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1: Rational Numbers: Multiplication and Division

# 1: Rational Numbers: Multiplication and Division

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2.1 • Rational Numbers: Multiplication and Division

43

Multiplying Rational Numbers

We define multiplication of rational numbers in common fractional form as follows:

Deﬁnition 2.1

If a, b, c, and d are integers, and b and d are not equal to zero, then

a

b

#

c

a

ϭ

d

b

#c

#d

To multiply rational numbers in common fractional form we simply multiply numerators

and multiply denominators. Furthermore, we see from the definition that the rational numbers

are commutative and associative with respect to multiplication. We are free to rearrange and

regroup factors as we do with integers. The following examples illustrate Definition 2.1:

1 #

3

3 #

4

Ϫ2

3

1 #

5

2

1 # 2

2

ϭ # ϭ

5

3 5

15

5

3 # 5

15

ϭ # ϭ

7

4 7

28

# 7 ϭ Ϫ2# # 7 ϭ Ϫ14

9

3 9

27

9

1 # 9

9

ϭ

ϭ

Ϫ11

51Ϫ112

Ϫ55

7

Ϫ3

ϭ

13

4

Ϫ

3

4

#

3

5

#

5

3

ϭ

3

5

#

or

or

14

27

9

Ϫ

55

Ϫ

7

Ϫ3 # 7

Ϫ21

ϭ #

ϭ

13

4 13

52

or

Ϫ

21

52

# 5 15

# 3 ϭ 15 ϭ 1

The last example is a very special case. If the product of two numbers is 1, the numbers are

said to be reciprocals of each other.

a

Using Definition 2.1 and applying the multiplication property of one, the fraction

b

where b and k are nonzero integers, simplifies as shown.

a

b

#k a#

#kϭb

k

a

ϭ

k

b

#k

# k,

# 1ϭa

b

This result is stated as Property 2.2.

Property 2.2 The Fundamental Principle of Fractions

If b and k are nonzero integers, and a is any integer, then

a

b

#k a

#kϭb

We often use Property 2.2 when we work with rational numbers. It is called the fundamental

principle of fractions and provides the basis for equivalent fractions. In the following examples, the property will be used for what is often called “reducing fractions to lowest terms”

or “expressing fractions in simplest or reduced form.”

Classroom Example

21

Reduce

to lowest terms.

35

EXAMPLE 1

Solution

12

2и6

2

ϭ

ϭ

18

3и6

3

Reduce

12

to lowest terms.

18

44

Chapter 2 • Real Numbers

Classroom Example

12

Change

to simplest form.

21

EXAMPLE 2

#7 2

#7ϭ5

EXAMPLE 3

A common factor of 7 has been divided out of both numerator and

denominator

Express

Ϫ24

in reduced form.

32

Solution

Ϫ24

3

ϭϪ

32

4

Classroom Example

63

Reduce Ϫ

.

105

14

to simplest form.

35

Solution

14

2

ϭ

5

35

Classroom Example

Ϫ18

Express

in reduced form.

42

Change

#8

3

# 8 ϭ Ϫ4 #

EXAMPLE 4

8

3

ϭϪ

8

4

Reduce Ϫ

# 1 ϭ Ϫ3

4

The multiplication property of 1 is

being used

72

.

90

Solution

72

2 # 2 # 2 # 3 # 3

4

Ϫ ϭϪ

ϭϪ

90

2 # 3 # 3 # 5

5

The prime factored forms of the numerator and

denominator may be used to help recognize common factors

The fractions may contain variables in the numerator or the denominator (or both), but

this creates no great difficulty. Our thought processes remain the same, as these next examples illustrate. Variables appearing in the denominators represent nonzero integers.

Classroom Example

8a

Reduce

.

15a

EXAMPLE 5

Reduce

9x

.

17x

Solution

9x

9 # x

9

ϭ

ϭ

17x

17 # x

17

Classroom Example

9c

Simplify

.

42d

EXAMPLE 6

Simplify

8x

.

