Download Adding and Subtracting Signed Numbers \ 3 2/ \ 1

1.3
Adding and Subtracting
Signed Numbers
1.3
OBJECTIVES
1. Find the median of a set of signed numbers
2. Find the difference of two signed numbers
3. Find the range of a set of signed numbers
In Section 0.4 we introduced the idea of signed numbers. Now we will examine the four
arithmetic operations (addition, subtraction, multiplication, and division) and see how
those operations are performed when signed numbers are involved. We start by considering
addition.
An application may help. As before, let’s represent a gain of money as a positive number and a loss as a negative number.
If you gain $3 and then gain $4, the result is a gain of $7:
347
If you lose $3 and then lose $4, the result is a loss of $7:
3 (4) 7
If you gain $3 and then lose $4, the result is a loss of $1:
3 (4) 1
If you lose $3 and then gain $4, the result is a gain of $1:
3 4 1
The number line can be used to illustrate the addition of signed numbers. Starting at the
origin, we move to the right for positive numbers and to the left for negative numbers.
Example 1
Adding Signed Numbers
(a) Add (3) (4).
4
3
© 2001 McGraw-Hill Companies
7
3
0
Start at the origin and move 3 units to the left. Then move 4 more units to the left to find the
sum. From the number line we see that the sum is
(3) (4) 7
2 2.
(b) Add 3
1
12
2
32
32
1
0
71
72
CHAPTER 1
THE LANGUAGE OF ALGEBRA
3
1
As before, we start at the origin. From that point move units left. Then move another
2
2
unit left to find the sum. In this case
2 2 2
3
1
CHECK YOURSELF 1
Add.
(a) (4) (5)
(b) (3) (7)
5
3
(d) 2
2
(c) (5) (15)
You have probably noticed some helpful patterns in the previous examples. These patterns will allow you to do the work mentally without having to use the number line. Look
at the following rule.
Rules and Properties: Adding Signed Numbers Case 1: Same Sign
of two positive numbers is
positive and the sum of two
negative numbers is negative.
We first encountered absolute
values in Section 0.4.
If two numbers have the same sign, add their absolute values. Give the sum the
sign of the original numbers.
Let’s again use the number line to illustrate the addition of two numbers. This time the
numbers will have different signs.
Example 2
Adding Signed Numbers
(a) Add 3 (6).
6
3
3
0
3
First move 3 units to the right of the origin. Then move 6 units to the left.
3 (6) 3
(b) Add 4 7.
7
4
4
0
3
© 2001 McGraw-Hill Companies
NOTE This means that the sum
ADDING AND SUBTRACTING SIGNED NUMBERS
SECTION 1.3
73
This time move 4 units to the left of the origin as the first step. Then move 7 units to the
right.
4 7 3
CHECK YOURSELF 2
Add.
(a) 7 (5)
1
16
(c) 3
3
(b) 4 (8)
(d) 7 3
You have no doubt noticed that, in adding a positive number and a negative number,
sometimes the sum is positive and sometimes it is negative. This depends on which of the
numbers has the larger absolute value. This leads us to the second part of our addition
rule.
Rules and Properties: Adding Signed Numbers Case 2:
Different Signs
NOTE Again, we first
encountered absolute values
in Section 0.4.
If two numbers have different signs, subtract their absolute values, the smaller
from the larger. Give the result the sign of the number with the larger absolute
value.
Example 3
Adding Signed Numbers
(a) 7 (19) 12
Because the two numbers have different signs, subtract the absolute values (19 7 12).
The sum has the sign () of the number with the larger absolute value, 19.
© 2001 McGraw-Hill Companies
(b) 13
7
3
2
2
Subtract the absolute values
2
13
with the larger absolute value, NOTE Remember, signed
numbers can be fractions and
decimals as well as integers.
7
6
3 . The sum has the sign () of the number
2
2
13
.
2
(c) 8.2 4.5 3.7
Subtract the absolute values (8.2 4.5 3.7). The sum has the sign () of the number
with the larger absolute value, 8.2.
74
CHAPTER 1
THE LANGUAGE OF ALGEBRA
CHECK YOURSELF 3
Add mentally.
(a) 5 (14)
(b) 7 (8)
(d) 7 (8)
2
7
(e) 3
3
(c) 8 15
(f) 5.3 (2.3)
In Section 1.2 we discussed the commutative, associative, and distributive properties.
There are two other properties of addition that we should mention. First, the sum of any
number and 0 is always that number. In symbols,
Rules and Properties: Additive Identity Property
For any number a,
NOTE No number loses its
a00aa
identity after addition with 0.
