4.5 Dividing Decimals 4.5 OBJECTIVES 1. 2. 3. 4. Divide a decimal by a whole number Divide a decimal by a decimal Divide a decimal by a power of ten Apply division to the solution of an application problem The division of decimals is very similar to our earlier work with dividing whole numbers. The only difference is in learning to place the decimal point in the quotient. Let’s start with the case of dividing a decimal by a whole number. Here, placing the decimal point is easy. You can apply the following rule. Step by Step: To Divide a Decimal by a Whole Number Step 1 Place the decimal point in the quotient directly above the decimal point of the dividend. Step 2 Divide as you would with whole numbers. Example 1 Dividing a Decimal by a Whole Number Divide 29.21 by 23. NOTE Do the division just as if you were dealing with whole numbers. Just remember to place the decimal point in the quotient directly above the one in the dividend. 1.27 2329.21 23 62 46 1 61 1 61 0 The quotient is 1.27. CHECK YOURSELF 1 Divide 80.24 by 34. © 2001 McGraw-Hill Companies Let’s look at another example of dividing a decimal by a whole number. Example 2 Dividing a Decimal by a Whole Number Divide 122.2 by 52. NOTE Again place the decimal point of the quotient above that of the dividend. 2.3 52122.2 104 18 2 15 6 26 355 356 CHAPTER 4 DECIMALS We normally do not use a remainder when dealing with decimals. Add a 0 to the dividend and continue. NOTE Remember that adding a 0 does not change the value of the dividend. It simply allows us to complete the division process in this case. 2.35 52 122.20 104 18 2 15 6 2 60 2 60 0 Add a 0. So 122.2 52 2.35. The quotient is 2.35. CHECK YOURSELF 2 Divide 234.6 by 68. Often you will be asked to give a quotient to a certain place value. In this case, continue the division process to one digit past the indicated place value. Then round the result back to the desired accuracy. When working with money, for instance, we normally give the quotient to the nearest hundredth of a dollar (the nearest cent). This means carrying the division out to the thousandths place and then rounding back. Example 3 Dividing a Decimal by a Whole Number and Rounding the Result place past the desired place, and then round the result. Find the quotient of 25.75 15 to the nearest hundredth. 1.716 1525.750 15 10 7 10 5 25 15 100 90 10 Add a 0 to carry the division to the thousandths place. So 25.75 15 1.72 (to the nearest hundredth). CHECK YOURSELF 3 Find 99.26 35 to the nearest hundredth. As we mentioned, problems similar to the one in Example 3 often occur when working with money. Example 4 is one of the many applications of this type of division. © 2001 McGraw-Hill Companies NOTE Find the quotient to one DIVIDING DECIMALS SECTION 4.5 357 Example 4 An Application Involving the Division of a Decimal by a Whole Number A carton of 144 items costs $56.10. What is the price per item to the nearest cent? To find the price per item, divide the total price by 144. NOTE You might want to review the rules for rounding decimals in Section 4.1. 0.389 14456.100 43 2 12 90 11 52 1 380 1 296 84 Carry the division to the thousandths place and then round back. The cost per item is rounded to $0.39, or 39¢. CHECK YOURSELF 4 An office paid $26.55 for 72 pens. What was the cost per pen to the nearest cent? We want now to look at division by decimals. Here is an example using a fractional form. Example 5 Rewriting a Problem That Requires Dividing by a Decimal Divide. 2.57 3.4 Write the division as a fraction. 2.57 10 3.4 10 25.7 34 We multiply the numerator and denominator by 10 so the divisor is a whole number. This does not change the value of the fraction. Multiplying by 10, shift the decimal point in the numerator and denominator one place to the right. © 2001 McGraw-Hill Companies 2.57 3.4 NOTE It’s always easier to rewrite a division problem so that you’re dividing by a whole number. Dividing by a whole number makes it easy to place the decimal point in the quotient. 