divisible

by 3)3 � 1 � 2 � 6

We address other divisibility rules for 4, 6, 8, and 9 in the Expanding Your Skills

portion of the exercises. However, these divisibility rules are harder to remember,

and it is often easier simply to perform division to test for divisibility.

Applying the Divisibility Rules

Determine whether the given number is divisible by 2, 3, 5, or 10.

a. 624 b. 82 c. 720

Solution:

Test for Divisibility

a. 624 By 2: Yes. The number 624 is even.

By 3: Yes. The sum 6 � 2 � 4 � 12 is divisible by 3.

By 5: No. The ones-place digit is not 5 or 0.

By 10: No. The ones-place digit is not 0.

b. 82 By 2: Yes. The number 82 is even.

By 3: No. The sum 8 � 2 � 10 is not divisible by 3.

By 5: No. The ones-place digit is not 5 or 0.

By 10: No. The ones-place digit is not 0.

c. 720 By 2: Yes. The number 720 is even.

By 3: Yes. The sum 7 � 2 � 0 � 9 is divisible by 3.

By 5: Yes. The ones-place digit is 0.

By 10: Yes. The ones-place digit is 0.

Example 2 Skill Practice

Determine whether the givennumber is divisible by 2, 3, 5,or 10.

2. 428 3. 75 4. 2100

TIP: When in doubtabout divisibility, you cancheck by division. When wedivide 624 by 3, the remain-der is zero.

BASIC—

2. Divisibility RulesThe number 20 is said to be divisible by 5 because 5 divides evenly into 20. To

determine whether one number is divisible by another, we can perform the divi-

sion and note whether the remainder is zero. However, there are several rules by

which we can quickly determine whether a number is divisible by 2, 3, 5, or 10.

These are called divisibility rules.

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3. Prime and Composite NumbersTwo important classifications of whole numbers are prime numbers and compos-

ite numbers.

DEFINITION Prime and Composite Numbers

• A prime number is a whole number greater than 1 that has only two fac-

tors (itself and 1).

• A composite number is a whole number greater than 1 that is not prime.

That is, a composite number will have at least one factor other than 1

and the number itself.

Note: The whole numbers 0 and 1 are neither prime nor composite.

Identifying Prime and Composite Numbers

Determine whether the number is prime, composite, or neither.

a. 19 b. 51 c. 1

Solution:

a. The number 19 is prime because its only factors are 1 and 19.

b. The number 51 is composite because . That is, 51 has factors

other than 1 and 51.

c. The number 1 is neither prime nor composite by definition.

3 � 17 � 51

Example 3

TIP: The number 2 isthe only even prime number.

Answers5. Composite 6. Neither 7. Prime8. No. The factor 10 is not a prime

number. The prime factorization of 60 is 2 2 3 5.��

Skill Practice

Determine whether the number isprime, composite, or neither.

5. 39 6. 0 7. 41

Prime numbers are used in a variety of ways in mathematics.We advise you to become

familiar with the first several prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, . . .

4. Prime FactorizationIn Example 1 we found four factorizations of 12.

The last factorization consists of only prime-number factors. Therefore, we

say is the prime factorization of 12.2 � 2 � 3

2 � 2 � 3

1 � 12

2 � 6

3 � 4

2 � 2 � 3

DEFINITION Prime Factorization

The prime factorization of a number is the factorization in which every factor

is a prime number.

Note: The order in which the factors are written does not affect the product.

BASIC—

Concept Connections

8. Is the product theprime factorization of 60?Explain.

2 � 3 � 10

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Prime factorizations of numbers will be particularly helpful when we add, subtract,

multiply, divide, and simplify fractions.

Determining the Prime Factorization of a Number

Find the prime factorization of 220.

Solution:

One method to factor a whole number is to make a factor tree. Begin by deter-

mining any two numbers that when multiplied equal 220. Then continue factor-

ing each factor until the branches “end” in prime numbers.

220

10 22

5 2 2 11 Branches end in prime numbers.

Therefore, the prime factorization of 220 is .2 � 2 � 5 � 11

Example 4

TIP: The prime factorization fromExample 4 can also be expressedby using exponents as 22 � 5 � 11.

In Example 4, note that the result of a prime factorization does not depend on

the original two-number factorization. Similarly, the order in which the factors are

written does not affect the product, for example,

220

11 20

2 10

2 5

220

4 55

2 2 5 11

220

2 110

2 55

5 11

220 � 2 � 2 � 5 � 11 220 � 2 � 2 � 5 � 11 220 � 11 � 2 � 2 � 5

TIP: You can check the prime factorization of any number by multiplying thefactors.

Skill Practice

9. Find the prime factorizationof 90.

Answer9. 2 � 3 � 3 � 5

Another technique to find the prime factorization of a number is to divide the

number by the smallest known prime factor. Then divide the quotient by its small-

est known prime factor. Continue dividing in this fashion until the quotient is a

prime number. The prime factorization is the product of divisors and the final quo-

tient. For example,

Therefore, the prime factorization of 220 is or .22 � 5 � 112 � 2 � 5 � 11

2 is the smallest prime factor of 220



the last quotient is prime

2)220

2)110

5)55

11

Avoiding Mistakes

Make sure that the end of eachbranch is a prime number.

BASIC—

miL58061_ch02_095-162.qxd 7/5/08 10:03 AM Page 109

5. Identifying All Factors of a Whole NumberSometimes it is necessary to identify all factors (both prime and other) of a num-

ber. Take the number 30, for example. A list of all factors of 30 is a list of all whole

numbers that divide evenly into 30.

Factors of 30: 1, 2, 3, 5, 6, 10, 15, and 30


Determining Prime Factorizations

Find the prime factorization.

a. 198 b. 153

Solution:

a.

The prime factorization of 198 is or

b.

The prime factorization of 153 is 3 � 3 � 17 or 32 � 17.

3)153

3)51

17

2 � 32 � 11.2 � 3 � 3 � 11

Since 198 is even, we

know it is divisible by 2.

2)198

3)99

3)33

11

The sum of the digits 9 � 9 � 18 is

divisible by 3.

Example 5

Listing All Factors of a Number

List all factors of 36.

Solution:

Begin by listing all the two-number factorizations of 36. This can be accom-

plished by systematically dividing 36 by 1, 2, 3, and so on. Notice, however, that

after the product , the two-number factorizations are repetitious, and we

can stop the process.

These products repeat the factorizations

above. Therefore, we can stop at

The list of all factors of 36 consists of the individual factors in the products.

The factors are 1, 2, 3, 4, 6, 9, 12, 18, 36.

36 � 1

6 � 6. 18 � 2

12 � 3

9 � 4

6 � 6

4 � 9

3 � 12

2 � 18

1 � 36

6 � 6

Example 6

⎫⎪⎬⎪⎭

Skill Practice

Find the prime factorization of thegiven number.

10. 168 11. 990

Skill Practice

12. List all factors of 45.

Answers10.

11.

12. 1, 3, 5, 9, 15, 452 � 3 � 3 � 5 � 11 or 2 � 32 � 5 � 112 � 2 � 2 � 3 � 7 or 23 � 3 � 7

TIP: When listing a setof factors, it is not necessaryto write the numbers in anyspecified order. However, ingeneral we list the factorsin order from smallest tolargest.

BASIC—

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1. In general, 2 to 3 hours of study time per week is needed for each 1 hour per week of class time. Based on the

number of hours you are in class this semester, how many hours per week should you be studying?


a. Factor b. Factorization c. Prime number

d. Composite number e. Prime factorization

Review Exercises

For Exercises 3–5, write two fractions, one representing the shaded area and one representing the unshaded area.

3. 4. 5.

6. Write a fraction with numerator 6 and denominator 5. Is this fraction proper or improper?

7. Write a fraction with denominator 12 and numerator 7. Is this fraction proper or improper?

8. Write a fraction with denominator 6 and numerator 6. Is this fraction proper or improper?

9. Write the improper fraction as a mixed number. 10. Write the mixed number 6 as an improper fraction.

Objective 1: Factors and Factorization

For Exercises 11–14, find two different factorizations of each number. (Answers may vary.) (See Example 1.)

11. 8 12. 20 13. 24 14. 14

15. Find two factors whose product is the number in the top row

and whose sum is the number in the bottom row. The first

column is done for you as an example.

16. Find two factors whose product is the number in the top row

and whose difference is the number in the bottom row. The

first column is done for you as an example.

27

235

10




• NetTutor


Product 36 42 45 72 24

Factor 9

Factor 4

Difference 5 1 12 14 5

Product 36 42 30 15 81

Factor 12

Factor 3

Sum 15 13 31 16 30

BASIC—

miL58061_ch02_095-114.qxd 8/27/08 12:04 AM Page 111


Objective 2: Divisibility Rules

17. State the divisibility rule for dividing by 2. 18. State the divisibility rule for dividing by 10.

19. State the divisibility rule for dividing by 3. 20. State the divisibility rule for dividing by 5.

For Exercises 21–28, determine if the number is divisible by a. 2 b. 3 c. 5 d. 10

(See Example 2.)

21. 45 22. 100 23. 137 24. 241

25. 108 26. 1040 27. 3140 28. 2115

29. Ms. Berglund has 28 students in her class. Can she distribute a package of 84 candies evenly to her

students?

30. Mr. Blankenship has 22 students in an algebra class. He has 110 sheets of graph paper. Can he distribute

the graph paper evenly among his students?

Objective 3: Prime and Composite Numbers

For Exercises 31–46, determine whether the number is prime, composite, or neither. (See Example 3.)

31. 7 32. 17 33. 10 34. 21

35. 51 36. 57 37. 23 38. 31

39. 1 40. 0 41. 121 42. 69

43. 19 44. 29 45. 39 46. 49

47. Are there any whole numbers that are 48. True or false? The square of any prime

not prime or composite? If so, list them. number is also a prime number.

49. True or false? All odd numbers are prime. 50. True or false? All even numbers are composite.

51. One method for finding prime numbers is the sieve of Eratosthenes. The natural numbers from 2 to 50 are

shown in the table. Start at the number 2 (the smallest prime number). Leave the number 2 and cross out

every second number after the number 2. This will eliminate all numbers that are multiples of 2. Then go

back to the beginning of the chart and leave the number 3, but cross out every third number after the

number 3 (thus eliminating the multiples of 3). Begin at the next open number and continue this process.

The numbers that remain are prime numbers. Use this process to find the prime numbers less than 50.

2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

BASIC—

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52. Use the sieve of Eratosthenes to find the prime numbers less than 80.

2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

Objective 4: Prime Factorization

For Exercises 53–56, determine whether or not the factorization represents the prime factorization. If not,

explain why.

53. 54. 55. 56.

For Exercises 57–68, find the prime factorization. (See Examples 4 and 5.)

57. 70 58. 495 59. 260 60. 175

61. 147 62. 102 63. 138 64. 231

65. 616 66. 364 67. 47 68. 41

Objective 5: Identifying All Factors of a Whole Number

For Exercises 69–76, list all the factors of the number. (See Example 6.)