Zero is called the additive
identity.
Example 4
Adding Signed Numbers
Add.
(a) 9 0 9
4 4
(b) 0 5
5
(c) (25) 0 25
CHECK YOURSELF 4
Add.
NOTE The opposite of a
number is also called the
additive inverse of that
number.
NOTE 3 and 3 are opposites.
8
(c) (36) 0
Recall that every number has an opposite. It corresponds to a point the same distance
from the origin as the given number, but in the opposite direction.
3
3
3
0
3
The opposite of 9 is 9.
The opposite of 15 is 15.
Our second property states that the sum of any number and its opposite is 0.
© 2001 McGraw-Hill Companies
3
(b) 0 (a) 8 0
ADDING AND SUBTRACTING SIGNED NUMBERS
SECTION 1.3
75
Rules and Properties: Additive Inverse Property
For any number a, there exists a number a such that
NOTE Here a represents the
opposite of the number a. The
sum of any number and its
opposite, or additive inverse,
is 0.
a (a) (a) a 0
Example 5
Adding Signed Numbers
(a) 9 (9) 0
(b) 15 15 0
(c) (2.3) 2.3 0
(d)
4
4
0
5
5
CHECK YOURSELF 5
Add.
(a) (17) 17
(c)
1
1
3
3
(b) 12 (12)
(d) (1.6) 1.6
In Section 0.4 we saw that the least and greatest elements of a set were called the minimum and maximum. The middle value of an ordered set is called the median. The median
is sometimes used to represent an average of the set of numbers.
Example 6
Finding the Median
Find the median for each set of numbers.
(a) 9, 5, 8, 3, 7
First, rewrite the set in ascending order.
© 2001 McGraw-Hill Companies
8, 5, 3, 7, 9
The median is then the element that has just as many numbers to its right as it has to its left.
In this set, 3 is the median, because there are two numbers that are larger (7 and 9) and two
numbers that are smaller (8 and 5).
(b) 3, 2, 18, 20, 13
First, rewrite the set in ascending order.
20, 13, 2, 3, 18
The median is then the element that is exactly in the middle. The median for this set is 2.
CHAPTER 1
THE LANGUAGE OF ALGEBRA
CHECK YOURSELF 6
Find the median for each set of numbers.
(a) 3, 2, 7, 6, 1
(b) 5, 1, 10, 2, 20
In the previous example, each set had an odd number of elements. If we had an even
number of elements, there would be no single middle number.
To find the median from a set with an even number of elements, add the two middle
numbers and divide their sum by 2.
Example 7
Finding the Median
Find the median for each set of numbers.
(a) 3, 3, 8, 4, 1, 7, 5, 9
First, rewrite the set in ascending order.
8, 7, 3, 1, 3, 4, 5, 9
Add the middle two numbers (1 and 3), then divide their sum by 2.
(1) (3)
2
1
2
2
The median is 1.
(b) 8, 3, 2, 4, 5, 7
Rewrite the set in ascending order.
7, 5, 2, 3, 4, 8
The median is one-half the sum of the middle two numbers.
1
2 3
0.5
2
2
CHECK YOURSELF 7
Find the median for each set of numbers.
(a) 2, 5, 15, 8, 3, 4
(b) 8, 3, 6, 8, 9, 7
To begin our discussion of subtraction when signed numbers are involved, we can look
back at a problem using natural numbers. Of course, we know that
853
(1)
© 2001 McGraw-Hill Companies
76
ADDING AND SUBTRACTING SIGNED NUMBERS
SECTION 1.3
77
From our work in adding signed numbers, we know that it is also true that
8 (5) 3
(2)
Comparing equations (1) and (2), we see that the results are the same. This leads us to
an important pattern. Any subtraction problem can be written as a problem in addition.
Subtracting 5 is the same as adding the opposite of 5, or 5. We can write this fact as
follows:
8 5 8 (5) 3
This leads us to the following rule for subtracting signed numbers.
Rules and Properties: Subtracting Signed Numbers
1. Rewrite the subtraction problem as an addition problem by
a. Changing the minus sign to a plus sign.
b. Replacing the number being subtracted with its opposite.
2. Add the resulting signed numbers as before.
In symbols,
NOTE This is the definition of
a b a (b)
subtraction.
Example 8 illustrates the use of this definition while subtracting.
Example 8
Subtracting Signed Numbers
Subtraction
Addition
Change the subtraction symbol ()
to an addition symbol ().