25.7 34 Our division problem is rewritten so that the divisor is a whole number. So 2.57 3.4 25.7 34 After we multiply the numerator and denominator by 10, we see that 2.57 3.4 is the same as 25.7 34. CHECK YOURSELF 5 Rewrite the division problem so that the divisor is a whole number. 3.42 2.5 358 CHAPTER 4 DECIMALS NOTE Of course, multiplying by any whole-number power of 10 greater than 1 is just a matter of shifting the decimal point to the right. Do you see the rule suggested by this example? We multiplied the numerator and the denominator (the dividend and the divisor) by 10. We made the divisor a whole number without altering the actual digits involved. All we did was shift the decimal point in the divisor and dividend the same number of places. This leads us to the following rule. Step by Step: To Divide by a Decimal Step 1 Move the decimal point in the divisor to the right, making the divisor a whole number. Step 2 Move the decimal point in the dividend to the right the same number of places. Add zeros if necessary. Step 3 Place the decimal point in the quotient directly above the decimal point of the dividend. Step 4 Divide as you would with whole numbers. Let’s look at an example of the use of our division rule. Example 6 Rounding the Result of Dividing by a Decimal Divide 1.573 by 0.48 and give the quotient to the nearest tenth. Write 0.48 1.57 ^ ^ 3 Shift the decimal points two places to the right to make the divisor a whole number. Now divide: statement is rewritten, place the decimal point in the quotient above that in the dividend. 3.27 48157.30 144 13 3 96 3 70 3 36 34 Note that we add a 0 to carry the division to the hundredths place. In this case, we want to find the quotient to the nearest tenth. Round 3.27 to 3.3. So 1.573 0.48 3.3 (to the nearest tenth) CHECK YOURSELF 6 Divide, rounding the quotient to the nearest tenth. 3.4 1.24 Let’s look at some applications of our work in dividing by decimals. © 2001 McGraw-Hill Companies NOTE Once the division DIVIDING DECIMALS SECTION 4.5 359 Example 7 Solving an Application Involving the Division of Decimals Andrea worked 41.5 hours in a week and earned $239.87. What was her hourly rate of pay? To find the hourly rate of pay we must use division. We divide the number of hours worked into the total pay. NOTE Notice that we must add a zero to the dividend to complete the division process. 5.78 41.5 239.8 70 ^ ^ 207 5 32 3 29 0 33 33 70 7 5 20 20 0 Andrea’s hourly rate of pay was $5.78. CHECK YOURSELF 7 A developer wants to subdivide a 12.6-acre piece of land into 0.45-acre lots. How many lots are possible? Example 8 Solving an Application Involving the Division of Decimals At the start of a trip the odometer read 34,563. At the end of the trip, it read 36,235. If 86.7 gallons (gal) of gas were used, find the number of miles per gallon (to the nearest tenth). First, find the number of miles traveled by subtracting the initial reading from the final reading. 36,235 34,563 1 672 Final reading Initial reading Miles covered Next, divide the miles traveled by the number of gallons used. This will give us the miles per gallon. 1 9.28 86.7 1672.0 00 ^ © 2001 McGraw-Hill Companies ^ 867 805 0 780 3 24 7 17 3 73 69 4 0 4 60 36 24 Round 19.28 to 19.3 mi/gal. CHAPTER 4 DECIMALS CHECK YOURSELF 8 John starts his trip with an odometer reading of 15,436 and ends with a reading of 16,238. If he used 45.9 gallons (gal) of gas, find the number of miles per gallon (mi/gal) (to the nearest tenth). Recall that you can multiply decimals by powers of 10 by simply shifting the decimal point to the right. A similar approach will work for division by powers of 10. Example 9 Dividing a Decimal by a Power of 10 (a) Divide. 3.53 10 35.30 30 53 50 30 30 0 The dividend is 35.3. The quotient is 3.53. The decimal point has been shifted one place to the left. Note also that the divisor, 10, has one zero. (b) Divide. 