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1.4
Multiplying and Dividing
Signed Numbers
1.4
OBJECTIVES
1. Find the product of two signed numbers
2. Find the quotient of two signed numbers
When you first considered multiplication in arithmetic, it was thought of as repeated
addition. Let’s see what our work with the addition of signed numbers can tell us about
multiplication when signed numbers are involved. For example,
3 4 4 4 4 12
We interpret multiplication as repeated
addition to find the product, 12.
Now, consider the product (3)(4):
(3)(4) (4) (4) (4) 12
Looking at this product suggests the first portion of our rule for multiplying signed
numbers. The product of a positive number and a negative number is negative.
Rules and Properties: Multiplying Signed Numbers Case 1:
Different Signs
The product of two numbers with different signs is negative.
To use this rule in multiplying two numbers with different signs, multiply their absolute
values and attach a negative sign.
Example 1
Multiplying Signed Numbers
Multiply.
(a) (5)(6) 30
The product is negative.
(b) (10)(10) 100
(c) (8)(12) 96
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NOTE Multiply together
numerators and then
denominators and reduce.
45 10
(d) 3
2
3
CHECK YOURSELF 1
Multiply.
(a) (7)(5)
(b) (12)(9)
(c) (15)(8)
7 5 (d) 4
14
89
90
CHAPTER 1
THE LANGUAGE OF ALGEBRA
The product of two negative numbers is harder to visualize. The following pattern may
help you see how we can determine the sign of the product.
(3)(2) 6
(2)(2) 4
NOTE This number is
(1)(2) 2
decreasing by 1.
Do you see that the product is
increasing by 2 each time?
(0)(2) 0
(1)(2) 2
NOTE (1)(2) is the opposite
of 2.
What should the product (2)(2) be? Continuing the pattern shown, we see that
(2)(2) 4
This suggests that the product of two negative numbers is positive. That is the case. We can
extend our multiplication rule.
NOTE If you would like a more
detailed explanation, see the
discussion at the end of this
section.
Rules and Properties: Multiplying Signed Numbers Case 2:
Same Sign
The product of two numbers with the same sign is positive.
Example 2
Multiplying Signed Numbers
Multiply.
(a) 9 7 63
The product of two positive numbers
(same sign, ) is positive.
(b) (8)(5) 40
The product of two negative numbers
(same sign, ) is positive.
(c)
23 6
1
1
1
CHECK YOURSELF 2
Multiply.
(b) (8)(9)
37
(c) 2
Two numbers, 0 and 1, have special properties in multiplication.
Rules and Properties: Multiplicative Identity Property
NOTE The number 1 is called
The product of 1 and any number is that number. In symbols,
the multiplicative identity for
this reason.
a11aa
6
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(a) 10 12
MULTIPLYING AND DIVIDING SIGNED NUMBERS
SECTION 1.4
91
Rules and Properties: Multiplicative Property of Zero
The product of 0 and any number is 0. In symbols,
a00a0
Example 3
Multiplying Signed Numbers
Find each product.
(1)(7) 7
(15)(1) 15
(7)(0) 0
0 12 0
4
(e) (0) 0
5
(a)
(b)
(c)
(d)
CHECK YOURSELF 3
Multiply.
(a) (10)(1)
(b) (0)(17)
(c)
7(1)
5
(d) (0)
4
3
Before we continue, consider the following equivalent fractions:
1
1
1
a
a
a
Any of these forms can occur in the course of simplifying an expression. The first form is
generally preferred.
To complete our discussion of the properties of multiplication, we state the following.
Rules and Properties: Multiplicative Inverse Property
1
is called the
a
multiplicative inverse, or the
reciprocal, of a. The product of
any nonzero number and its
reciprocal is 1.
NOTE
For any number a, where a 0, there is a number
a
1
1
a
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Example 4 illustrates this property.
Example 4
Multiplying Signed Numbers
1
1
3
1
(b) 5 1
5
2 3
(c) 1
3 2
(a) 3 1
The reciprocal of 3 is .
3
The reciprocal of 5 is
The reciprocal of
1
1
or .
5
5
2 1
3
is , or .
3 23
2
1
such that
a
CHAPTER 1
THE LANGUAGE OF ALGEBRA
CHECK YOURSELF 4
Find the multiplicative inverse (or the reciprocal) of each of the following numbers.
(a) 6
(b) 4
(c)
1
4
(d) 3
5
You know from your work in arithmetic that multiplication and division are related operations. We can use that fact, and our work of the last section, to determine rules for the
division of signed numbers. Every division problem can be stated as an equivalent
multiplication problem. For instance,
15
3
5
because
15 5 3
24
4
6
because
24 (6)(4)
30
6
5
because
30 (5)(6)
The examples above illustrate that because the two operations are related, the rule of
signs that we stated in the last section for multiplication is also true for division.
