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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

pillows can be made?



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

receive?

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

in form, check your answers for Problems 1–12 and

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)



Answers to the Concept Quiz

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



Addition

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

Add ϩ .

4

5



EXAMPLE 1



Add



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



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