69. 12 70. 18 71. 32 72. 55

73. 81 74. 60 75. 48 76. 72


For Exercises 77–80, determine whether the number is divisible by 4. Use the following divisibility rule: A whole

number is divisible by 4 if the number formed by its last two digits is divisible by 4.

77. 230 78. 1046 79. 4616 80. 10,264


number is divisible by 8 if the number formed by its last three digits is divisible by 8.

81. 1032 82. 2520 83. 17,126 84. 25,058


number is divisible by 9 if the sum of its digits is divisible by 9.

85. 396 86. 414 87. 8453 88. 1587


number is divisible by 6 if it is divisible by both 2 and 3 (use the divisibility rules for 2 and 3 together).

89. 522 90. 546 91. 5917 92. 6394

126 � 3 � 7 � 3 � 2210 � 5 � 2 � 7 � 348 � 2 � 3 � 836 � 2 � 2 � 9

BASIC—

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1. Equivalent FractionsThe fractions all represent the same portion of a whole. See Figure 2-5.

Therefore, we say that the fractions are equivalent.

36,

24, and 12

Section 2.3 Simplifying Fractions to Lowest Terms

Objectives

1. Equivalent Fractions

2. Simplifying Fractions to

Lowest Terms

3. Applications of Simplifying

Fractions� �

� �3

6

2

4

1

2

Figure 2-5

One method to show that two fractions are equivalent is to calculate their cross

products. For example, to show that we have

Yes. The fractions are equivalent. 12 � 12

3 � 4 �?

6 � 2

3

6 �

2

4

36 � 2

4,Avoiding Mistakes

The test to determine whether twofractions are equivalent is not thesame process as multiplyingfractions. Multiplying of fractions is covered in Section 2.4.

Determining Whether Two Fractions Are Equivalent

Fill in the blank with or

a. b.

Solution:

a. b.

Therefore, Therefore, .

2. Simplifying Fractions to Lowest TermsIn Figure 2-5, we see that all represent equal quantities. However, the frac-

tion is said to be in lowest terms because the numerator and denominator share

no common factors other than 1.

To simplify a fraction to lowest terms, we apply the following important principle.

12

36,

24, and 12

7

9�

5

7

6

13.�

18

39

45 � 49 234 � 234

5 � 9 �?

7 � 7 18 � 13 �?

39 � 6

7

9�

5

7

6

13�

18

39

7

9

5

7

6

13

18

39

�.�


Fill in the blank withor

1.

2.5424

94

611

1324

�.

�

Answers1. 2. ��

BASIC—

miL58061_ch02_095-162.qxd 7/5/08 10:03 AM Page 114

Section 2.3 Simplifying Fractions to Lowest Terms 115

Answers

3. 4. 5.107

215

37

PROPERTY Fundamental Principle of Fractions

Suppose that a number, c, is a common factor in the numerator and denom-

inator of a fraction. Then

� � �ab

ab

� 1ab

�cc

a � cb � c

To simplify a fraction, we begin by factoring the numerator and denominator into

prime factors. This will help identify the common factors.

Simplifying a Fraction to Lowest Terms

Simplify to lowest terms.

a. b. c.

Solution:

a. Factor the numerator and denominator. Notice that 2 is a

common factor.

Apply the fundamental principle of fractions.

Any nonzero number divided by itself is 1.

b. Factor the numerator and denominator.


Any nonzero number divided by itself is 1.

c. Factor the numerator and denominator.


�5

6

�5

6� 1 � 1

�5

3 � 2�

2

2�

2

2

20

24�

5 � 2 � 2

3 � 2 � 2 � 2

�5

3

�5

3� 1 � 1

�5

3�

2

2�

17

17

170

102�

5 � 2 � 17

3 � 2 � 17

�3

5

�3

5� 1

�3

5�

2

2

6

10�

3 � 2

5 � 2

20

24

170

102

6

10


Simplify to lowest terms.

3. 4. 5.150105

26195

1535

BASIC—

miL58061_ch02_095-162.qxd 7/5/08 10:04 AM Page 115


Answers

6. 7. 13 8.16

79

In Example 2, we show numerous steps to simplify fractions to lowest terms. How-

ever, the process is often made easier. For instance, we sometimes “divide out”

common factors, and replace them with the new common factor of 1.

The largest number that divides evenly into the numerator and denominator is

called their greatest common factor. By identifying the greatest common factor

you can simplify the process even more.

Notice that “dividing out” the common factor of

4 has the same effect as dividing the numerator

and denominator by 4. This is often done mentally.

20 divided by 4 equals 5.


205

246

�5

6

20

24�

5 � 41

6 � 41

�5

6

20

24�

5 � 21

� 21

3 � 2 � 21

� 21

�5

6

TIP: Simplifying a fractionis also called reducing afraction to lowest terms. Forexample, the simplified (orreduced) form of is .5

62024

Simplifying Fractions to Lowest Terms

Simplify the fraction. Write the answer as a fraction or whole number.

a. b. c.

Solution:

a.

Or alternatively:

b.

Or alternatively:

c.

Or alternatively:12 divided by 12 equals 1.

60 divided by 12 equals 5.�

121

605

�1

5

The greatest common factor in the

numerator and denominator is 12.

12

60�

1 � 121

5 � 121

�1

5

� 3



753

251

�3

1

The greatest common factor in the numerator

and denominator is 25.

75

25�

3 � 251

1 � 251

�3

1� 3



11010

999

�10

9

The greatest common factor in the

numerator and denominator is 11.

110

99�

10 � 111

9 � 111

�10

9

12

60

75

25

110

99


Simplify the fraction. Write theanswer as a fraction or wholenumber.

6. 7. 8.1590

393

2836

BASIC—

TIP: Recall that any fraction ofthe form � n. Therefore, � 3.3

1n1

Avoiding Mistakes

Do not forget to write the “1” in thenumerator of the fraction .1

5

miL58061_ch02_095-162.qxd 7/5/08 10:04 AM Page 116


Answers

9. 10. 11.

12. Four zeros; the numerator anddenominator are both divisible by 10,000.

35

140

6319

Simplifying Fractions by 10, 100, and 1000

Simplify each fraction to lowest terms by first reducing by 10, 100, or 1000. Write

the answer as a fraction.

a. b. c.

Solution:

a. Both 170 and 30 are divisible by 10. “Strike through”

one zero.

The fraction is simplified completely.

b. Both 2500 and 7500 are divisible by 100. Strike

through two zeros.

Simplify further.

c. The numbers 5000 and 130,000 are both divisible by

1000. Strike through three zeros.

Simplify further.

�1

26

�51

13026

5000

130,000�

5000

130,000

�1

3

�25

1

753

2500

7500�

2500

7500

173 �

17

3

170

30�

170

30

5000

130,000

2500

7500

170

30


Simplify to lowest terms by firstreducing by 10, 100, or 1000.

9. 10.

11.21,00035,000

130052,000

630190

Avoiding Mistakes

Suppose that you do not recognize the greatest common factor in the numerator and denominator. Youcan still divide by any common factor. However, you will have to repeat this process more than once tosimplify the fraction completely. For instance, consider the fraction from Example 3(c).

�21

105

�15

12

2

6010

�2

10Dividing by the common factor of 6 leaves a fraction that can be simplifiedfurther.

Divide again, this time by 2. The fraction is now simplified completely becausethere are no other common factors in the numerator and denominator.

BASIC—

Concept Connections

12. How many zeros may beeliminated from the numera-tor and denominator of thefraction 430,000

154,000,000?

Avoiding Mistakes

The “strike through” method onlyworks for the digit 0 at the end ofthe numerator and denominator.

miL58061_ch02_095-162.qxd 7/5/08 10:04 AM Page 117


3. Applications of Simplifying Fractions

Simplifying Fractions in an Application

Madeleine got 28 out of 35 problems correct on an algebra exam. David got 27

out of 45 questions correct on a different algebra exam.

a. What fractional part of the exam did each student answer correctly?

b. Which student performed better?

Solution:

a. Fractional part correct for Madeleine:

or equivalently

Fractional part correct for David:

or equivalently

b. From the simplified form of each fraction, we see that Madeleine

performed better because That is, 4 parts out of 5 is greater than

3 parts out of 5. This is also easily verified on a number line.

10

MadeleineDavid

35

45

45 7 3

5.

273

455

�3

5

27

45

284

355

�4

5

28

35


13. Joanne planted 77 seeds inher garden and 55 sprouted.Geoff planted 140 seeds and80 sprouted.a. What fractional part of the

seeds sprouted forJoanne and what partsprouted for Geoff?

b. For which person did agreater portion of seedssprout?

Answers13. a. Joanne: Geoff:

b. Joanne had a greater portion ofseeds sprout.

47

57 ;

BASIC—


1. Sometimes, test anxiety can be greatly reduced by adequate preparation and practice. List some places in the

text where you can find extra problems for practice.


a. Lowest terms b. Greatest common factor

Review Exercises

For Exercises 3–10, write the prime factorization for each number.

3. 145 4. 114 5. 92 6. 153

7. 85 8. 120 9. 195 10. 180




• NetTutor


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BASIC—

Objective 1: Equivalent Fractions

For Exercises 11–14, shade the second figure so that it expresses a fraction equivalent to the first figure.

11. 12.

13. 14.

15. True or false? The fractions and are equivalent.

16. In your own words, explain the concept of equivalent fractions.

For Exercises 17–24, determine if the fractions are equivalent. Then fill in the blank with either

(See Example 1.)

17. 18. 19. 20.

21. 22. 23. 24.

Objective 2: Simplifying Fractions to Lowest Terms

For Exercises 25–52, simplify the fraction to lowest terms. Write the answer as a fraction or a whole number.

(See Examples 2–3.)

25. 26. 27. 28.

29. 30. 31. 32.

33. 34. 35. 36.

37. 38. 39. 40.

41. 42. 43. 44.65

5

33

11

84

126

105

140

2

2

9

9

24

6

50

25

8

8

14

14

8

16

20

25

30

25

15

12

49

42

36

20

21

24

6

18

15

18

12

24

12

18

5

6

20

27

8

9

12

15

4

5

3

4

12

16

3

8

6

16

3

6

1

2

2

9

1

4

3

5

2

3

� or �.

54

45

miL58061_ch02_095-162.qxd 7/5/08 10:04 AM Page 119


45. 46. 47. 48.

49. 50. 51. 52.

For Exercises 53–60, use the order of operations to simplify.

53. 54. 55. 56.

57. 58. 59. 60.

For Exercises 61–68, simplify to lowest terms by first reducing the powers of 10. (See Example 4.)

61. 62. 63. 64.

65. 66. 67. 68.9800

28,000

5100

30,000

50,000

65,000

42,000

22,000

2000

1500

3000

1800

720

800

120

160

15 � 3

15 � 3

8 � 2

8 � 2

4 � 7

11 � 11

7 � 2

5 � 5

11 � 11

4 � 7

5 � 5

7 � 2

9 � 1

15 � 3

6 � 2

10 � 4

69

92

34

85

39

130

385

195

70

120

130

150

85

153

77

110

BASIC—

71. a. What fraction of the alphabet is made up of

vowels? (Include the letter y as a vowel, not a

consonant.)

b. What fraction of the alphabet is made up of

consonants?