Rules and Properties: Dividing Signed Numbers
1. The quotient of two numbers with different signs is negative.
2. The quotient of two numbers with the same sign is positive.
Again, the rule is easy to use. To divide two signed numbers, divide their absolute values. Then attach the proper sign according to the rule above.
Example 5
Dividing Signed Numbers
Divide.
(a)
Positive
Positive
(b)
Negative
Negative
(c)
Negative
Positive
(d)
Positive
Negative
(e)
Positive
Negative
28
4
7
Positive
36
9
4
Positive
42
6
7
Negative
75
25
3
15.2
4
3.8
Negative
Negative
© 2001 McGraw-Hill Companies
92
MULTIPLYING AND DIVIDING SIGNED NUMBERS
SECTION 1.4
93
CHECK YOURSELF 5
Divide.
55
11
(a)
(b)
80
20
(c)
48
8
(d)
144
12
(e)
13.5
2.7
You should be very careful when 0 is involved in a division problem. Remember that 0
divided by any nonzero number is just 0. Recall that
0
0
7
0 (7)(0)
because
However, if zero is the divisor, we have a special problem. Consider
9
?
0
This means that 9 0 ?.
Can 0 times a number ever be 9? No, so there is no solution.
9
Because cannot be replaced by any number, we agree that division by 0 is not allowed.
0
We say that
Rules and Properties: Division by Zero
Division by 0 is undefined.
Example 6
Dividing Signed Numbers
© 2001 McGraw-Hill Companies
Divide, if possible.
(a)
7
is undefined.
0
(b)
9
is undefined.
0
(c)
0
0
5
(d)
0
0
8
0
is called an indeterminate form. You will learn more about this
0
in later mathematics classes.
Note: The expression
CHECK YOURSELF 6
Divide if possible.
(a)
0
3
(b)
5
0
(c)
7
0
(d)
0
9
CHAPTER 1
THE LANGUAGE OF ALGEBRA
The fraction bar serves as a grouping symbol. This means that all operations in the numerator and denominator should be performed separately. Then the division is done as the
last step. Example 7 illustrates this property.
Example 7
Dividing Signed Numbers
Evaluate each expression.
(a)
(6)(7)
42
14
3
3
Multiply in the numerator, then
divide.
(b)
3 (12)
9
3
3
3
Add in the numerator, then
divide.
(c)
4 (2)(6)
4 (12)
6 2
6 2
Multiply in the numerator. Then
add in the numerator and
subtract in the denominator.
16
2
8
Divide as the last step.
CHECK YOURSELF 7
Evaluate each expression.
(a)
4 (8)
6
(b)
3 (2)(6)
5
(c)
(2)(4) (6)(5)
(4)(11)
Evaluating fractions with a calculator poses a special problem. Example 8 illustrates this
problem.
Example 8
Using a Calculator to Divide
Use your scientific calculator to evaluate each fraction.
(a)
4
23
As you can see, the correct answer should be 4. To get this answer with your calculator,
you must place the denominator in parentheses. The key stroke sequence will be
4
( 23 ) © 2001 McGraw-Hill Companies
94
MULTIPLYING AND DIVIDING SIGNED NUMBERS
(b)
SECTION 1.4
95
7 7
3 10
In this problem, the correct answer is 2. This can be found on your calculator by placing the
numerator in parentheses and then placing the denominator in parentheses. The key stroke
sequence will be
( 7 /
7 ) ( 3 10 ) When evaluating a fraction with a calculator, it is safest to use parentheses in both the
numerator and the denominator.
CHECK YOURSELF 8
Evaluate using your calculator.
(a)
8
57
(b)
3 2
13 23
Example 9
Multiplying Signed Numbers
Evaluate each expression.
(a) 7(9 12)
Evaluate inside the parentheses first.
7(3) 21
(b) (8)(7) 40
Multiply first, then subtract.
56 40
16
(c) (5)2 3
(5)(5) 3
25 3
Evaluate the power first.
Note that (5)2 (5)(5)
25
22
(d) 52 3
Note that 52 25. The power applies only to the 5.
25 3
© 2001 McGraw-Hill Companies
28
CHECK YOURSELF 9
Evaluate each expression.
(a) 8(9 7)
(c) (4)2 (4)
(b) (3)(5) 7
(d) 42 (4)
96
CHAPTER 1
THE LANGUAGE OF ALGEBRA
Rules and Properties: The Product of Two Negative Numbers
NOTE Here is a more detailed
From our earlier work, we know that the sum of a number and its opposite is 0:
explanation of why the product
of two negative numbers is
positive.