(a) 15 7 15 (7)
Replace 7 with its opposite, 7.
8
(b) 9 12 9 (12) 3
© 2001 McGraw-Hill Companies
(c) 6 7 6 (7) 13
3
7
3
7
10
(d) 2
5
5
5
5
5
(e) 2.1 3.4 2.1 (3.4) 1.3
(f) Subtract 5 from 2. We write the statement as 2 5 and proceed as before:
2 5 2 (5) 7
CHAPTER 1
THE LANGUAGE OF ALGEBRA
CHECK YOURSELF 8
Subtract.
(a) 18 7
7
5
(d) 6
6
(b) 5 13
(c) 7 9
(e) 2 7
(f) 5.6 7.8
The subtraction rule is used in the same way when the number being subtracted is negative. Change the subtraction to addition. Replace the negative number being subtracted
with its opposite, which is positive. Example 9 will illustrate this principle.
Example 9
Subtracting Signed Numbers
Subtraction
Addition
Change the subtraction
to an addition.
(a) 5 (2) 5 (2) 5 2 7
Replace 2 with its opposite, 2 or 2.
(b) 7 (8) 7 (8) 7 8 15
(c) 9 (5) 9 5 4
(d) 12.7 (3.7) 12.7 3.7 9
3
7
3
7
4
(e) 1
4
4
4
4
4
(f) Subtract 4 from 5. We write
5 (4) 5 4 1
CHECK YOURSELF 9
Subtract.
(a) 8 (2)
(d) 9.8 (5.8)
(b) 3 (10)
(e) 7 (7)
(c) 7 (2)
Given a set of numbers, the range is the difference between the maximum and the
minimum.
© 2001 McGraw-Hill Companies
78
ADDING AND SUBTRACTING SIGNED NUMBERS
SECTION 1.3
79
Example 10
Finding the Range
Find the range for each set of numbers.
(a) 5, 2, 7, 9, 3
Rewrite the set in ascending order. The maximum is 9, the minimum is 7. The range is the
difference.
9 (7) 9 7 16
The range is 16.
(b) 3, 8, 17, 12, 2
Rewrite the set in ascending order. The maximum is 12. The minimum is 17. The range is
12 (17) 29.
CHECK YOURSELF 10
Find the range for each set of numbers.
(a) 2, 4, 7, 3, 1
NOTE Some graphing
calculators have a negative sign
() that acts to change the
sign of a number.
(b) 3, 4, 7, 5, 9, 4
Your scientific calculator can be used to do arithmetic with signed numbers. Before we
look at an example, there are some keys on your calculator with which you should become
familiar.
There are two similar keys you must find on the calculator. The first is used for subtraction ( ) and is usually found in the right column of calculator keys. The second will
“change the sign” of a number. It is usually a and is found on the bottom row.
We will use these keys in our next example.
Example 11
© 2001 McGraw-Hill Companies
Subtracting Signed Numbers
NOTE If you have a graphing
Using your calculator, find the difference.
calculator, the key sequence
will be
(a) 12.43 3.516
() 12.43 3.516 ENTER
Enter the 12.43 and push the to make it negative. Then push 3.516 . The result
should be 15.946.
(b) 23.56 (4.7)
The key sequence is
23.56 4.7 The answer should be 28.26.
CHAPTER 1
THE LANGUAGE OF ALGEBRA
CHECK YOURSELF 11
Use your calculator to find the difference.
(a) 13.46 5.71
(b) 3.575 (6.825)
CHECK YOURSELF ANSWERS
1. (a) 9; (b) 10; (c) 20; (d) 4
2. (a) 2; (b) 4; (c) 5; (d) 4
8
3. (a) 9; (b) 15; (c) 7; (d) 1; (e) 3; (f ) 3
4. (a) 8; (b) ; (c) 36
3
5. (a) 0; (b) 0; (c) 0; (d) 0
6. (a) 1; (b) 1
7. (a) 2.5; (b) 4.5
8. (a) 11; (b) 8; (c) 16; (d) 2; (e) 9; (f ) 2.2
9. (a) 10; (b) 13; (c) 5; (d) 4; (e) 14
10. (a) 7 (4) 11;
(b) 9 (7) 16
11. (a) 19.17; (b) 3.25
© 2001 McGraw-Hill Companies
80
Name
Exercises
1.3
Section
Date
Add.
1. 3 6
ANSWERS
2. 5 9
1.
3. 11 5
2.
4. 8 7
3.
