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Transcript
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