5 (5) 0
Multiply both sides of the equation by 3:
(3)[5 (5)] (3)(0)
Because the product of 0 and any number is 0, on the right we have 0.
(3)[5 (5)] 0
We use the distributive property on the left.
(3)(5) (3)(5) 0
We know that (3)(5) 15, so the equation becomes
15 (3)(5) 0
We now have a statement of the form
15 0
in which
is the value of (3)(5). We also know that
is the number that
must be added to 15 to get 0, so
is the opposite of 15, or 15. This means
that
(3)(5) 15
The product is positive!
It doesn’t matter what numbers we use in this argument. The resulting product
of two negative numbers will always be positive.
© 2001 McGraw-Hill Companies
CHECK YOURSELF ANSWERS
8
4
1. (a) 35; (b) 108; (c) 120; (d) 2. (a) 120; (b) 72; (c)
5
7
5
5
1
1
3. (a) 10; (b) 0; (c) ; (d) 0
4. (a) ; (b) ; (c) 4; (d) 7
6
4
3
5. (a) 5; (b) 4; (c) 6; (d) 12; (e) 5
6. (a) 0; (b) undefined; (c) undefined; (d) 0
1
7. (a) 2; (b) 3; (c)
8. (a) 4; (b) 0.5
9. (a) 16; (b) 22;
2
(c) 20; (d) 12
Name
Exercises
1.4
Section
Date
Multiply.
1. 4 10
2. 3 14
ANSWERS
1.
3. (5)(12)
4. (10)(2)
2.
3.
5. (8)(9)
6. (12)(3)
4.
5.
2
7. (4) 3
1
9. (8)
4
3
8. (9) 2
6.
7.
8.
3
10. (4)
2
9.
10.
11. (3.25)(4)
12. (5.4)(5)
11.
12.
13. (8)(7)
14. (9)(8)
13.
14.
15. (5)(12)
16. (7)(3)
15.
16.
17. (9) 2
3
18. (6) 3
2
17.
18.
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19.
19. (1.25)(12)
20. (1.5)(20)
21. (0)(18)
22. (17)(0)
20.
21.
22.
23.
23. (15)(0)
24. (0)(25)
24.
11
25. (0)
12
8
26. (0)
9
25.
26.
97
ANSWERS
27.
27. (3.57)(0)
28. (2.37)(0)
28.
29.
29.
23
30.
54
31.
74
32.
98
30.
31.
3
4
2
7
4
8
5
9
32.
Divide.
33.
34.
33.
20
4
34.
70
14
35.
48
6
36.
24
8
37.
50
5
38.
32
8
39.
52
4
40.
56
7
41.
75
3
42.
60
15
43.
0
8
44.
125
25
45.
9
1
46.
10
0
47.
96
8
48.
20
2
49.
18
0
50.
0
8
51.
17
1
52.
27
1
53.
144
16
54.
150
6
55.
29.4
4.9
56.
25.9
3.7
35.
36.
37.
38.
39.
40.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
98
© 2001 McGraw-Hill Companies
41.
ANSWERS
57.
8
32
58.
6
30
57.
58.
24
59.
16
61.
28
42
25
60.
10
62.
125
75
59.
60.
61.
Perform the indicated operations.
63.
(6)(3)
2
62.
64.
(9)(5)
3
63.
64.
65.
(8)(2)
4
66.
(7)(8)
14
65.
66.
67.
69.
71.
73.
24
4 8
68.
12 12
3
70.
55 19
12 6
72.
75
22
74.
36
7 3
67.
14 4
6
69.
11 7
14 8
71.
10 6
44
73.
68.
70.
72.
74.
75.
Do the indicated operations. Remember the rules for the order of operations.
© 2001 McGraw-Hill Companies
76.
75. 5(7 2)
76. 7(8 5)
77. 2(5 8)
78. 6(14 16)
77.
78.
79.
79. 3(9 7)
80. 6(12 9)
80.
81.
81. 3(2 5)
82. 2(7 3)
83. (2)(3) 5
84. (6)(8) 27
82.
83.
84.
99
ANSWERS
85.
85. 4(7) 5
86. (3)(9) 11
87. (5)(2) 12
88. (7)(3) 25
88.
89. (3)(7) 20
90. (2)(6) 8
89.
91. 4 (3)(6)
92. 5 (2)(3)
93. 7 (4)(2)
94. 9 (2)(7)
92.