3.785 100 378.500 300 78 5 70 0 8 50 8 00 500 500 0 Here the dividend is 378.5, whereas the quotient is 3.785. The decimal point is now shifted two places to the left. In this case the divisor, 100, has two zeros. CHECK YOURSELF 9 Perform each of the following divisions. (a) 52.6 10 (b) 267.9 100 Example 9 suggests the following rule. Rules and Properties: To Divide a Decimal by a Power of 10 Move the decimal point to the left the same number of places as there are zeros in the power of 10. © 2001 McGraw-Hill Companies 360 DIVIDING DECIMALS SECTION 4.5 361 Example 10 Dividing a Decimal by a Power of 10 Divide. 27.3 10 2 (a) ^ 7.3 Shift one place to the left. 2.73 (b) 57.53 100 0 ^ 57.53 Shift two places to the left. 0.5753 NOTE As you can see, we may have to add zeros to correctly place the decimal point. (c) 39.75 1000 0 ^ 039.75 Shift three places to the left. 0.03975 85 1000 0 (d) ^ 085. The decimal after the 85 is implied. 0.085 REMEMBER: 104 is a 1 (e) 235.72 104 0 followed by four zeros. ^ 0235.72 Shift four places to the left. 0.023572 CHECK YOURSELF 10 Divide. (a) 3.84 10 (b) 27.3 1000 Let’s look at an application of our work in dividing by powers of 10. Example 11 Solving an Application Involving a Power of 10 © 2001 McGraw-Hill Companies To convert from millimeters (mm) to meters (m), we divide by 1000. How many meters does 3450 mm equal? 3450 mm 3 ^ 450. m 3.450 m Shift three places to the left to divide by 1000. CHAPTER 4 DECIMALS CHECK YOURSELF 11 A shipment of 1000 notebooks cost a stationery store $658. What was the cost per notebook to the nearest cent? Recall that the order of operations is always used to simplify a mathematical expression with several operations. You should recall the order of operations as the following. Rules and Properties: 1. 2. 3. 4. The Order of Operations Perform any operations enclosed in parentheses. Apply any exponents. Do any multiplication and division, moving from left to right Do any addition and subtraction, moving from left to right. Example 12 Applying the Order of Operations Simplify each expression. (a) 4.6 (0.5 4.4)2 3.93 4.6 (2.2)2 3.93 parentheses 4.6 4.84 3.93 exponent 9.44 3.93 add (left of the subtraction) 5.51 subtract (b) 16.5 (2.8 0.2)2 4.1 2 16.5 (3)2 4.1 2 parentheses 16.5 9 4.1 2 exponent 16.5 9 8.2 multiply 7.5 8.2 subtraction (left of the addition) 15.7 add CHECK YOURSELF 12 Simplify each expression. (a) 6.35 (0.2 8.5)2 3.7 (b) 2.52 (3.57 2.14) 3.2 1.5 CHECK YOURSELF ANSWERS 1. 2.36 2. 3.45 3. 2.84 4. $0.37, or 37¢ 5. 34.2 25 6. 2.7 7. 28 lots 8. 17.5 mi/gal 9. (a) 5.26; (b) 2.679 10. (a) 0.384; (b) 0.0273 11. 66¢ 12. (a) 5.54; (b) 9.62 © 2001 McGraw-Hill Companies 362 Name 4.5 Exercises Section Date Divide. 1. 16.68 6 2. 43.92 8 ANSWERS 1. 3. 1.92 4 4. 5.52 6 2. 3. 4. 5. 5.48 8 6. 2.76 8 5. 6. 7. 13.89 6 8. 21.92 5 7. 8. 9. 185.6 32 10. 165.6 36 9. 10. 11. 79.9 34 12. 179.3 55 11. 12. 13. 13. 5213.78 14. 7626.22 14. 15. 15. 0.611.07 16. 0.810.84 16. 17. 17. 3.87.22 18. 2.913.34 18. 19. 19. 5.211.622 20. 6.43.616 20. © 2001 McGraw-Hill Companies 21. 21. 0.271.8495 22. 0.0380.8132 22. 23. 23. 0.0461.587 24. 0.523.2318 24. 25. 25. 0.658 2.8 26. 0.882 0.36 26. 363 ANSWERS 27. Divide by moving the decimal point. 28. 27. 5.8 10 28. 5.1 10 30. 29. 4.568 100 30. 3.817 100 31. 31. 24.39 1000 32. 8.41 100 32. 33. 6.9 1000 34. 7.2 1000 35. 7.8 102 36. 3.6 103 37. 45.2 105 38. 57.3 104 29. 33. 34. 35. 36. Divide and round the quotient to the indicated decimal place. 37. 39. 23.8 9 38. tenth 40. 5.27 8 hundredth 41. 38.48 46 hundredth 42. 3.36 36 43. 125.4 52 tenth 44. 2.563 54 thousandth 39. thousandth 40. 41. 45. 0.7 1.642 hundredth 46. 0.6 7.695 42. 47. 4.5 8.415 tenth 48. 5.8 16 49. 3.12 4.75 hundredth 50. 64.2 16.3 43. tenth hundredth thousandth 44. 45. Solve the following applications. 46. 51. Cost of CDs. Marv paid $40.41 for three CDs on sale. What was the cost per CD? 47. 52. Contributions. Seven employees of an office donated $172.06 during a charity drive. What was the average donation? 48. 53. Book purchases. A shipment of 72 paperback books cost a store $190.25. What was the average cost per book to the nearest cent? © 2001 McGraw-Hill Companies 49. 50. 51. 52. 53. 364 ANSWERS 54. Cost. A restaurant bought 50 glasses at a cost of $39.90. What was the cost per glass to the nearest cent? 55. Cost. The cost of a case of 48 items is $28.20. What is the cost of an individual item 54. 