72. Of the 88 constellations that can be seen in the

night sky, 12 are associated with astrological

horoscopes. The names of as many as

36 constellations are associated with animals

or mythical creatures.

a. Of the 88 constellations, what fraction is

associated with horoscopes?

b. What fraction of the constellations have

names associated with animals or mythical

creatures?

Objective 3: Applications of Simplifying Fractions

69. André tossed a coin 48 times and heads came up

20 times. What fractional part of the tosses came

up heads? What fractional part came up tails?

70. At Pizza Company, Lee made 70 pizzas one day.

There were 105 pizzas sold that day. What

fraction of the pizzas did Lee make?

miL58061_ch02_095-162.qxd 7/5/08 10:04 AM Page 120


73. Jonathan and Jared both sold candy bars for a

fund-raiser. Jonathan sold 25 of his 35 candy bars,

and Jared sold 24 of his 28 candy bars. (See Example 5.)

a. What fractional part of his total number of

candy bars did each boy sell?

b. Which boy sold the greater fractional part?

74. Lisa and Lynette are taking online courses. Lisa

has completed 14 out of 16 assignments in her

course while Lynette has completed 15 out of

24 assignments.

a. What fractional part of her total number of

assignments did each woman complete?

b. Which woman has completed more of her

course?

75. Raymond read 720 pages of a 792-page book.

His roommate, Travis, read 540 pages from a

660-page book.

a. What fractional part of the book did each

person read?

b. Which of the roommates read a greater

fraction of his book?

76. Mr. Bishop and Ms. Waymire both gave exams

today. By mid-afternoon, Mr. Bishop had finished

grading 16 out of 36 exams, and Ms. Waymire had

finished grading 15 out of 27 exams.

a. What fractional part of her total has

Ms. Waymire completed?

b. What fractional part of his total has

Mr. Bishop completed?

77. For a recent year, the population of the United States was reported to be 296,000,000. During the same year,

the population of California was 36,458,000.

a. Round the U.S. population to the nearest hundred million.

b. Round the population of California to the nearest million.

c. Using the results from parts (a) and (b), write a simplified fraction showing the portion of the U.S.

population represented by California.

BASIC—

78. For a recent year, the population of the United States was reported to be 296,000,000. During the same year,

the population of Ethiopia was 75,067,000.

a. Round the U.S. population to the nearest hundred million.

b. Round the population of Ethiopia to the nearest million.

c. Using the results from parts (a) and (b), write a simplified fraction comparing the population of the

United States to the population of Ethiopia.

d. Based on the result from part (c), how many times greater is the U.S. population than the population

of Ethiopia?

miL58061_ch02_095-162.qxd 7/5/08 10:04 AM Page 121



79. Write three fractions equivalent to



82. Write three fractions equivalent to 80100.

1218.

13.

34.

BASIC—

Calculator Exercises

For Exercises 83–90, use a calculator to simplify the

fractions. Write the answer as a proper or improper

fraction.

83. 84.

85. 86.

87. 88.

89. 90.713

437

969

646

871

469

493

510

462

220

779

969

728

784

792

891

Topic: Simplifying Fractions on a Calculator

Some calculators have a fraction key, . To enter

a fraction, follow this example.

Expression:

Keystrokes: 3 4

Result:

To simplify a fraction to lowest terms, follow this

example.

Expression:

Keystrokes: 22 10

Result:

To convert to an improper

fraction, press

22

10

3

4

Calculator Connections

numerator denominatorc c

2 _ 1 _| 5 � 21

5

whole number fractionc

}

�11

511 _| 5

3_| 4

miL58061_ch02_095-162.qxd 7/5/08 10:04 AM Page 122

Section 2.4 Multiplication of Fractions and Applications 123

Elija takes 1

3

Max gets

of � 1

3

1

6

1

2

Figure 2-6

From the illustration, the product Notice that the product is found

by multiplying the numerators and multiplying the denominators. This is true in

general to multiply fractions.

PROCEDURE Multiplying Fractions

To multiply fractions, write the product of the numerators over the product

of the denominators. Then simplify the resulting fraction, if possible.

provided b and d are not equal to 0

Multiplying Fractions

Multiply.

a. b.

Solution:

a.

Notice that the product is simplified completely because there are no

common factors shared by 8 and 35.

b. First write the whole number as a fraction.

Multiply the numerators. Multiply the denominators.

�40

3

�8 � 5

3 � 1

8

3� 5 �

8

3�

5

1

835

2

5�

4

7�

2 � 4

5 � 7�

8

35

Multiply the numerators.

Multiply the denominators.

8

3� 5

2

5�

4

7

Example 1

ab

�cd

�a � cb � d

16

12 � 1

3 � 16.

The product is not reducible because there are no

common factors shared by 40 and 3.

Answers

1. 2. 3.7712

1027

18

1. Multiplication of FractionsSuppose Elija takes of a cake and then gives of this portion to his friend

Max. Max gets of of the cake. This is equivalent to the expression See

Figure 2-6.

12 � 1

3.13

12

12

13

Concept Connections

1. What fraction is of of awhole?

14

12

Skill Practice

Multiply. Write the answer as afraction.

2. 3.7

12� 11

23

�59

Objectives

1. Multiplication of Fractions

2. Fractions and the Order of

Operations

3. Area of a Triangle

4. Applications of Multiplying

Fractions

Section 2.4Multiplication of Fractions and Applications

BASIC—

miL58061_ch02_095-162.qxd 7/5/08 10:04 AM Page 123


Multiplying and Simplifying Fractions

Multiply the fraction and simplify if possible.

Solution:


Simplify by first dividing 20 and 420 by 10.

Simplify further by dividing 2 and 42 by 2.

It is often easier to simplify before multiplying. Consider the product from

Example 2.

1

2 and 6 share a common factor of 2.

3

As a general rule, this method is used most often in the text.


Multiply and simplify.

Solution:

1

3

�7

33

We can simplify further because 2

and 6 share a common factor of 2.�

102

186

�21

7

5511



10

18�

21

55�

102

186

�21

7

5511

10

18�

21

55

Example 3

�1

21

�42

306

�51

147


30 and 5 share a common factor of 5. 4

30�

5

14�

42

306

�51

147

�1

21

�21

4221

�20

420

4

30�

5

14�

4 � 5

30 � 14

4

30�

5

14

Example 2

Answers

4. 5.15

715

Skill Practice


4.7

20�

43

Skill Practice


5.625

�1518

BASIC—

Example 2 illustrates a case where the product of fractions must be simplified.

miL58061_ch02_095-162.qxd 7/5/08 10:04 AM Page 124



Multiply and simplify. Write the answers as fractions.

a. b.

Solution:

a. Write the whole number as a fraction.

Reduce before multiplying.

Multiply.

b. Multiply the first two fractions.

Simplify.

2. Fractions and the Order of OperationsFor problems with more than one operation, recall the order of operations.

�7

8

�21

7

651

�65

1

248

� a21

65b �

65

24

21

25�

15

39�

65

24� a21

7

255

�15

3

3913

b �65

24

�9

4

�63

1�

3

84

6 a3

8b �

6

1�

3

8

21

25�

15

39�

65

246 a3

8b

Example 4

Answers

6. 7.110

65

Skill Practice

Multiply and simplify. Write theanswers as fractions.

6.

7. a 316b a4

9b a6

5b

9 a 215b

BASIC—

PROCEDURE Using the Order of Operations

Step 1 Perform all operations inside parentheses first.

Step 2 Simplify any expressions containing exponents or square roots.

Step 3 Perform multiplication or division in the order that they appear

from left to right.

Step 4 Perform addition or subtraction in the order that they appear

from left to right.

miL58061_ch02_123-133.qxd 8/22/08 5:36 AM Page 125


Solution:

a. With an exponent of 3, multiply 3 factors of the base.


b.

Multiply fractions within parentheses.

Square by multiplying times itself.

In Section 1.7 we learned to recognize powers of 10. These are

and so on. In this section, we learn to recognize the powers of one-tenth, that is,

raised to a whole-number power. For example, consider the following expressions.

From these examples, we see that a power of one-tenth results in a fraction with

a 1 in the numerator. The denominator has a 1 followed by the same number

of zeros as the exponent on the base of

3. Area of a TriangleRecall that the area of a rectangle with length l and width w is given by

w

l

A � l � w

110.

a1

10b

4

�1

10�

1

10�

1

10�

1

10�

1

10,000

a1

10b

3

�1

10�

1

10�

1

10�

1

1000

a1

10b

2

�1

10�

1

10�

1

100

a1

10b

1

�1

10

110

101 � 10, 102 � 100,

�1

100

110

110 �

1

10�

1

10

� a1

10b

2

Perform the multiplication within the

parentheses. Simplify. a

2

15�

3

4b

2

� a21

155

�31

42

b2

�8

125

�2 � 2 � 2

5 � 5 � 5

a2

5b

3

�2

5�

2

5�

2

5

BASIC—

Simplifying Expressions

Simplify.

a. b. a2

15�

3

4b

2

a2

5b

3


Simplify. Write the answer as afraction.

8. 9. a65

�1

12b

3

a43b

2

Answers

8. 9.1

1000169

miL58061_ch02_095-162.qxd 7/5/08 10:04 AM Page 126


Answer

10. 20 m2

The value of b is the measure of the base of the triangle. The value of h is

the measure of the height of the triangle. The base b can be chosen as the length

of any of the sides of the triangle. However, once you have chosen the base, the

height must be measured as the shortest distance from the base to the opposite

vertex (or point) of the triangle.

Figure 2-7 shows the same triangle with different choices for the base. Fig-

ure 2-8 shows a situation in which the height must be drawn “outside” the tri-

angle. In such a case, notice that the height is drawn down to an imaginary

extension of the base line.

Finding the Area of a Triangle

Find the area of the triangle.

Solution:

Identify the measure of the base and the height.

Apply the formula for the area of a triangle.

Write the whole numbers as fractions.

Simplify.

Multiply numerators. Multiply denominators.

� 12 m2

�12

1 m2

�1

21

a63

1 mb a

4

1 mb

�1

2 a

6

1 mb a

4

1 mb

�1

216 m2 14 m2

A �1

2 bh

b � 6 m and h � 4 m

4 m

6 m

Example 6

Base

Opposite vertexh h

Base

Opposite vertex

Base

Opposite vertex

h

Figure 2-7 Figure 2-8

Skill Practice


10.

5 m

8 m

The area of the triangle is 12 square

meters (m2).

BASIC—

FORMULA Area of a Triangle

The formula for the area of a triangle is given by read “one-half

base times height.”

b

h

A � 12bh,

miL58061_ch02_095-162.qxd 7/5/08 10:04 AM Page 127


Solution:

Identify the measure of the base and the height.

Apply the formula for the area of a triangle.

Simplify.

The area of the triangle is square feet (ft2).

Find the Area of a Composite Geometric Figure

Find the area.