4.
5.
3
5
4
4
6.
7
8
3
3
5.
6.
7.
1
4
2
5
8.
2
5
3
9
7.
8.
9. (2) (3)
10. (1) (9)
9.
10.
11.
3
12
12. 5
5
4
3
14. 7
14
3
7
11. 5
5
12.
13.
1
3
13. 2
8
14.
15.
15. (1.6) (2.3)
16. (3.5) (2.6)
16.
17.
© 2001 McGraw-Hill Companies
17. 9 (3)
18.
18. 10 (4)
19.
19.
3
1
4
2
20.
2
1
3
6
20.
21.
21.
5 20
4
9
22.
6 12
11
5
22.
81
ANSWERS
23.
23. 11.4 13.4
24. 5.2 9.2
25. 3.6 7.6
26. 2.6 4.9
27. 9 0
28. 15 0
29. 7 (7)
30. 12 (12)
31. 4.5 4.5
32.
33. 7 (9) (5) 6
34. (4) 6 (3) 0
35. 7 (3) 5 (11)
36. 24.
25.
26.
27.
28.
29.
30.
31.
3 3
2
2
32.
33.
34.
35.
6
13
4
5
5
5
36.
37. 37.
3
7
1
2
4
4
38.
1
5
1
3
6
2
38.
39. 2.3 (5.4) (2.9)
39.
40. 5.4 (2.1) (3.5)
40.
41.
Subtract.
42.
41. 21 13
42. 36 22
43. 82 45
44. 103 56
43.
44.
46.
45.
15
8
7
7
46.
17
9
8
8
47.
48.
47. 7.9 5.4
48. 11.7 4.5
49. 8 10
50. 14 19
49.
50.
82
© 2001 McGraw-Hill Companies
45.
ANSWERS
51. 24 45
52. 136 352
51.
52.
53.
53.
7
19
6
6
54.
5
32
9
9
54.
55.
55. 7.8 11.6
56. 14.3 25.5
56.
57.
57. 5 3
58. 15 8
58.
59.
59. 9 14
60. 8 12
60.
61.
61. 2
7
5
10
62. 5
7
9
18
62.
63.
63. 3.4 4.7
64. 8.1 7.6
65. 5 (11)
66. 7 (5)
64.
65.
66.
67. 7 (12)
68. 3 (10)
67.
68.
69.
70.
11
7
72.
16
8
3
3
4
2
© 2001 McGraw-Hill Companies
6
5
71.
7
14
5
7
6
6
69.
70.
71.
72.
73.
73. 8.3 (5.7)
74. 6.5 (4.3)
75. 8.9 (11.7)
76. 14.5 (24.6)
74.
75.
76.
77. 36 (24)
78. 28 (11)
77.
78.
83
ANSWERS
79.
79. 19 (27)
80.
81.
81.
4 4 3
11
80. 11 (16)
82. 1
5
2
8
83. 12.7 (5.7)
84. 5.6 (2.6)
85. 6.9 (10.1)
86. 3.4 (7.6)
82.
83.
Use your calculator to evaluate each expression.
84.
85.
87. 4.1967 5.2943
88. 5.3297 (4.1897)
86.
89. 4.1623 (3.1468)
90. (3.6829) 4.5687
87.
91. 6.3267 8.6789
92. 6.6712 5.3245
88.
Find the median for each of the following sets.
89.
93. 1, 3, 5, 7, 9
94. 2, 4, 6, 8, 10
90.
95. 8, 7, 2, 25, 5, 13, 3
96. 53, 23, 34, 21, 32, 30, 32
91.
Determine the range for each of the following sets.
92.
97. 2, 7, 9, 15, 24
93.
99. 4, 3, 2, 7, 9
94.
101.
95.
7
1
3
, 2, , 8,
8
2
4
103. 3, 2, 5, 6, 3
96.
98. 4, 8, 11, 15, 27
100. 7, 2, 1, 8, 11
102. 3,
5
1 2
, 7, ,
6
3 3
104. 1, 9, 7, 2, 3
97.
Solve the following problems.
98.
105. Checking account. Amir has $100 in his checking account. He writes a check for
$23 and makes a deposit of $51. What is his new balance?
99.
106. Checking account. Olga has $250 in her checking account. She deposits $52 and
then writes a check for $77. What is her new balance?
100.
102.
Bal:
Dep:
CK # 1111:
103.
104.
105.
106.
84
© 2001 McGraw-Hill Companies
101.