95. (7)2 17
96. (6)2 20
93.
97. (5)2 18
98. (2)2 10
86.
87.
90.
91.
94.
99. 62 4
100. 52 3
95.
101. (4)2 (2)(5)
102. (3)3 (8)(2)
97.
103. (8)2 52
104. (6)2 42
98.
105. (6)2 (3)2
106. (8)2 (4)2
107. 82 52
108. 62 32
101.
109. 82 (5)2
110. 92 (6)2
102.
111. Basketball. You score 23 points a game for 11 straight games. What is the total
96.
99.
100.
number of points that you scored?
103.
112. Gambling. In Atlantic City, Nick played the slot machines for 12 hours. He lost
104.
$45 an hour. Use signed numbers to represent the change in Nick’s financial status
at the end of the 12 hours.
105.
106.
© 2001 McGraw-Hill Companies
107.
108.
109.
110.
111.
112.
100
ANSWERS
113. Stocks. Suppose you own 35 shares of stock. If the price increases $1.25 per share,
how much money have you made?
113.
114.
114. Checking account. Your bank charges a flat service charge of $3.50 per month on
your checking account. You have had the account for 3 years. How much have you
paid in service charges?
115. Temperature. The temperature is 6F at 5:00 in the evening. If the temperature
drops 2F every hour, what is the temperature at 1:00 A.M.?
116. Dieting. A woman lost 42 pounds (lb). If she lost 3 lb each week, how long has she
115.
116.
117.
118.
been dieting?
117. Mowing lawns. Patrick worked all day mowing lawns and was paid $9 per hour. If
he had $125 at the end of a 9-hour day, how much did he have before he started
working?
119.
120.
121.
118. Unit pricing. A 4.5-lb can of food costs $8.91. What is the cost per pound?
119. Investment. Suppose that you and your two brothers bought equal shares of an
investment for a total of $20,000 and sold it later for $16,232. How much did each
person lose?
120. Temperature. Suppose that the temperature outside is dropping at a constant rate.
At noon, the temperature is 70F and it drops to 58F at 5:00 P.M. How much did the
temperature change each hour?
121. Test tube count. A chemist has 84 ounces (oz) of a solution. He pours the solution
2
oz. How many test tubes can he fill?
3
© 2001 McGraw-Hill Companies
into test tubes. Each test tube holds
101
ANSWERS
122.
Use your calculator to evaluate each expression.
7
8
122.
123.
45
4 2
123.
124.
6 9
4 1
124.
125.
125.
10 4
7 10
126. Some animal ecologists in Minnesota are planning to reintroduce a group of
animals into a wilderness area. The animals, a mammal on the endangered species
list, will be released into an area where they once prospered and where there is an
abundant food supply. But, the animals will face predators. The ecologists expect
the number of mammals to grow about 25 percent each year but that 30 of the
animals will die from attacks by predators and hunters.
The ecologists need to decide how many animals they should release to establish
a stable population. Work with other students to try several beginning populations
and follow the numbers through 8 years. Is there a number of animals that will
lead to a stable population? Write a letter to the editor of your local newspaper
explaining how to decide what number of animals to release. Include a formula for
the number of animals next year based on the number this year. Begin by filling out
this table to track the number of animals living each year after the release:
126.
a.
b.
c.
d.
e.
f.
No.
Initially
Released
Year
1
20
______
________
100
______
________
200
______
________
2
3
4
5
6
7
8
Getting Ready for Section 1.5 [Sections 1.3 and 1.4]
628
53
32 (4 1)
(d)
2 2
(a)
(b)
458
842
(e) 8 4 3 2
(c)
8 3 2
12 6
(f) 62 18 2 3
1. 40
15. 60
29. 1
43. 0
3. 60
17. 6
31. 1
45. 9
5. 72
19. 15
33. 5
47. 12
7. 6
9. 2
11. 13
13. 56
21. 0
23. 0
25. 0
27. 0
35. 8
37. 10
39. 13
41. 25
49. Undefined
51. 17
53. 9
55. 6
2
1
3
59. 61.
63. 9
65. 4
67. 2
69. 8
4
2
3
71. 2
73. Undefined
75. 25
77. 6
79. 6
81. 21
83. 11
85. 33
87. 2
89. 1
91. 22
93. 1
95. 32
97. 43
99. 40
101. 6
103. 39
105. 27
107. 89
109. 89
111. 253 points
113. $43.75
115. 22F
117. $44
119. $1256
121. 126
123. 4
125. 2
a. 10
b. 2
c. 1
d. 1
e. 4
f. 9
57. 102
© 2001 McGraw-Hill Companies
Answers