Example 8

5

8 �

5

8 ft2

�1

2 a

5

31

ftb a31

4 ftb

�1

2 a

5

3 ftb a

3

4 ftb

A �1

2 bh

b �5

3 ft and h �

3

4 ft

Answers

11. or ft2 12. 14 ft2113

43

10 in.

8 in.

in.72

�

Area of rectangle: (10 in.)(8 in.)

� 80 in.2

The total area of the region is 94 square inches (in.2).

Total area � 80 in.2 � 14 in.2 � 94 in.2

� 14 in.2

�14

1 in.2

� a42

1 in.b a

7

21

in.b

�1

21

a84

1 in.b a

7

2 in.b

Area of triangle: 1

2 18 in.2 a

7

2 in.b

112bh21l � w21area of triangle21area of rectangle2

Skill Practice

12. Find the area of the kite.

ft32

ft112

4 ft

Solution:

The total area is the sum of the areas of the rectangular region and the

triangular region. That is,

Total area �

BASIC—

Finding the Area of a Triangle


ft53

ft34


11. Find the area of the triangle.

2 ft

ft43

8 in.�

10 in.

8 in.

in.72

miL58061_ch02_095-162.qxd 7/5/08 10:04 AM Page 128


1. Write down the page number(s) for the Chapter Summary for this chapter. _______

Describe one way in which you can use the summary found at the end of each chapter.

2. Define the key term powers of one-tenth.

Review Exercises

For Exercises 3–6, identify the numerator and the denominator. Then simplify the fraction to lowest terms.

3. 4. 5. 6.

Objective 1: Multiplication of Fractions

7. Shade the portion of the figure 8. Shade the portion of the figure

that represents of that represents of 12.

12

12.

16

2100

7000

25

15

32

36

10

14


4. Applications of Multiplying Fractions

Multiplying Fractions in an ApplicationThe population of Texas comprises roughly of the population of the United

States. If the U.S. population is approximately 296,000,000, approximate the

population of Texas.

Solution:

We must find of the U.S. population. This translates to

Divide.

The population of Texas is approximately 22,200,000.

� 22,200,000

�88,800,000

4

Simplify by dividing:

88,800,000 � 4.

�888,000,000

40 Simplify by a factor of 10.

�888,000,000

40

Multiply the numerators.

Multiply the denominators.

3

40� 296,000,000 �

3

40�

296,000,000

1

340

340

Example 9

Answer

13. 5,920,000

Skill Practice

13. Find the population ofTennessee if it is approximately of the U.S. population. Assume that the U.S. population isapproximately 296,000,000.

150

TIP: To take a fractionalportion of a quantity, multiplythe quantity by the fraction.




• NetTutor


BASIC—

miL58061_ch02_123-133.qxd 8/22/08 5:36 AM Page 129

9. Shade the portion of the figure 10. Shade the portion of the figure

that represents of that represents of

11. Find of . 12. Find of . 13. Find of 8. 14. Find of 20.

For Exercises 15–26, multiply the fractions. Write the answer as a fraction. (See Example 1.)

15. 16. 17. 18.

19. 20. 21. 22.

23. 24. 25. 26.

For Exercises 27–50, multiply the fractions and simplify to lowest terms. Write the answer as a fraction or whole

number. (See Examples 2–4.)

27. 28. 29. 30.

31. 32. 33. 34.

35. 36. 37. 38.

39. 40. 41. 42.

43. 44. 45. 46.

47. 48. 49. 50.

Objective 2: Fractions and the Order of Operations

For Exercises 51–54, simplify the powers of

51. 52. 53. 54. a1

10b

9

a1

10b

6

a1

10b

4

a1

10b

3

1

10.

38

22� 11 �

5

19

100

49� 21 �

14

25

11

18�

2

20� 15

7

10�

3

28� 5

55

9�

18

32�

24

11

5

2�

10

21�

7

5

49

8�

4

5�

20

7

9

15�

16

3�

25

8

4 �8

9212 �

15

42

85

6�

12

10

21

4�

16

7

a39

11b a

11

13ba

17

9b a

72

17ba

12

45b a

5

4ba

6

11b a

22

15b

49

24�

6

7

24

15�

5

3

16

25�

15

32

21

5�

25

12

7

12�

18

5

5

6�

3

4

1

8�

4

7

2

9�

3

5

6

5�

7

5

13

9�

5

4

5

8� 5

4

5� 6

3 � a2

7b8 � a

1

11ba

9

10b a

7

4ba

12

7b a

2

5b

1

8�

9

8

14

9�

1

9

2

3�

1

3

1

2�

3

8

25

34

15

23

14

12

14.

13

14.

14


BASIC—

miL58061_ch02_095-162.qxd 7/5/08 10:04 AM Page 130

67. 68. 69. 70.

For Exercises 71–80, find the area of the figure. (See Examples 6–7.)

71. 72. 73. 74.

75. 76. 77. 78.

79. 80.

ft2324

ft34

in.1516

in.1316

m83

3 m

cm34

cm13

3 mmmm169

5 yd

yd85

ft74

1 ft

8 m

8 m

12 in.15 in.

8 cm

11 cm


b

h

For Exercises 55–66, simplify. Write the answer as a fraction or whole number. (See Example 5.)

55. 56. 57. 58.

59. 60. 61. 62.

63. 64. 65. 66.

Objective 3: Area of a Triangle

For Exercises 67–70, label the height with h and the base with b, as shown in the figure.

28

6� a3

2b216

9� a1

2b31

6� a24

5�

30

8b1

3� a21

4�

8

7b

a10

3�

1

100b2a1

9�

3

5b2a5 �

2

5b3a4 �

3

4b3

a4

3b3a3

2b3a1

4b2a1

9b2

BASIC—

miL58061_ch02_95-162.qxd 7/7/08 4:51 PM Page 131


For Exercises 81–84, find the area of the shaded region. (See Example 8.)

81. 82.

83. 84.

Objective 4: Applications of Multiplying Fractions

m94

m154

8 m

cm73

cm23

6 cm

8 m

3 m

3 m

8 yd

4 yd

4 yd

BASIC—

85. Ms. Robbins’ car holds 16 gallons (gal) of gas. If

the fuel gauge indicates that there is of a tank

left, how many gallons of gas are left in the

tank? (See Example 9.)

58

86. Land in a rural part of Bowie County, Texas,

sells for $11,000 per acre. If Ms. Anderson

purchased acre, how much did it cost?34

87. Jim has half a pizza left over from dinner. If he

eats of this for breakfast, how much pizza did

he eat for breakfast?

14

88. In a certain sample of individuals, are known

to have blood type O. Of the individuals with

blood type O, are Rh-negative. What fraction

of the individuals in the sample have O negative

blood?

14

25

89. One week, Nielsen Media Research reported

that a new episode of C.S.I. was the second most

watched television show with 9,825,000 total

viewers. The following week, a repeat episode

was shown, but only of the viewers returned to

watch. How many viewers tuned in to watch the

repeat episode?

23

90. Nancy spends hour 3 times a day walking and

playing with her dog. What is the total time she

spends walking and playing with the dog each

day?

34

91. The Bishop Gaming Center hosts a football

pool. There is $1200 in prize money. The first-

place winner receives of the prize money. The

second-place winner receives of the prize

money, and the third-place winner receives of

the prize money. How much money does each

person get?

112

14

23

92. Frankie’s lawn measures 40 yd by 36 yd. In the

morning he mowed of the lawn. How many

square yards of lawn did he already mow? How

much is left to be mowed?

23

miL58061_ch02_95-162.qxd 7/7/08 4:51 PM Page 132

Section 2.5 Division of Fractions and Applications 133


93. Evaluate. 94. Evaluate.

a. b. a. b.

For Exercises 95–98, evaluate the square roots.

95. 96. 97. 98.

99. Find the next number in the sequence:

100. Find the next number in the sequence:

101. Which is greater, of or of ? 102. Which is greater, of or of ?23

14

14

23

12

18

18

12

23,

29,

227,

12,

14,

18,

116,

B9

4B64

81B1

100B1

25

B4

49a

2

7b

2

B1

36a

1

6b

2

BASIC—

Objectives

1. Reciprocal of a Fraction

2. Division of Fractions

3. Order of Operations

4. Applications of Multiplication

and Division of Fractions

Section 2.5Division of Fractions and Applications

1. Reciprocal of a FractionTwo numbers whose product is 1 are said to be reciprocals of each other. For exam-

ple, consider the product of and .

Because the product equals 1, we say that is the reciprocal of and vice versa.

To divide fractions, first we need to learn how to find the reciprocal of a

fraction.

83

38

3

8�

8

3�

31

81

�81

31

� 1

83

38

PROCEDURE Finding the Reciprocal of a Fraction

To find the reciprocal of a nonzero fraction, interchange the numerator and

denominator of the fraction. Thus, the reciprocal of is (provided

and ).b � 0

a � 0ba

ab

Concept Connections

Fill in the blank.

1. The product of a number andits reciprocal is .

Answer1. 1

miL58061_ch02_095-162.qxd 7/5/08 10:04 AM Page 133


PROCEDURE Dividing Fractions

To divide two fractions, multiply the dividend (the “first” fraction) by the

reciprocal of the divisor (the “second” fraction).

The process to divide fractions can be written symbolically as

provided b, c, and d are not 0.ab

�cd �

ab

�dc

Change division to

multiplication.

Take the reciprocal

of the divisor.

Concept Connections

Fill in the box to make the right-and left-hand sides equal.

6.

7. �51

23

�15

�

47

�35

�47

�

Solution:

a. The reciprocal of is .

b. The reciprocal of is , or 9.

c. First write the whole number 5 as the improper fraction .

The reciprocal of is .

d. The number 0 has no reciprocal because is undefined.10

15

51

51

91

19

52

25

2. Division of FractionsTo understand the division of fractions, we compare it to the division of whole

numbers. The statement 6 ÷ 2 asks, “How many groups of 2 can be found among

6 wholes?” The answer is 3.

In fractional form, the statement 6 ÷ 2 = 3 can be written as This result can

also be found by multiplying.

That is, to divide by 2 is equivalent to multiplying by the reciprocal

In general, to divide two nonzero numbers we can multiply the dividend by the

reciprocal of the divisor. This is how we divide by a fraction.

12.

6 �1

2�

6

1�

1

2�

6

2� 3

62 � 3.

6 � 2 � 3

Answers

2. 3. 4 4.

5. 1 6. 7.23

53

17

107

BASIC—

Finding Reciprocals

Find the reciprocal.

a. b. c. 5 d. 01

9

2

5


Find the reciprocal.

2. 3.

4. 7 5. 1

14

710

miL58061_ch02_095-162.qxd 7/5/08 10:05 AM Page 134


Solution:

a.

Multiply numerators. Multiply denominators.

b.

Simplify.

Multiply.

c.

Multiply by the reciprocal of the divisor.

Simplify.

Multiply.

d.


Simplify.