55. 56. to the nearest cent? 57. 56. Office supplies. An office bought 18 hand-held calculators for $284. What was the cost per calculator to the nearest cent? 57. Monthly payments. Al purchased a new refrigerator that cost $736.12 with interest included. He paid $100 as a down payment and agreed to pay the remainder in 18 monthly payments. What amount will he be paying per month? 58. Monthly payments. The cost of a television set with interest is $490.64. If you make a down payment of $50 and agree to pay the balance in 12 monthly payments, what will be the amount of each monthly payment? 58. 59. 60. 61. 62. 63. 64. 65. 66. 59. Mileage. In five readings, Lucia’s gas mileage was 32.3, 31.6, 29.5, 27.3, and 33.4 miles per gallon (mi/gal). What was her average gas mileage to the nearest tenth of a mile per gallon? 60. Pollution. Pollution index readings were 53.3, 47.8, 41.9, 55.8, 43.7, 41.7, and 52.3 for a 7-day period. What was the average reading (to the nearest tenth) for the 7 days? 61. Label making. We have 91.25 inches (in.) of plastic labeling tape and wish to make labels that are 1.25 in. long. How many labels can be made? 62. Wages. Alberto worked 32.5 hours (h), earning $306.15. How much did he make per hour? 63. Cost per pound. A roast weighing 5.3 pounds (lb) sold for $14.89. Find the cost per © 2001 McGraw-Hill Companies pound to the nearest cent. 64. Weight. One nail weighs 0.025 ounce (oz). How many nails are there in 1 lb? (1 lb is 16 oz.) 65. Mileage. A family drove 1390 miles (mi), stopping for gas three times. If they purchased 15.5, 16.2, and 10.8 gallons (gal) of gas, find the number of miles per gallon (the mileage) to the nearest tenth. 66. Mileage. On a trip an odometer changed from 36,213 to 38,319. If 136 gal of gas were used, find the number of miles per gallon (to the nearest tenth). 365 ANSWERS 67. 67. Conversion. To convert from millimeters (mm) to inches, we can divide by 25.4. If film is 35 mm wide, find the width to the nearest hundredth of an inch. 68. 69. 68. Conversion. To convert from centimeters (cm) to inches, we can divide by 2.54. The rainfall in Paris was 11.8 cm during 1 week. What was that rainfall to the nearest hundredth of an inch? 70. 71. 69. Construction. A road-paving project will cost $23,500. If the cost is to be shared by 72. 100 families, how much will each family pay? 70. Conversion. To convert from milligrams (mg) to grams (g), we divide by 1000. A tablet is 250 mg. What is its weight in grams? 73. 71. Conversion. To convert from milliliters (mL) to liters (L), we divide by 1000. If a bottle of wine holds 750 mL, what is its volume in liters? 74. 72. Unit cost. A shipment of 100 calculators cost a store $593.88. Find the cost per calculator (to the nearest cent). 73. The blood alcohol content (BAC) of a person who has been drinking is determined by the formula BAC oz of alcohol % of alcohol 0.075 of body wt. (hours of drinking 0.015) A 125-lb person is driving and is stopped by a policewoman on suspicion of driving under the influence (DUI). The driver claims that in the past 2 hours he consumed only six 12-oz bottles of 3.9% beer. If he undergoes a breathalyzer test, what will his BAC be? Will this amount be under the legal limit for your state? 74. Four brands of soap are available in a local store. Ounces Total Price Squeaky Clean Smell Fresh Feel Nice Look Bright 5.5 7.5 4.5 6.5 $0.36 0.41 0.31 0.44 Compute the unit price, and decide which brand is the best buy. 366 Unit Price © 2001 McGraw-Hill Companies Brand ANSWERS 75. Sophie is a quality control expert. She inspects boxes of #2 pencils. Each pencil weighs 4.4 grams (g). The contents of a box of pencils weigh 66.6 g. If a box is labeled CONTENTS: 16 PENCILS, should Sophie approve the box as meeting specifications? Explain your answer. 75. 76. 76. Write a plan to determine the number of miles per gallon (mpg) your car (or your family car) gets. Use this plan to determine your car’s actual mpg. 77. 77. Express the width and length of a $1 bill in centimeters (cm). Then express this same length in millimeters (mm). 