ANSWERS
107. Football yardage. On four consecutive running plays, Ricky Watters of the Seattle
Seahawks gained 23 yards, lost 5 yards, gained 15 yards, and lost 10 yards. What
was his net yardage change for the series of plays?
107.
108.
109.
108. VISA balance. Ramon owes $780 on his VISA account. He returns three items
costing $43.10, $36.80, and $125.00 and receives credit on his account. Next, he
makes a payment of $400. He then makes a purchase of $82.75. How much does
Tom still owe?
110.
111.
112.
109. Temperature. The temperature at noon on a June day was 82. It fell by 12 in the
next 4 hours. What was the temperature at 4:00 P.M.?
113.
114.
110. Mountain climbing. Chia is standing at a point 6000 feet (ft) above sea level. She
descends to a point 725 ft lower. What is her distance above sea level?
115.
111. Checking account. Omar’s checking account was overdrawn by $72. He wrote
another check for $23.50. How much was his checking account overdrawn after
writing the check?
112. Personal finance. Angelo owed his sister $15. He later borrowed another $10.
What positive or negative number represents his current financial condition?
113. Education. A local community college had a decrease in enrollment of 750
students in the fall of 1999. In the spring of 2000, there was another decrease of
425 students. What was the total decrease in enrollment for both semesters?
114. Temperature. At 7 A.M., the temperature was 15F. By 1 P.M., the temperature
had increased by 18F. What was the temperature at 1 P.M.?
115. Education. Ezra’s scores on five tests taken in a mathematics class were 87, 71, 95,
81, and 90. What was the range of his scores?
© 2001 McGraw-Hill Companies
Name:___________
1+5
2x5
4+5
15 - 2
4x3
3+6
9+4
3+9
1x2
13 - 4
5+6
=
=
=
=
=
=
=
=
=
=
=
_____
______
Name:
= ____
4x3
= ____
= ____
= ____
2x5
= ____
1+5
= ____
= ____
4+5
= ____
2x5
____
-2 =
= ____
Nam
15
5
+
4
= ____
e:____
= ____
= ____
____
8x3
2
__
___
= ____
15
____
6 = __
+
=
3
3
4x
= ____
__
= ____
__
6
+
=
1
5
+5
3+6
= ____
= 9__= ____
__
__
__
+
=
2 x 56
9+4
____ 4 x
= ____
= __
__
=
__
2
3 =
=
4 + 5 1 x __
____
3+9
= __ 4 = ____2 x
= ____
-__
5 =
15 = ____
2 = 13
____
1x2
= ____
= __4__+ 5
____
= __4__
9+4
= __
x3
13 - 4
__
15 = ____
____ = ____
2 =
______
+6 =3+6 =
__
5
___
__
:__
8x3
____
Name
9+4
= __
= __
__
3+6
__
3+9
= __ 1 + 5 =
= __
__
5+6
__
1x2
= __
= __
__ 2 x 5 =
6+9
__
13 = __
____
4 =
4x3 =
__ 4 + 5 =
1x2
____
____
5+6
= __
____
1+5 =
= __
__ 15 - 2 =
2x5 =
13 _
__
4
___
_
=
=
___
____
2x5
4x3 =
4+5 =
9+4
____
= __
____
4+5 =
__
3+6 =
15 - 2 =
____
____
15 - 2 =
8x3 =
9+4 =
____
_
4x3 =
+ 6 = ___
3
3
+9 =
____
____
3+6 =
5+6 =
1x2 =
____
____
9+4 =
6+9 =
13 - 4 =
____
_
___
=
3+9 =
2
1x
5+6 =
____
____
1x2 =
13 - 4 =
____
____
13 - 4 =
9+4 =
____
5+6 =
____
____
____
____
____
____
____
____
____
____
____
4x3
2x5
4+5
15 - 2
8x3
3+6
5+6
6+9
1x2
13 - 4
9+4
Name:___________
____
____
____
____
____
____
____
____
____
____
____
4x3
2x5
4+5
15 - 2
8x3
3+6
5+6
6+9
1x2
13 - 4
9+4
=
=
=
=
=
=
=
=
=
=
=
____
____
____
____
____
____
____
____
____
____
____
85
ANSWERS
116. The bar chart shown represents the total yards gained by the leading passer in the
NFL from 1993 to 1997. Use a signed number to represent the change in total
passing yards from one year to the next.
116.
(a) from 1993 to 1994
(c) from 1995 to 1996
(b) from 1994 to 1995
(d) from 1996 to 1997
117.
5000
118.