36y

Solution

8x

2 # 2 # 2 # x

2x

ϭ # # # # ϭ

36y

2 2 3 3 y

9y

Classroom Example

Ϫ6ab

Express

in reduced form.

39b

EXAMPLE 7

Express

Ϫ9xy

in reduced form.

30y

Solution

Ϫ9xy

9xy

3и3иxиy

3x

ϭϪ

ϭϪ

ϭϪ

2и3и5иy

10

30y

30y

2.1 • Rational Numbers: Multiplication and Division

Classroom Example

Ϫ3xyz

Reduce

.

Ϫ8yz

EXAMPLE 8

Reduce

45

Ϫ7abc

.

Ϫ9ac

Solution

Ϫ7abc

7abc

7abc

7b

ϭ

ϭ

ϭ

Ϫ9ac

9ac

9ac

9

We are now ready to consider multiplication problems with the agreement that the final

answer should be expressed in reduced form. Study the following examples carefully; we use

different methods to handle the various problems.

Classroom Example

3 5

Multiply # .

8 9

EXAMPLE 9

5

7 # 5

ϭ #

ϭ

14

9 14

3

#

EXAMPLE 10

5

.

14

#

7 # 5

5

# 3 # 2 # 7 ϭ 18

Find the product of

1

2

18

2

ϭ

24

3

#

1

3

A common factor of 8 has been divided out of 8 and 24, and a common factor

of 9 has been divided out of 9 and 18

6 14

Multiply aϪ ba b.

8 32

EXAMPLE 11

Solution

3

7

и 14 ϭ Ϫ 21

64

и 32

4

16

6 14

6

aϪ ba b ϭ Ϫ

8 32

8

Classroom Example

10

12

Multiply aϪ baϪ b.

3

18

Divide a common factor of 2 out of 6 and 8, and a

common factor of 2 out of 14 and 32

9

14

Multiply aϪ baϪ b .

4

15

EXAMPLE 12

Solution

9

14

3и3

aϪ b aϪ b ϭ

4

15

2и2

Classroom Example

6m

15n

Multiply a

ba

b.

5n

26

8

18

and .

9

24

Solution

8

9

Classroom Example

4 33

Multiply aϪ ba b.

9 40

7

9

Solution

7

9

Classroom Example

21

6

Find the product of and .

7

30

Multiply

EXAMPLE 13

и 2 и 7 ϭ 21

и 3 и 5 10

Multiply

9x

7y

Solution

2

9x

7y

#

14y

9 # x # 14 # y

2x

ϭ

ϭ

7 # y # 45

5

45

5

#

Immediately we recognize that a negative times a

negative is positive

14y

.

45

46

Chapter 2 • Real Numbers

Classroom Example

Ϫ3z 16x

Multiply

и 9z .

8xy

EXAMPLE 14

Multiply

Ϫ6c

7ab

#

14b

.

5c

Solution

Ϫ6c

7ab

#

12

14b

2 # 3 # c # 2 # 7 # b

ϭϪ

ϭϪ

7 # a # b # 5 # c

5a

5c

Dividing Rational Numbers

The following example motivates a definition for division of rational numbers in fractional

form.

3

3

3

3

3

a ba b

4

2

4

4

2

3

3

9

ϭ ± ≤± ≤ ϭ

ϭ a ba b ϭ

2

2

3

1

4

2

8

3

3

2

c

3

2

Notice that this is a form of 1, and is the reciprocal of

2

3

3

2

3

3

In other words, divided by is equivalent to times . The following definition for divi4

3

4

2

sion should seem reasonable:

Deﬁnition 2.2

If b, c, and d are nonzero integers and a is any integer, then

c

a

a

Ϭ ϭ

b

d

b

d

c

#

a

c

a

c

d

by , we multiply times the reciprocal of , which is . The

c

b

d

b

d

Notice that to divide

following examples demonstrate the important steps of a division problem.