78. 79. 78. If the perimeter of a square is 19.2 cm, how long is each side? 80. 81. 82. P 19.2 cm 83. 84. 79. If the perimeter of an equilateral triangle (all sides have equal length) is 16.8 cm, how long is each side? P 16.8 cm 80. If the perimeter of a regular (all sides have equal length) pentagon is 23.5 in., how long is each side? © 2001 McGraw-Hill Companies P 23.5 in. Simplify each expression. 81. 4.2 3.1 1.5 (3.1 0.4)2 82. 150 4.1 1.5 (2.5 1.6)3 2.4 83. 17.9 1.1 (2.3 1.1)2 (13.4 2.1 4.6) 84. 6.892 3.14 2.5 (4.1 3.2 1.6)2 367 Answers 1. 2.78 3. 0.48 5. 0.685 7. 2.315 9. 5.8 11. 2.35 13. 0.265 15. 18.45 17. 1.9 19. 2.235 21. 6.85 23. 34.5 25. 0.235 27. 0.58 29. 0.04568 31. 0.02439 33. 0.0069 35. 0.078 37. 0.000452 39. 2.6 41. 0.84 43. 2.4 45. 2.35 47. 1.9 49. 1.52 51. $13.47 53. $2.64 55. $0.59, or 59¢ 57. $35.34 59. 30.8 mi/gal 61. 73 labels 63. $2.81 65. 32.7 mi/gal 67. 1.38 in. 69. $235 71. 0.75 L 73. 75. 79. 5.6 cm 81. 11.8 83. 17.0291 © 2001 McGraw-Hill Companies 77. 368 Using Your Calculator to Divide Decimals It would be most surprising if you had reached this point without using your calculator to divide decimals. It is a good way to check your work, and a reasonable way to solve applications. Let’s first use the calculator for a straightforward problem. Example 1 Dividing Decimals Use your calculator to find the quotient 211.56 82 Enter the problem in the calculator to find that the answer is 25.8. CHECK YOURSELF 1 Use your calculator to find the quotient 304.32 9.6 Now that you’re convinced that it is easy to divide decimals on your calculator, let’s introduce a twist. Example 2 An Application Involving the Division of Decimals © 2001 McGraw-Hill Companies Omar drove 256.3 miles on a tank of gas. When he filled up the tank, it took 9.1 gallons. What was his gas mileage? Here’s where students get into trouble when they use a calculator. Entering these values, you may be tempted to answer “28.16483516 miles per gallon.” The difficulty is that there is no way you can compute gas mileage to the nearest hundred-millionth mile. How do you decide where to round off the answer that the calculator gives you? A good rule of thumb is to never report more digits than the least number of digits in any of the numbers that you are given in the problem. In this case, you were given a number with four digits and another with two digits.Your answer should not have more than two digits. Instead of 28.16483516, the answer could be 28 miles per gallon. Think about the question. If you were asked for gas mileage, how precise an answer would you give? The best answer to this question would be to give the nearest whole number of miles per gallon: 28 miles per gallon. CHECK YOURSELF 2 Emmet gained a total of 857 yards (yd) in 209 times that he carried the football. How many yards did he average for each time he carried the ball? CHECK YOURSELF ANSWERS 1. 31.7 2. 4.10 yd 369 Name Section Date Calculator Exercises Divide and check. ANSWERS 1. 8.901 2.58 2. 16.848 0.288 3. 99.705 34.5 4. 171.25 2.74 5. 0.01372 0.056 6. 0.200754 0.00855 1. 2. 3. 4. 5. Divide and round to the indicated place. 6. 7. 2.546 1.38 hundredth 8. 45.8 9.4 tenth 7. 9. 0.5782 1.236 8. thousandth 10. 1.25 0.785 hundredth 9. 11. 1.34 2.63 10. two decimal places 12. 12.364 4.361 three decimal places 11. Solve the following applications. 12. 13. Salary. In 1 week, Tom earned $178.30 by working 36.25 hours (h). What was his hourly rate of pay to the nearest cent? 13. 14. 14. Area. An 80.5-acre piece of land is being subdivided into 0.35-acre lots. How many 15. lots are possible in the subdivision? 16. 18. 15. 3.925 cm 16. 8.3838 in. 17. 1.7584 cm 18. 13.0624 mm Answers 1. 3.45 13. $4.92 370 3. 2.89 5. 0.245 7. 1.84 15. 1.25 cm 17. 0.56 cm 9. 0.468 11. 0.51 © 2001 McGraw-Hill Companies If the circumference of a circle is known, the diameter can be determined by dividing the circumference by p. In each of the following, determine the diameter for the given circumference. Use 3.14 for p. 17.

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