4453
Yards
4000
4030
3000
3328
3281
1996
1997
2575
2000
1000
0
1993
1994
1995
Year
117. In this chapter, it is stated that “Every number has an opposite.” The opposite of 9 is
9. This corresponds to the idea of an opposite in English. In English an opposite is
often expressed by a prefix, for example, un- or ir-.
(a) Write the opposite of these words: unmentionable, uninteresting, irredeemable,
irregular, uncomfortable.
(b) What is the meaning of these expressions: not uninteresting, not irredeemable,
not irregular, not unmentionable?
(c) Think of other prefixes that negate or change the meaning of a word to its
opposite. Make a list of words formed with these prefixes, and write a sentence
with three of the words you found. Make a sentence with two words and
phrases from each of the lists above. Look up the meaning of the word
irregardless.
What is the value of [(5)]? What is the value of (6)? How
does this relate to the above examples? Write a short description about this
relationship.
118. The temperature on the plains of North Dakota can change rapidly, falling or rising
Day
Mon.
Tues.
Wed.
Thurs.
Fri.
Sat.
Sun.
Temp.
Change
from 10 A.M.
to 3 P.M.
13
20
18
10
25
5
15
Write a short speech for the TV weather reporter that summarizes the daily
temperature change. Use the median as you characterize the average daily midday
change.
86
© 2001 McGraw-Hill Companies
many degrees in the course of an hour. Here are some temperature changes during
each day over a week.
ANSWERS
119. Science. The daily average temperatures in degrees Fahrenheit for a week in
February were 1, 3, 5, 2, 4, 12, and 10. What was the range of temperatures for
that week?
119.
120.
120. How long ago was the year 1250 B.C.E.? What year was 3300 years ago? Make a
number line and locate the following events, cultures, and objects on it. How long
ago was each item in the list? Which two events are the closest to each other? You
may want to learn more about some of the cultures in the list and the mathematics
and science developed by that culture.
121.
122.
a.
Inca culture in Peru—1400 A.D.
The Ahmes Papyrus, a mathematical text from Egypt—1650 B.C.E.
Babylonian arithmetic develops the use of a zero symbol—300 B.C.E.
First Olympic Games—776 B.C.E.
Pythagoras of Greece dies—580 B.C.E.
Mayans in Central America independently develop use of zero—500 A.D.
The Chou Pei, a mathematics classic from China—1000 B.C.E.
The Aryabhatiya, a mathematics work from India—499 A.D.
Trigonometry arrives in Europe via the Arabs and India—1464 A.D.
Arabs receive algebra from Greek, Hindu, and Babylonian sources and develop it
into a new systematic form—850 A.D.
Development of calculus in Europe—1670 A.D.
Rise of abstract algebra—1860 A.D.
Growing importance of probability and development of statistics—1902 A.D.
b.
c.
d.
e.
f.
121. Complete the following statement: “3 (7) is the same as ____ because . . . ”
Write a problem that might be answered by doing this subtraction.
122. Explain the difference between the two phrases: “a number subtracted from 5”
and “a number less than 5.” Use algebra and English to explain the meaning of these
phrases. Write other ways to express subtraction in English. Which ones are
confusing?
© 2001 McGraw-Hill Companies
Getting Ready for Section 1.4 [Section 0.3]
Add.
(a)
(b)
(c)
(d)
(e)
(f)
(1) (1) (1) (1)
33333
999
(10) (10) (10)
(5) (5) (5) (5) (5)
(8) (8) (8) (8)
87
Answers
1. 9
15. 3.9
29. 0
43. 37
57. 8
3. 16
5. 2
17. 6
7.
19.
1
4
33. 1
47. 2.5
31. 0
45. 1
59. 23
13
10
61. 9. 5
21. 35. 2
49. 2
11
10
7
20
11. 2
23. 2
37. 3
51. 21
63. 8.1
13. 25. 4
39. 6
53. 2
65. 16
7
8
27. 9
41. 8
55. 3.8
67. 19
9
17
71.
73. 14
75. 20.6
77. 12
79. 8
81. 2
4
14
83. 7
85. 3.2
87. 9.491
89. 1.0155
91. 2.3522
93. 5
95. 7
97. 22
99. 13
101. 10
103. 11
105. $128
107. 23 yards
109. 70
111. $95.50
113. 1175
115. 24
69.
117.
e. 25
121.
a. 4
b. 15
c. 27
f. 32
© 2001 McGraw-Hill Companies
d. 30
119. 14F
88