2

1

2

Ϭ ϭ

3

2

3

#

2

4

ϭ

1

3

5

3

5

Ϭ ϭ

6

4

6

#

4

5

ϭ

3

6

3

3

9

9

Ϫ Ϭ ϭϪ

12

6

12

# 4 5 # 2 # 2 10

#3ϭ2#3#3ϭ 9

1

и

2

6

3

ϭϪ

3

2

1

9

9

7

11

27

33

27

72

27 и 72

81

aϪ b Ϭ aϪ b ϭ aϪ baϪ b ϭ

ϭ

56

72

56

33

56 и 33

77

3

6

6

Ϭ2ϭ

7

7

#

1

6

ϭ

2

7

#

1

3

ϭ

2

7

1

2

4

10

5x

5x

Ϭ

ϭ

7y

28y

7y

#

28y

5 # x # 28 # y

ϭ

ϭ 2x

10

7 # y # 10

2

2.1 • Rational Numbers: Multiplication and Division

47

EXAMPLE 15

Classroom Example

Lynn purchased 24 yards of fabric

3

for her sewing class. If of a yard is

4

needed for each pillow, how many

2

Frank has purchased 50 candy bars to make s’mores for the Boy Scout troop. If he uses of a

3

candy bar for each s’more, how many s’mores will he be able to make?

Solution

To find how many s’mores can be made, we need to divide 50 by

2.

3

25

2

50 Ϭ ϭ 50

3

#

3

50

ϭ

2

1

#

3

50

ϭ

2

1

#

3

75

ϭ

ϭ 75

2

1

1

Frank can make 75 s’mores.

Concept Quiz 2.1

For Problems 1–10, answer true or false.

1. 6 is a rational number.

1

2. is a rational number.

8

Ϫ2

2

ϭ

3.

Ϫ3

3

Ϫ5

5

ϭ

4.

3

Ϫ3

5. The product of a negative rational number and a positive rational number is a positive

rational number.

6. If the product of two rational numbers is 1, the numbers are said to be reciprocals.

Ϫ3 7

7. The reciprocal of

is .

7

3

10

8.

is reduced to lowest terms.

25

4ab

9.

is reduced to lowest terms.

7c

p

q

m

m

10. To divide by , we multiply by .

n

q

n

p

Problem Set 2.1

For Problems 1–24, reduce each fraction to lowest terms.

(Objective 1)

5.

15

9

6.

48

36

1.

8

12

2.

12

16

7.

Ϫ8

48

8.

Ϫ3

15

3.

16

24

4.

18

32

9.

27

Ϫ36

10.

9

Ϫ51

48

Chapter 2 • Real Numbers

11.

Ϫ54

Ϫ56

12.

Ϫ24

Ϫ80

49. aϪ

24y

7x

baϪ

b

12y

35x

13.

24x

44x

14.

15y

25y

50. aϪ

10a

45b

b aϪ

b

15b

65a

15.

9x

21y

16.

4y

30x

51.

6

3

Ϭ

x

y

52.

14

6

Ϭ

x

y

17.

14xy

35y

18.

55xy

77x

53.

5x

13x

Ϭ

9y

36y

54.

3x

7x

Ϭ

5y

10y

19.

Ϫ20ab

52bc

20.

Ϫ23ac

41c

55.

Ϫ7

9

Ϭ

x

x

56.

8

28

Ϭ

y

Ϫy

Ϫ21xy

22.

Ϫ14ab

57.

Ϫ4

Ϫ18

Ϭ

n

n

58.

Ϫ34

Ϫ51

Ϭ

n

n

Ϫ56yz

21.

Ϫ49xy

23.

65abc

91ac

24.

68xyz

85yz

For Problems 59–74, perform the operations as indicated,

and express answers in lowest terms. (Objective 2)

For Problems 25–58, multiply or divide as indicated, and

express answers in reduced form. (Objective 2)

59.

3

4

#8#

9

12

20

4

5

2

3

Ϭ

7

5

28.

5

11

Ϭ

6

13

7

5

18

62. aϪ b a b aϪ b

9

11

14

29.

3

8

30.

4

9

#

3

2

63. a

12y

3x

8

ba ba

b

4y

9x

5

31.

Ϫ6

13

32.

3

4

#

Ϫ14

12

64. a

5y

2x

9

ba ba b

x

3y

4x

33.