Multiply. �9

2

�12

3

1�

3

82

�12

1�

3

8

Write the whole number 12 as an

improper fraction. 12 �

8

3�

12

1�

8

3

�5

14

�35

5

14�

1

71

�35

14�

1

7

35

14� 7 �

35

14�

7

1

�5

36

�21

279

�15

5

84

Multiply by the reciprocal of the divisor

(“second” fraction). 2

27�

8

15�

2

27�

15

8

�8

35

�2 � 4

5 � 7

Multiply by the reciprocal of the divisor

(“second” fraction). 2

5�

7

4�

2

5�

4

7

Write the whole number 7 as an improper fraction

before multiplying by the reciprocal.

Skill Practice

Divide and simplify. Write theanswer as a fraction.

8. 9.

10. 11. 20 �125

154

� 10

38

�910

14

�25

Avoiding Mistakes

Do not try to simplify until aftertaking the reciprocal of the divisor.In Example 2(a) it would be incor-rect to “cancel” the 2 and the 4 inthe expression 25 � 7

4.

Dividing Fractions

Divide and simplify, if possible. Write the answer as a fraction.

a. b. c. d. 12 �8

3

35

14� 7

2

27�

8

15

2

5�

7

4

Example 2

3. Order of OperationsWhen simplifying fractional expressions with more than one operation, be sure to

follow the order of operations. Simplify within parentheses first. Then simplify

expressions with exponents, followed by multiplication or division in the order of

appearance from left to right.

BASIC—

Answers

8. 9. 10. 11.253

38

512

58

miL58061_ch02_095-162.qxd 7/5/08 10:05 AM Page 135


Solution:

Perform operations within parentheses first.


Simplify.

Multiply within parentheses.

With an exponent of 2, multiply 2 factors of the base.

Multiply. �81

4

�9

2�

9

2

� a9

2b

2

� a3

51

�15

3

2b

2

� a3

5�

15

2b

2

a3

5�

2

15b

2

Applying the Order of Operations

Simplify. Write the answer as a fraction.

Solution:

Simplify within parentheses.


Simplify.

Multiply. �1

4

�31

2�

1

62

� a3

2b �

1

6

Simplify within parentheses and write the whole

number as an improper fraction. � a

3

2b �

6

1

� a21

31

�93

42

b � 6

We will divide from left to right. To emphasize this

order, we can insert parentheses around the first

two fractions.

� a2

3�

4

9b � 6

2

3�

4

9� 6

2

3�

4

9� 6

Example 3

Applying the Order of Operations

Simplify. Write the answer as a fraction.

a3

5�

2

15b

2

Example 4

Skill Practice

12. Simplify.67

� 2 �3

14

TIP: In Example 3 wecould also have writteneach division as multiplica-tion of the reciprocal rightfrom the start.

1

Simplify.

Multiply. �14

�21

31

�93

42

�162

23

�49

� 6 � a23

�94b �

16

Skill Practice

13. Simplify.

a47

�83b

2

Answers

12. 2 13.9

196

BASIC—

miL58061_ch02_095-162.qxd 7/5/08 10:05 AM Page 136


4. Applications of Multiplication and Division of Fractions

Sometimes it is difficult to determine whether multiplication or division is

appropriate to solve an application problem. Division is generally used for a prob-

lem that requires you to separate or “split up” a quantity into pieces. Multiplication

is generally used if it is necessary to take a fractional part of a quantity.

Answers14. 50 batches15. 10 segments of piping

Using Division in an Application

A road crew must mow the grassy median along a stretch of highway I-95. If

they can mow mile (mi) in 1 hr, how long will it take them to mow a 15-mi

stretch?

Solution:

Read and familiarize.

Strategy/operation: From the figure, we must separate or “split up” a 15-mi

stretch of highway into pieces that are mi in length. Therefore, we must

divide 15 by .

The 15-mi stretch of highway will take 24 hr to mow.

� 24

�15

3

1�

8

51

Write the whole number as a fraction. Multiply

by the reciprocal of the divisor. 15 ÷

5

8 �

15

1�

8

5

58

58

mi58

mi58

mi58

15 mi

58


14. A cookie recipe requires package of chocolate chips

for each batch of cookies. If arestaurant has 20 packagesof chocolate chips, how manybatches of cookies can itmake?

25

Using Division in an Application

A -ft length of wire must be cut into pieces of equal length that are ft long.

How many pieces can be cut?

38

94


15. A -yd ditch will be dug toput in a new water line. Ifpiping comes in segments of

yd, how many segmentsare needed to line the ditch?

54

252

BASIC—

miL58061_ch02_095-162.qxd 7/5/08 10:05 AM Page 137


Using Multiplication in an Application

Carson estimates that his total cost for college for 1 year is $12,600. He has

financial aid to pay of the cost.

a. How much money is the financial aid worth?

b. How much money will Carson have to pay?

c. If Carson’s parents help him by paying of the amount not paid by financial

aid, how much money will be paid by Carson’s parents?

Solution:

a. Carson’s financial aid will pay of $12,600. Because we are looking for a

fraction of a quantity, we multiply.

Financial aid will pay $8400.

b. Carson will have to pay the remaining portion of the cost. This can be

found by subtraction.

$12,600 � $8400 $4200

Carson will have to pay $4200.

�

� 8400

�2

31

�12,600

4200

1

2

3� 12,600 �

2

3�

12,600

1

23

13

23


16. A new school will cost$20,000,000 to build, and thestate will pay of the cost.a. How much will the

state pay?b. How much will the

state not pay?c. The county school district

issues bonds to pay ofthe money not covered by the state. How muchmoney will be covered by bonds?

45

35

Answer16. a. $12,000,000

b. $8,000,000c. $6,400,000

TIP: The answer to Example 7(b) could also have been found by noting thatfinancial aid paid of the cost. This means that Carson must pay of the cost, or

� $4200

13

�$12,600

1�

131

�$12,600

4200

1

13

23

Solution:


Operation: Here we divide the total length of wire into pieces of equal length.


Simplify.

Six pieces of wire can be cut.

� 6

�93

41

�82

31

9

4�

3

8�

9

4�

8

3

BASIC—

miL58061_ch02_095-162.qxd 7/5/08 10:05 AM Page 138


c. Carson’s parents will pay of $4200.

Carson’s parents will pay $1400.

1

31

�4200

1400

1

13

Using Multiplication in an Application

Three-fifths of the students in the freshman class are female. Of these students,

are over the age of 25. What fraction of the freshman class is female over the

age of 25?

59

Example 8

Solution:


Strategy/operation: We must find of of one whole freshman class. This implies

multiplication.

One-third of the freshman class consists of female students over the age of 25.

5

9�

3

5�

51

93

�31

51

�1

3

35

59

Skill Practice

17. In a certain police department, of the department is female. Of thefemale officers, werepromoted within the last year.What fraction of the policedepartment consists offemales who were promotedwithin the last year?

27

13

Answer

17.2

21


1. Write down the page number(s) for the Problem Recognition Exercises for this chapter. How do you think

that the Problem Recognition Exercises can help you?

2. Define the key term reciprocal.

Review Exercises

For Exercises 3–11, multiply and simplify to lowest terms. Write the answer as a fraction or whole number.

3. 4. 5.34

5�

5

17

24

7�

7

8

9

11�

22

5




• NetTutor


BASIC—

miL58061_ch02_134-143.qxd 8/27/08 12:05 AM Page 139

6. 7. 8.

9. 10. 11.

Objective 1: Reciprocal of a Fraction

12. For each number, determine whether the number has a reciprocal.

a. b. c. 6 d. 0

For Exercises 13–20, find the reciprocal of the number, if it exists. (See Example 1.)

13. 14. 15. 16.

17. 4 18. 9 19. 0 20.

Objective 2: Division of Fractions

For Exercises 21–24, fill in the blank.

21. Dividing by 3 is the same as multiplying by _____. 22. Dividing by 5 is the same as multiplying by _____.

23. Dividing by 8 is the same as ____________ by . 24. Dividing by 12 is the same as ____________ by .

For Exercises 25–48, divide and simplify the answer to lowest terms. Write the answer as a fraction or whole

number. (See Example 2.)

25. 26. 27. 28.

29. 30. 31. 32.

33. 34. 35. 36.

37. 38. 39. 40.

41. 42. 43. 44.

45. 46. 47. 48.13

5�

17

10

36

5�

9

25

1000

17�

10

3

9

100�

13

1000

30

40�

15

8

9

50�

18

25

20

6� 5

12

5� 4

24 �8

512 �

3

4

4

3�

1

3

10

9�

1

18

4 �3

57 �

2

3

6

5�

6

5

3

4�

3

4

9

10�

9

2

15

2�

3

2

11

2�

3

4

14

3�

6

5

8

7�

3

10

7

13�

2

5

11

3�

6

5

2

15�

5

12

1

12

1

8

0

4

14

5

10

9

5

6

7

8

5

3

1

2

1

3� 3

1

10� 10a

9

5b a

5

9b

a2

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7

2b8 � a

5

24b3 � a

7

6b


BASIC—

miL58061_ch02_095-162.qxd 7/5/08 10:05 AM Page 140

Mixed Exercises

For Exercises 49–64, multiply or divide as indicated. Write the answer as a fraction or whole number.

49. 50. 51. 52.

53. 54. 55. 56.

57. 58. 59. 60.

61. 62. 63. 64.

Objective 3: Order of Operations

65. Explain the difference in the process to evaluate versus Then evaluate each expression.

66. Explain the difference in the process to evaluate versus . Then evaluate each expression.

For Exercises 67–78, simplify by using the order of operations. Write the answer as a fraction or whole number.


67. 68. 69. 70.

71. 72. 73. 74.

75. 76. 77. 78.

Objective 4: Applications of Multiplication and Division of Fractions

79. How many eighths are in ? 80. How many sixths are in ?

81. During the month of December, a department 82. A developer sells lots of land in increments

store wraps packages free of charge. Each of acre. If the developer has 60 acres,

package requires yd of ribbon. If Li used up how many lots can be sold?

a 36-yd roll of ribbon, how many packages

were wrapped? (See Example 5.)

83. If one cup is gal, how many cups of orange juice can be filled from gal? (See Example 6.)32

116

23

34

43

94

8

27� a

3

4b

2

�13

18

15

16� a

2

3b

2

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21a

25

3�

50

9b

2

� 8a63

8�

9

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2

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a5

12�

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3b

2

a2

5�

8

3b

27

8� a

1

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2

a3

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2

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14

5

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35

16�

1

4

3

5�

6

7�

5

3

48

56�

3

8� 8

54

21�

2

3� 9

8 � 238 � 2

3

23 � 6.2

3 � 6

5 �15

48 �

16

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40

21�

18

25

22

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5

16

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1

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17

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42

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16

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5

66 �

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9

7

12�

5

3

7

8�

1

4


BASIC—

miL58061_ch02_095-162.qxd 7/5/08 10:05 AM Page 141

84. If 1 centimeter (cm) is meter (m), how many centimeters are in a -m

piece of rope?

85. Dorci buys 16 sheets of plywood, each in. thick, to cover her

windows in the event of a hurricane. She stacks the wood in the garage.

How high will the stack be?

