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