5

7

Ϭ

9

9

34.

7

3

Ϭ

11

11

2

3

1

65. aϪ b a b Ϭ

3

4

8

35.

1

Ϫ5

Ϭ

4

6

36.

14

7

Ϭ

8

Ϫ16

67.

3

4

27.

5

7

#

12

15

#

37. aϪ

26

9

#

8

10

baϪ b

10

32

39. Ϫ9 Ϭ

7y

3x

41.

5x

9y

43.

6a

14b

45.

10x

Ϫ9y

#

#

2

b

47. ab

1

3

#

#

3

11

#

5

6

#

9

10

66.

3

4

#

4

1

Ϭ

5

6

6

21

38. aϪ baϪ b

7

24

40. Ϫ10 Ϭ

1

4

6b

7a

3

4

1

68. aϪ b Ϭ aϪ b a b

8

5

2

6

5

5

69. aϪ b Ϭ a b aϪ b

7

7

6

4

4

3

70. aϪ b Ϭ a b a b

3

5

5

4a

11b

#

16b

18a

44.

5y

8x

#

14z

15y

72. aϪ b a b Ϭ aϪ b

15

20x

46.

3x

4y

#

Ϫ8w

9z

5

2

1

73. a b a b Ϭ aϪ b Ϭ 1Ϫ32

2

3

4

#

8

7

5

5

6

Ϭ aϪ b aϪ b

7

6

7

42.

48. 3xy

#

3

13

12

61. aϪ b a b aϪ b

8

14

9

26.

25.

60.

4

x

4

9

3

71. a b aϪ b Ϭ aϪ b

9

8

4

7

8

74.

4

7

3

2

1

3

1

Ϭ a ba b Ϭ 2

3

4

2

2.1 • Rational Numbers: Multiplication and Division

For Problems 75–81, solve the word problems. (Objective 3)

3

of all of the accounts

4

within the ABC Advertising Agency. Maria is per1

sonally responsible for

of all accounts in her

3

department. For what portion of all of the accounts at

75. Maria’s department has

ABC is Maria personally responsible?

1

feet long, and he wants

2

to cut it into three pieces of the same length (see

Figure 2.1). Find the length of each of the three

pieces.

76. Pablo has a board that is 4

4

1

ft

2

49

3

cup of sugar.

4

How much sugar is needed to make 3 cakes?

77. A recipe for a birthday cake calls for

78. Jonas left an estate valued at \$750,000. His will states

that three-fourths of the estate is to be divided equally

among his three children. How much should each

1

cups of milk. If

2

she wants to make one-half of the recipe, how much

milk should she use?

79. One of Arlene’s recipes calls for 3

2

80. The total length of the four sides of a square is 8 yards.

3

How long is each side of the square?

1

81. If it takes 3 yards of material to make one dress, how

4

much material is needed for 20 dresses?

82. If your calculator is equipped to handle rational numbers

a

b

59–74.

Figure 2.1

Thoughts Into Words

83. State in your own words the property

Ϫ

a

Ϫa

a

ϭ

ϭ

b

b

Ϫb

84. Explain how you would reduce

72

to lowest terms.

117

85. What mistake was made in the following simplification

process?

1

2

3

1

1

1

Ϭ a ba b Ϭ 3 ϭ Ϭ Ϭ 3 ϭ

2

3

4

2

2

2

#2#

1

1

ϭ

3

3

How would you correct the error?

Further Investigations

86. The division problem 35 Ϭ 7 can be interpreted as

“how many 7s are there in 35?” Likewise, a division

1

problem such as 3 Ϭ can be interpreted as, “how

2

many one-halves in 3?” Use this how-many interpretation to do the following division problems.

(a) 4 Ϭ

1

2

(b) 3 Ϭ

1

4

1

(c) 5 Ϭ

8

1

(d) 6 Ϭ

7

5

1

(e)

Ϭ

6

6

7

1

(f)

Ϭ

8

8

87. Estimation is important in mathematics. In each

of the following, estimate whether the answer is larger

than 1 or smaller than 1 by using the how-many idea

from Problem 86.