86. Davey built a bookshelf 36 in. long. Can the shelf hold a set of encyclopedias

if there are 24 books and each book averages in. thick? Explain your

answer.

87. A radio station allows 18 minutes (min) of 88. A television station has 20 min of advertising each

advertising each hour. How many 40-second hour. How many 30-second ( -min) commercials

( -min) commercials can be run in can be run in

a. 1 hr b. 1 day a. 1 hr b. 1 day

89. Ricardo wants to buy a new house for $240,000. The bank requires of the cost of the house as a down

payment. As a gift, Ricardo’s mother will pay of the down payment. (See Example 7.)

a. How much money will Ricardo’s mother pay toward the down payment?

b. How much money will Ricardo have to pay toward the down payment?

c. How much is left over for Ricardo to finance?

90. Althea wants to buy a Toyota Camry for a total cost of $18,000. The dealer requires of the money as a

down payment. Althea’s parents have agreed to pay one-half of the down payment for her.

a. How much money will Althea’s parents pay toward the down payment?

b. How much will Althea pay toward the down payment?

c. How much will Althea have to finance?

91. A landowner has acres of land. She plans to sell of the land. (See Example 8.)

a. How much land will she sell? b. How much land will she retain?

92. Josh must read 24 pages for his English class and 18 pages for psychology.

He has read of the pages.

a. How many pages has he read?

b. How many pages does he still have to read?

93. A lab technician has liters (L) of alcohol. If she needs samples of L,

how many samples can she prepare?

94. Troy has a -in. nail that he must hammer into a board. Each strike of the

hammer moves the nail in. into the board. How many strikes of the

hammer must he make?


95. The rectangle shown here has an area of 30 ft2. Find the length.

96. The rectangle shown here has an area of 8 m2. Find the width. ?

14 m

?

ft52

116

78

18

74

16

13

94

112

23

110

23

12

54

34

54

1100


BASIC—

miL58061_ch02_095-162.qxd 7/5/08 10:05 AM Page 142

Problem Recognition Exercises 143

2. a. b.

c. d.

4. a. b.

c. d.

6. a. b.

c. d.

8. a. b.

c. d.

10. a. b.

c. d.

12. a. b.

c. d.

14. a. b.

c. d.

16. a. b.

c. d.1

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2

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7�

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1

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1

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10 � 66 � 10

10 � 66 � 10

4

5�

1

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4

5�

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4

5�

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4

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7

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7

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7

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0 �9

8

9

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8

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5

12

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10

3

10

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7

12

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7

Multiply or divide as indicated. Write the answer as a whole number or a fraction.

1. a. b.

c. d.

3. a. b.

c. d.

5. a. b.

c. d.

7. a. b.

c. d.

9. a. b.

c. d.

11. a. b.

c. d.

13. a. b.

c. d.

15. a. b.

c. d. 4 � a1

6b

2

4 � a1

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1

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8 � 48 � 4

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1

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9

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9

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9

8

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5

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Problem Recognition Exercises

Multiplication and Division of Fractions

BASIC—

miL58061_ch02_095-162.qxd 7/5/08 10:05 AM Page 143


Answer

1. 2656

TIP: To check whether the answer from Example 1 is reasonable, we canround each factor and estimate the product.

3 rounds to 3.

4 rounds to 5.34

15

⎫⎪⎬⎪⎭

Thus, which is close to 15

15

.a3

15b a4

34b � 132 152 � 15,

Section 2.6 Multiplication and Division of Mixed Numbers

Objectives

1. Multiplication of Mixed

Numbers

2. Division of Mixed Numbers

3. Applications of

Multiplication and Division

of Mixed Numbers

1. Multiplication of Mixed NumbersTo multiply mixed numbers, we follow these steps.

PROCEDURE Multiplying Mixed Numbers

Step 1 Change each mixed number to an improper fraction.

Step 2 Multiply the improper fractions and simplify to lowest terms, if

possible (see Section 2.4).

Answers greater than 1 may be written as an improper fraction or as a mixed

number, depending on the directions of the problem.

Example 1 demonstrates this process.

Multiplying Mixed Numbers

Multiply and write the answer as a mixed number.

Solution:

Write each mixed number as an improper fraction.

Simplify.

Multiply.

Write the improper fraction as a mixed number.

15

5�76

�5

26

�25

1

� 151

5

�76

5

�16

4

5�

19

41

a3

1

5b a4

3

4b �

16

5�

19

4

a3

1

5b a4

3

4b


1. Multiply and write the answeras a mixed number.

a4

35b a5

56b

BASIC—

miL58061_ch02_095-162.qxd 7/5/08 10:06 AM Page 144

Section 2.6 Multiplication and Division of Mixed Numbers 145

Multiplying Mixed Numbers

Multiply and write the answer as a mixed number or whole number.

a. b.

Solution:

a.

Simplify.

Multiply.

b. Write the whole number and mixed number as

improper fractions.

Simplify.

Multiply.


2. Division of Mixed NumbersTo divide mixed numbers, we use the following steps.

PROCEDURE Dividing Mixed Numbers

Step 1 Change each mixed number to an improper fraction.

Step 2 Divide the improper fractions and simplify to lowest terms, if

possible. Recall that to divide fractions, we multiply the dividend

by the reciprocal of the divisor (see Section 2.5).

Answers greater than 1 may be written as an improper fraction or as a mixed

number, depending on the directions of the problem.

105

3�316

�3

16

�15

1

� 105

1

3

�316

3

�12

4

1�

79

93

12 � a8

7

9b �

12

1�

79

9

� 119

�119

1

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21

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7

31

25

1

2� 4

2

3�

51

2�

14

3

12 � a8

7

9b 25

1

2� 4

2

3


Multiply and write the answer as amixed number or whole number.

2.

3. 716

� 10

16

12

� 3 7

11

Avoiding Mistakes

Do not try to multiply mixed numbers by multiplying the whole-number parts and multiplying thefractional parts. You will not get thecorrect answer.

For the expression , itwould be incorrect to multiply (25)(4)and Notice that these valuesdo not equal 119.

12 � 2

3.

25 12 � 4

23

BASIC—

Write each mixed

number as an

improper fraction.

Answers

2. 60 3. 71 23

miL58061_ch02_095-162.qxd 7/5/08 10:06 AM Page 145


Dividing Mixed Numbers

Divide and write the answer as a mixed number or whole number.

a. b. c.

Solution:

a. Write the mixed numbers as improper fractions.


Multiply.


b. Write the whole number and mixed number as

improper fractions.


Simplify.

Multiply.


c. Write the whole number and mixed number as

improper fractions.


Multiply.


3. Applications of Multiplication and Division ofMixed Numbers

Examples 4 and 5 demonstrate multiplication and division of mixed numbers in

day-to-day applications.

� 141

42

�83

42

�83

6�

1

7

135

6� 7 �

83

6�

7

1

� 11

6

�7

6

�61

1�

7

366

�6

1�

7

36

6 � 51

7�

6

1�

36

7

� 117

28

�45

28

�15

2�

3

14

7

1

2� 4

2

3�

15

2�

14

3

13

5

6� 76 � 5

1

77

1

2� 4

2

3

Example 3

Avoiding Mistakes

Be sure to take the reciprocal of thedivisor after the mixed number ischanged to an improper fraction.

BASIC—

Answers

4. 5. 6. 159

123

3

1117

Skill Practice

Divide and write the answer as amixed number or whole number.

4. 5.

6. 12

49

� 8

8 � 4

45

10

13

� 2

56

miL58061_ch02_095-162.qxd 7/5/08 10:06 AM Page 146


Applying Multiplication of Mixed Numbers

Antonio has a painting company, and he recently won a contract to paint

eight large classrooms at a university. Each room requires gal of paint.

How much paint is needed for this job?

Solution:

Amount needed for 8 rooms:

The painter requires gal of paint.

Applying Division of Mixed Numbers

A construction site brings in 6 tons of soil. Each truck holds ton. How many

truckloads are necessary?

Solution:

The 6 tons of soil must be distributed in -ton increments. This will require

division.

Write the mixed number as an improper fraction.


Simplify.

Multiply.

A total of 10 truckloads of soil will be required.

� 10

�10

1

�2010

31

�31

21

�20

3�

3

2

6

2

3�

2

3�

20

3�

2

3

23

23

23

23

Example 5

46

2

5

� 46

2

5 gal

�232

5 gal

�8

1 �

29

5 gal

8 � a5

4

5 galb � 8 � a

29

5b gal

5 45

Example 4

Answers

7. cups of flour 8. 64 packages678

tons6 23

ton23

ton23

ton23

� � �

Skill Practice

7. A recipe calls for cups offlour. How much flour isrequired for times therecipe?

212

234

Skill Practice

8. A department store wrapspackages for $2 each. Ribbon

ft long is used to wrapeach package. How manypackages can be wrappedfrom a roll of ribbon 168 ftlong?

2 58

TIP: The solution toExample 4 can beestimated. In this case, thepainter might want to over-estimate the answer to besure that he doesn’t runshort of paint. We can round

gal to 6 gal. Therefore,5 45

BASIC—

our estimate is 8(6 gal) �48 gal of paint.

miL58061_ch02_095-162.qxd 7/5/08 10:06 AM Page 147


Study Skills Exercise

1. Find the page numbers for the Chapter Review Exercises, Chapter Test, and Cumulative Review Exercises

for this chapter.

Chapter Review Exercises, page(s) .

Chapter Test, page(s) .

Cumulative Review Exercises, page(s) .

Compare these features and state the advantages of each.

Review Exercises

For Exercises 2–7, multiply or divide the fractions. Write your answers as fractions.

2. 3. 4.

5. 6. 7.

8. Explain the process to change a mixed number to an improper fraction.

For Exercises 9–12, write the mixed number as an improper fraction.

9. 10. 11. 12.

For Exercises 13–16, write the improper fraction as a mixed number.

13. 14. 15. 16.

Objective 1: Multiplication of Mixed Numbers

For Exercises 17–32, multiply the mixed numbers. Write the answer as a mixed number or whole number.


17. 18. 19. 20.

21. 22. 23. 24.

25. 26. 27. 28.

29. 30. 31. 32. a6

1

8b a2

3

4b a8

7ba5

2

5b a2

9b a1

4

5ba1

3

10b a1

1

4ba3

1

2b a2

1

7b

0 � 6

1

104

5

8� 0a2

2

3b � 3a7

1

4b � 10

a8

2

3b a2

1

13ba5

3

16b a5

1

3b3

1

3� 64

2

9� 9

6

1

8�

4

72

1

3�

5

7a5

1

5b a3

3

4ba2

2

5b a3

1

12b

31

2

39

4

57

11

77

6

4

1

81

4

72

7

103

2

5

52

18� 13

32

15� 4

42

11�

7

2

20

9�

10

3

13

5�

10

9

5

6�

2

9




• NetTutor


BASIC—

miL58061_ch02_144-151.qxd 8/27/08 12:06 AM Page 148


Objective 2: Division of Mixed Numbers

For Exercises 33–48, divide the mixed numbers. Write the answer as a mixed number, proper fraction, or whole

number. (See Example 3.)

33. 34. 35. 36.

37. 38. 39. 40.

41. 42. 43. 44.

45. 46. 47. 48.

Objective 3: Applications of Multiplication and Division of Mixed Numbers

49. Tabitha charges $8 per hour for baby sitting. 50. Kurt bought acres of land. If land costs

If she works for hr, how much should $10,500 per acre, how much will the land cost him?

she be paid? (See Example 4.)

51. According to the U.S. Census Bureau’s 52. Florida voters approved an amendment to the state

Valentine’s Day Press Release, the average constitution to set Florida’s minimum wage at $

American consumes lb of chocolate per hour. Kayla earns minimum wage while working

in a year. Over the course of 25 years, how at the campus bookstore. If she works for hr,

many pounds of chocolate would the how much should she be paid?

average American consume?

53. The age of a small kitten can be approximated by the following rule. The

kitten’s age is given as 1 week for every quarter pound of weight. (See Example 5.)

a. Approximately how old is a -lb kitten?

b. Approximately how old is a -lb kitten?

54. Richard’s estate is to be split equally among his three children. If his estate

is worth million, how much will each child inherit?

55. Lucy earns $14 per hour and Ricky earns $10 per hour. Suppose

Lucy worked hr last week and Ricky worked hr.

a. Who earned more money and by how much?

b. How much did they earn altogether?

56. A roll of wallpaper covers an area of . If the roll is ft wide, how

long is the roll?

Mixed Exercises

For Exercises 57–76, perform the indicated operation. Write the answer as a

mixed number, proper fraction, or whole number.

57. 58. 59. 60.

61. 62. 63. 64. 5

1

3� 64

1

12� 0

4

3� 5

1

8

2

3� 2

7

10

8 � 2

1

3 6 � 1

1

8 3

3

4� 1

5

62

1

5� 1

1

10

1172428 ft2

4212351

2

$134

218

134

15 34

25 710

6 23

434

223

4

2

3� 33

1

2� 22

1

7�

5

131

1

3�

2

7

6

1

2�

1

22

5

6�

1

60 � 1

9

110 � 6

7

12

5

5

6� 2

1

34

1

2� 2

1

47

3

5� 1

7

122

1

2� 1

1

16

12

4

5� 2

3

55

8

9� 1

1

3 5

1

10�

3

41

7

10� 2

3

4

BASIC—

miL58061_ch02_095-162.qxd 7/5/08 10:06 AM Page 149


65. 66. 67. 68.

69. 70. 71. 72.

73. 74. 75. 76.


77. A landscaper will use a decorative concrete border around the cactus garden

shown. Each concrete brick is ft long and costs $3. What is the total cost of

the border?

78. Sara drives a total of 64 mi to work and back. Her car gets 21 miles per gallon

(mpg) of gas. If gas is $5 per gallon, how much does it cost her to commute to

and from work each day?

12

12

114

15 ft

20 ft

3

1

8� 5

5

7� 1

5

167

1

8� 1

1

3� 2

1

4a5

1

6b a1

4

7b a

14

33ba3

2

5b a

7

34b a3

3

4b

0 � 2

1

86

8

9� 0 20 �

2

15 12 �

1

8

3

8� 2

1

20 � 9

2

3

2

7� 1

8

9 10

1

2� 9

Topic: Multiplying and Dividing Fractions and Mixed Numbers

Expression Keystrokes Result

8 15 � 25 28

2 3 4 1 6

To convert the result to an improper fraction, press

23

4�

1

6

8

15�

25

28

Calculator Connections

This is how you enterthe mixed number 23

4.

⎫⎪⎪⎪⎬⎪⎪⎪⎭

Calculator Exercises

For Exercises 79–86, use a calculator to multiply or divide and simplify. Write the answer as a mixed number.

BASIC—

79. 80.

83. 84. 1061

9� 41

5

632

7

12� 12

1

6

38

1

3� 12

1

212

2

3� 25

1

881. 82.

85. 86. 9

8

9� 28

1

311

1

2� 41

3

4

25

1

5� 18

1

256

5

6� 3

1

6

10 _| 21 �10

21

16 _ 1 _| 2 � 161

2

33 _| 2 �33

2

miL58061_ch02_095-162.qxd 7/5/08 10:06 AM Page 150

Group Activity 151

Cooking for Company

Estimated time: 15 minutes

Group Size: 3

This recipe for chili serves 6 people.

CHILI

1 tablespoon oil teaspoon salt

2 onions

lb ground beef teaspoon cayenne pepper

14 oz canned tomatoes 4 whole cloves

teaspoons chili powder 15 oz canned kidney beans

teaspoon cumin3

8

1

1

4

1

81

1

2

1

1

8

1. Suppose that this recipe is to be used for a Super Bowl party for 30 people. Each student in the group

should select three ingredients and determine the amount needed to make this recipe for 30 people. Then

write the revised recipe.

2. Suppose that this recipe is to be used for a dinner party for 9 guests. Each student in the group should

select three ingredients and determine the amount needed to make this recipe for 9 people. Then write the

revised recipe.

Group Activity

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Key Concepts

A fraction represents a part of a whole unit. For exam-

ple, represents one part of a whole unit that is divided

into 3 equal pieces.

In the fraction , the “top” number, 1, is the numerator,

and the “bottom” number, 3, is the denominator.

A fraction in which the numerator is less than the de-

nominator is called a proper fraction. A fraction in

which the numerator is greater than or equal to the de-

nominator is called an improper fraction.

An improper fraction can be written as a mixed numberby dividing the numerator by the denominator. Write

the quotient as a whole number, and write the remain-

der over the divisor.

A mixed number can be written as an improper fraction

by multiplying the whole number by the denominator

and adding the numerator.Then write that total over the

denominator.

Fractions can be represented on a number line. For

example,represents

3

10.

10

13

13

Examples

Example 1

of the

pie is shaded.

Example 2

For the fraction , the numerator is 7 and the denomi-

nator is 9.

Example 3

is an improper fraction, is a proper fraction, and is

an improper fraction.

Example 4

can be written as because

Example 5

can be written as because

Example 6

810

10

2 � 5 � 4

5�

14

5.14

5245

3

3�10

�9

1

313

103

33

35

53

79

13

Chapter 2 Summary

Introduction to Fractions and Mixed NumbersSection 2.1

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Summary 153

Prime Numbers and FactorizationSection 2.2

Key Concepts

A factorization of a number is a product of factors that

equals the number.

A number is divisible by another if their quotient leaves

no remainder.

Divisibility Rules for 2, 3, 5, and 10

• A whole number is divisible by 2 if the ones-place

digit is 0, 2, 4, 6, or 8.

• A whole number is divisible by 3 if the sum of the

digits of the number is divisible by 3.


digit is 0 or 5.


digit is 0.

A prime number is a whole number greater than 1 that

has exactly two factors, 1 and itself.

Composite numbers are whole numbers that have more

than two factors.The numbers 0 and 1 are neither prime

nor composite.

Prime Factorization

The prime factorization of a number is the factorization

in which every factor is a prime number.

A factor of a number n is any number that divides

evenly into n. For example, the factors of 80 are 1, 2, 4, 5,

8, 10, 16, 20, 40, and 80.

Examples

Example 1

and are two factorizations of 16.

Example 2

15 is divisible by 5 because 5 divides into 15 evenly.

Example 3

382 is divisible by 2.

640 is divisible by 2, 5, and 10.

735 is divisible by 3 and 5.

Example 4

9 is a composite number.

2 is a prime number.

1 is neither prime nor composite.

Example 5

The prime factorization of 756 is

2 � 2 � 3 � 3 � 3 � 7 or 22 � 33 � 7

2)756

2)378

3)189

3)63

3)21

7

8 � 24 � 4

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Simplifying Fractions to Lowest TermsSection 2.3

Key Concepts

Equivalent fractions are fractions that represent the

same portion of a whole unit.

To determine if two fractions are equivalent, calculate

the cross products. If the cross products are equal, then

the fractions are equivalent.

To simplify fractions to lowest terms, use the funda-

mental principle of fractions:

Consider the fraction and the nonzero number c.

Then

To simplify fractions with common powers of 10,

“strike through” the common zeros first.

a � cb � c

�ab

�cc

�ab

� 1 �ab

ab

Examples

Example 1

Example 2

a. Compare b. Compare

Fractions are Fractions are

not equivalent equivalent

Example 3

Example 4

3, 0 0 0

12, 0 0 0

�3

12�

31

124

�1

4

25

15�

5 � 5

3 � 5�

5

3�

5

5�

5

3� 1 �

5

3

40 � 4020 � 18

8

10�4

5

6

4�5

3

4

5 and

8

10.

5

3 and

6

4.

�

�14

28

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Summary 155

Multiplication of Fractions and ApplicationsSection 2.4

Key Concepts

Multiplication of Fractions

To multiply fractions, write the product of the numera-

tors over the product of the denominators. Then sim-

plify the resulting fraction, if possible.

When multiplying a whole number and a fraction, first

write the whole number as a fraction by writing the

whole number over 1.

Powers of one-tenth can be expressed as a fraction with

1 in the numerator. The denominator has a 1 followed

by the same number of zeros as the exponent of the

base of

The formula for the area of a triangle is given by

Area is expressed in square units such as ft2, in.2, yd2,

m2, and cm2.

b

h

A � 12bh.

110.

Examples

Example 1

Example 2

Example 3

Example 4

The area of the triangle is

The area is 4 m2.

� 4

�71

41

�16

4

71

� a1

2�

7

2b �

16

7

A � 12bh

a1

10b

4

�1

10,000

8 � a5

6b �

84

1�

5

63

�20

3

a15

16b a

4

5b �

153

164

�41

51

�3

4

4

7�

6

5�

24

35

m72

m167

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Division of Fractions and ApplicationsSection 2.5

Key Concepts

The reciprocal of is for The product of

a fraction and its reciprocal is 1. For example,

.

Dividing Fractions

To divide two fractions, multiply the dividend (the

“first” fraction) by the reciprocal of the divisor (the

“second” fraction).

When dividing by a whole number, first write the

whole number as a fraction by writing the whole num-

ber over 1. Then multiply by its reciprocal.

When simplifying expressions with more than one

operation, follow the order of operations.

6

11�

11

6� 1

a, b � 0.ba

ab

Examples

Example 1

The reciprocal of is

The reciprocal of 4 is

The number 0 does not have a reciprocal because is

undefined.

Example 2

Example 3

Example 4

�5

427

�12

2

1�

10

7

�5

42�

1

12

5

42� c a

1

6b

2

� 3 d �5

42� c

1

3612

�31

1d

9

8� 4 �

9

8�

4

1�

9

8�

1

4�

9

32

18

25�

30

35�

183

255

�35

7

305

�21

25

10

14.

85.

58

Multiplication and Division of Mixed NumbersSection 2.6

Key Concepts

Multiply Mixed Numbers

Step 1 Change each mixed number to an improper

fraction.

Step 2 Multiply the improper fractions and reduce to

lowest terms, if possible.

Divide Mixed Numbers

Step 1 Change each mixed number to an improper

fraction.