(a)

3

1

Ϭ

4

2

(b) 1 Ϭ

7

8

(c)

1

3

Ϭ

2

4

(d)

8

7

Ϭ

7

8

(e)

2

1

Ϭ

3

4

(f)

3

3

Ϭ

5

4

88. Reduce each of the following to lowest terms. Don’t forget that we reviewed some divisibility rules in Problem

Set 1.2.

(a)

99

117

(b)

175

225

50

Chapter 2 • Real Numbers

(c)

Ϫ111

123

(d)

Ϫ234

270

(e)

270

495

(f)

324

459

(g)

1. True

2. True

3. True

4. True

9. True

10. True

2.2

5. False

6. True

91

143

7. False

(h)

187

221

8. False

Addition and Subtraction of Rational Numbers

OBJECTIVES

1

Add and subtract rational numbers in fractional form

2

Combine similar terms whose coefﬁcients are rational numbers in fractional form

3

Solve application problems that involve the addition and subtraction of rational

numbers in fractional form

Suppose that it is one-fifth of a mile between your dorm and the student center, and twofifths of a mile between the student center and the library along a straight line as indicated

in Figure 2.2. The total distance between your dorm and the library is three-fifths of a mile,

1

2

3

and we write ϩ ϭ .

5

5

5

1

mile

5

Dorm

2

mile

5

Student Center

Library

Figure 2.2

A pizza is cut into seven equal pieces and you eat two of the pieces. How

7

much of the pizza (Figure 2.3) remains? We represent the whole pizza by and then con7

7

2

5

clude that Ϫ ϭ of the pizza remains.

7

7

7

Figure 2.3

These examples motivate the following definition for addition and subtraction of rational

numbers in

a

form:

b

2.2 • Addition and Subtraction of Rational Numbers

51

Deﬁnition 2.3

If a, b, and c are integers, and b is not zero, then

c

aϩc

a

ϩ ϭ

b

b

b

a

c

aϪc

Ϫ ϭ

b

b

b

Subtraction

We say that rational numbers with common denominators can be added or subtracted by adding

or subtracting the numerators and placing the results over the common denominator. Consider

the following examples:

3

2

3 ϩ 2

5

ϩ

ϭ

ϭ

7

7

7

7

7

2

7 Ϫ 2

5

Ϫ ϭ

ϭ

8

8

8

8

2

1

2 ϩ 1

3

1

ϩ

ϭ

ϭ

ϭ

6

6

6

6

2

We agree to reduce the final answer

3

5

3Ϫ5

Ϫ2

2

Ϫ

ϭ

ϭ

or    Ϫ

11

11

11

11

11

5

7

5 ϩ 7

12

ϩ

ϭ

ϭ

x

x

x

x

9

3

9 Ϫ 3

6

Ϫ ϭ

ϭ

y

y

y

y

In the last two examples, the variables x and y cannot be equal to zero in order to exclude

division by zero. It is always necessary to restrict denominators to nonzero values, although we

will not take the time or space to list such restrictions for every problem.

How do we add or subtract if the fractions do not have a common denominator?

We use the fundamental principle of fractions,

a

aиk

, and obtain equivalent fractions

ϭ

b

bиk

that have a common denominator. Equivalent fractions are fractions that name the same

number. Consider the following example, which shows the details.

Classroom Example

1

1

4

5

EXAMPLE 1

1

1

ϩ .

2

3

Solution

1

1и3

3

ϭ

ϭ

2

2и3

6

1и2

2

1

ϭ

ϭ

3

3и2

6

3

1

and are equivalent fractions naming the same number

2

6

1

2

and are equivalent fractions naming the same number

2

6

1

1

3

2

3 ϩ 2

5

ϩ

ϭ ϩ

ϭ

ϭ

2

3

6

6

6

6

Notice that we chose 6 as the common denominator, and 6 is the least common multiple

of the original denominators 2 and 3. (Recall that the least common multiple is the smallest ### Tài liệu bạn tìm kiếm đã sẵn sàng tải về

1: Rational Numbers: Multiplication and Division

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