Step 2 Divide the improper fractions and reduce to

lowest terms, if possible. Recall that to divide

fractions, multiply the dividend by the reciprocal

of the divisor.

Examples

Example 1

Example 2

62

3� 2

7

9�

20

3�

25

9�

204

31

�93

255

�12

5� 2

2

5

44

5� 2

1

2�

2412

51

�51

21

�12

1� 12

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Review Exercises 157

Section 2.1For Exercises 1–2, write a fraction that represents the

shaded area.

1. 2.

3. a. Write a fraction that has denominator 3 and

numerator 5.

b. Label this fraction as proper or improper.

4. a. Write a fraction that has numerator 1 and

denominator 6.

b. Label this fraction as proper or improper.

5. In an office supply store, 7 of the 15 computers

displayed are laptops. Write a fraction represent-

ing the computers that are laptops.

For Exercises 6–7, write a fraction and a mixed

number that represent the shaded area.

6. 7.

For Exercises 8–9, convert the mixed number to a

fraction.

8. 9.

10. How many fourths are in ?

For Exercises 11–12, convert the improper fraction to

a mixed number.

11. 12.23

21

47

9

414

112

56

1

7

10

For Exercises 13–15, locate the numbers on the

number line.

13. 14. 15.

For Exercises 16–17, divide. Write the answer as a

mixed number.

16. 17.

Section 2.2For Exercises 18–20, refer to this list of numbers:

21, 43, 51, 55, 58, 124, 140, 260, 1200.

18. List all the numbers that are divisible by 3.



For Exercises 21–24, determine if the number is

prime, composite, or neither.

21. 61 22. 44

23. 1 24. 0

For Exercises 25–27, find the prime factorization.

25. 64 26. 330

27. 900

For Exercises 28–29, list all the factors of the number.

28. 48 29. 80

Section 2.3For Exercises 30–31, determine if the fractions are

equivalent. Fill in the blank with or .

30. 31.10

14

15

21

5

9

3

6

Z�

26�15827�941

13

8

7

8

10

5

Chapter 2 Review Exercises

43210

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For Exercises 32–39, simplify the fraction to lowest

terms. Write the answer as a fraction.

32. 33. 34.

35. 36. 37.

38. 39.

40. On his final exam, Gareth got 42 out of 45

questions correct. What fraction of the test

represents correct answers? What fraction

represents incorrect answers?

41. Isaac proofread 6 pages of his 10-page term paper.

Yulisa proofread 6 pages of her 15-page term

paper.

a. What fraction of his paper did Isaac proofread?

b. What fraction of her paper did Yulisa

proofread?

Section 2.4For Exercises 42–47, multiply the fractions and

simplify to lowest terms. Write the answer as a

fraction or whole number.

42. 43. 44.

45. 46. 47.

For Exercises 48–51, evaluate by using the order of

operations.

48. 49.

50. 51.

52. Write the formula for the area of a triangle.

b

h

a1

10b

3

a1000

17ba

3

20�

2

3b

3

a2

5b

2

� a1

10b

2

a1

10b

4

45

7�

6

10�

28

63

2

9�

5

8�

36

2533 �

5

11

14 �9

2

4

3�

8

3

3

5�

2

7

1400

2000

120

150

42

21

17

17

63

27

24

16

14

49

5

20

53. Write the formula for the area of a rectangle.

For Exercises 54–56, find the area of the shaded

region.

54. 55.

56.

57. Maximus wants to build a workbench for his

garage. He needs four boards, each cut into -yd

pieces. How many yards of lumber does he

require?

For Exercises 58–61, refer to the graph. The graph

represents the distribution of the students at a college

by race/ethnicity.

58. If the college has 3600 students, how many are

African American?


Asian American?

Distribution of Student Body byRace/Ethnicity

Hispanic

Other

Asian American

112

112

512

14

16

African Am

casian

78

3 yd

yd203

6 yd

m54

m83

12 ft

ft172

w

l

BASIC—

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ft43

ft43

Review Exercises 159


Hispanic females (assume that one-half of the

Hispanic student population is female)?


Caucasian males (assume that one-half of the

Caucasian student population is male)?

Section 2.5For Exercises 62–63, multiply.

62. 63.

For Exercises 64–67, find the reciprocal of the

number, if it exists.

64. 65. 7

66. 0 67.

68. Dividing by 5 is the same as multiplying

by ___________.

69. Dividing by is the same as by .

For Exercises 70–75, divide and simplify the answer to

lowest terms. Write the answer as a fraction or whole

number.

70. 71.

72. 73.

74. 75.

For Exercises 76–79, simplify by using the order of

operations. Write the answer as a fraction.

76. 77.

78. 79.4

13� a

1

2b

3

� 281

55�

3

11�

3

2

a12

5b

2

�36

5a

2

19�

8

19b

3

12 �6

7

200

51�

25

17

3

10�

9

5

6

7� 18

7

9�

35

63

28

15�

21

20

92

29

1

6

7

2

1

12� 12

3

4�

4

3

For Exercises 80–81, translate to a mathematical

statement. Then simplify.

80. How much is of 20?

81. How many ’s are in 18?

82. How many -lb bags of candy can be filled from

a 24-lb sack of candy?

83. Amelia worked only of her normal 40-hr

workweek. If she makes $18 per hour, how much

money did she earn for the week?

84. A small patio floor will be made from square

pieces of tile that are ft on a side. See the

figure. Find the area of the patio (in square feet)

if its dimensions are 10 tiles by 12 tiles.

43

45

23

23

45

85. Chuck is an elementary school teacher and

needs 22 pieces of wood, ft long, for a class

project. If he has a 9-ft board from which to cut

the pieces, will he have enough -ft pieces for his

class? Explain.

Section 2.6For Exercises 86–96, multiply or divide as indicated.

86. 87.

88. 89.

90. 91. 45

16� 2

7

845

5

13� 0

4 � a55

8b6

1

2� 1

3

13

a111

3b a2

3

34ba3

2

3b a6

2

5b

38

38

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92. 93.

94. 95.

96. 0 � 35

12

10

1

5� 174

6

11� 2

7 � 15

93

5

11� 3

4

5

97. It takes 1 gal of paint for Neva to paint her

living room. If her great room (including the

dining area) is 2 times larger than the living

room, how many gallons will it take to paint that

room?

98. A roll of ribbon contains 12 yd. How many

pieces of length 1 yd can be cut from this roll?14

12

12

14

1. a. Write a fraction that represents the shaded

portion of the figure.

b. Is the fraction proper or improper?

2. a. Write a fraction that represents the total

shaded portion of the three figures.

b. Is the fraction proper or improper?

3. Write an improper fraction and a mixed number

that represent the shaded region.

4. Is the fraction a proper or an improper

fraction? Explain.

5. a. Write as a mixed number.

b. Write as an improper fraction.379

113

77

For Exercises 6–9, plot the fraction on the number line.

6.

7.

8.

9.

10. Label the following numbers as prime,

composite, or neither.

a. 15 b. 0

c. 53 d. 1

e. 29 f. 39

11. a. List all the factors of 45.

b. Write the prime factorization of 45.

12. a. What is the divisibility rule for 3?

b. Is 1,981,011 divisible by 3?

13. Determine whether 1155 is divisible by

a. 2 b. 3

c. 5 d. 10

13

5

7

12

3

4

1

2

Chapter 2 Test

43210

10

10

10

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Cumulative Review Exercises 161

For Exercises 14–15, determine if the fractions are

equivalent. Then fill in the blank with either = or .

14. 15.

For Exercises 16–17, simplify the fractions to lowest

terms.

16. 17.

18. Christine and Brad are putting their photographs

in scrapbooks. Christine has placed 15 of her 25

photos and Brad has placed 16 of his 20 photos.

a. What fractional part of the total photos has

each person placed?

b. Which person has a greater fractional part

completed?

For Exercises 19–26, multiply or divide as indicated.

Simplify the fraction to lowest terms.

19. 20.

21. 22.

23. 24.600

1200�

50

65�

13

15

2

18�

9

25�

40

6

105

42� 5

28

24�

21

8

a75

24b � 4

2

9�

57

46

1,200,000

1,400,000

150

105

4

25

2

5

5

4

15

12

Z 25. 26.

27. Perform the order of operations. Simplify the

fraction to lowest terms.


29. Which is greater, or ?

30. How many “quarter-pounders” can be made

from 12 lb of ground beef?

31. The Humane Society has 120 dogs. Of the 120,

are female. Among the female dogs, are pure

breeds. How many of the dogs are female and

pure breeds?

32. A zoning requirement indicates that a house

built on less than 1 acre of land may take up no

more than one-half of the land. If Liz and

George purchased a -acre lot of land, what is the

maximum land area that they can use to build

the house?

45

115

58

20 � 1420 � 1

4

8 cm

cm113

52

72� c a

1

2b

2

�8

3d

44

17� 2

4

15

10

21� 4

1

6

1. Fill in the table with either the word name for

the number or the number in standard form.

For Exercises 2–13, perform the indicated operation.

2. 432 � 998 3. 572�433

4. 4122 � 52 5. 384 � 16

6. 23(81) 7.

8. 9.

10. 11. 6 � 2 8

12. 52 � 32 13. (5 � 3)2

�48

8

1007

� 823

3,000,000

� 40,000

4�74

Chapters 1–2 Cumulative Review Exercises

Height (ft)

StandardMountain Form Words

Mt. Foraker Seventeen thousand, four

(Alaska) hundred

Mt. Kilimanjaro

(Tanzania) 19,340

El Libertador Twenty-two thousand,

(Argentina) forty-seven

Mont Blanc

(France-Italy) 15,771

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For Exercises 14–18, match the algebraic statement with

the property that it demonstrates.

14. a. Commutative property

of addition

15. b. Associative property

of addition

16. (12 � 3) � 5 c. Distributive property

� 12 � (3 � 5) of multiplication over

addition

17. d. Commutative property

of multiplication

18. 32 � 9 e. Associative property

� 9 � 32 of multiplication

19. Write a fraction that represents the shaded area.

a.

b.

20. Identify each fraction as proper or improper.

a. b. c.

21. a. List all the factors of 30.

b. Write the prime factorization of 30.

8

8

8

7

7

8

� 18 � 72 � 2

8 � 17 � 22

� 4 � 3 � 4 � 2

413 � 22

5 � 8 � 8 � 5

22. Simplify the fractions to lowest terms.

a. b.

23. Multiply and simplify to lowest terms.

24. Divide and simplify to lowest terms.

25. Is multiplication of fractions a commutative

operation? Explain, using the fractions and .

26. Is multiplication of fractions an associative

operation? Explain, using the fractions , , and

27. Simplify.

28. Find the area of the rectangle.


30. At one college of the students are male, and of

the males, are from out of state. What fraction

of the students are males who are from out of

state?

110

34

8 ft

ft252

m115

m59

a5

6�

12

25b

2

�2

3

53.

29

12

516

813

523 � 64

5

35

27�

51

95

60,000

150,000

144

84

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Fractions and Mixed Numbers: Multiplication and Division