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Dividing Decimals4.5

4.5 OBJECTIVES

1. Divide a decimal by a whole number2. Divide a decimal by a decimal3. Divide a decimal by a power of ten4. Apply division to the solution of an application problem

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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 thecase of dividing a decimal by a whole number. Here, placing the decimal point is easy. Youcan apply the following rule.

Step 1 Place the decimal point in the quotient directly above the decimalpoint of the dividend.

Step 2 Divide as you would with whole numbers.

Step by Step: To Divide a Decimal by a Whole Number

Example 1

Dividing a Decimal by a Whole Number

Divide 29.21 by 23.

1.27

236 24 61 611 61

0

The quotient is 1.27.

23 �29.21

C H E C K Y O U R S E L F 1

Divide 80.24 by 34.

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.

2.3

10418 215 62 6

52 �122.2NOTE Again place the decimalpoint of the quotient abovethat of the dividend.

NOTE Do the division just as ifyou were dealing with wholenumbers. Just remember toplace the decimal point in thequotient directly above the onein the dividend.

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We normally do not use a remainder when dealing with decimals. Add a 0 to the dividendand continue.

2.35Add a 0.

10418 215 62 602 60

0

So 122.2 � 52 � 2.35. The quotient is 2.35.

52 �122.20NOTE Remember that addinga 0 does not change the valueof the dividend. It simply allowsus to complete the divisionprocess in this case.

NOTE Find the quotient to oneplace past the desired place,and then round the result.

Often you will be asked to give a quotient to a certain place value. In this case, continuethe division process to one digit past the indicated place value. Then round the result backto the desired accuracy.

When working with money, for instance, we normally give the quotient to the nearesthundredth of a dollar (the nearest cent). This means carrying the division out to the thou-sandths place and then rounding back.

Example 3

Dividing a Decimal by a Whole Number and Rounding the Result

Find the quotient of 25.75 � 15 to the nearest hundredth.

1.716

1510 710 5

2515100

9010

So 25.75 � 15 � 1.72 (to the nearest hundredth).

15 �25.750

Add a 0 to carry the division tothe thousandths place.

As we mentioned, problems similar to the one in Example 3 often occur when workingwith money. Example 4 is one of the many applications of this type of division.

C H E C K Y O U R S E L F 2

Divide 234.6 by 68.

C H E C K Y O U R S E L F 3

Find 99.26 � 35 to the nearest hundredth.

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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.

0.389

43 212 9011 521 3801 296

84

The cost per item is rounded to $0.39, or 39¢.

144 �56.100NOTE You might want toreview the rules for roundingdecimals in Section 4.1.

C H E C K Y O U R S E L F 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.

So

2.57 � 3.4 � 25.7 � 34

� 25.7 � 34

�25.7

34

�2.57 � 10

3.4 � 10

�2.57

3.42.57 � 3.4

Write the division as a fraction.

We multiply the numerator and denominator by 10so the divisor is a whole number. This does notchange the value of the fraction.

Multiplying by 10, shift the decimal point in thenumerator and denominator one place to the right.

Our division problem is rewritten so that the divisoris a whole number.

After we multiply the numerator and denominatorby 10, we see that 2.57 � 3.4 is the same as 25.7 � 34.

NOTE It’s always easier torewrite a division problem sothat you’re dividing by a wholenumber. Dividing by a wholenumber makes it easy to placethe decimal point in thequotient.

C H E C K Y O U R S E L F 5

Rewrite the division problem so that the divisor is a whole number.

3.42 � 2.5

Carry the division to the thousandthsplace and then round back.

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NOTE Once the divisionstatement is rewritten, placethe decimal point in thequotient above that in thedividend.

^

Rounding the Result of Dividing by a Decimal

Divide 1.573 by 0.48 and give the quotient to the nearest tenth.

Write

Now divide:

3.27

14413 39 63 703 36

34

Round 3.27 to 3.3. So

1.573 � 0.48 � 3.3 (to the nearest tenth)

48 �157.30

0.48 �1.57 3

Step 1 Move the decimal point in the divisor to the right, making the divisor awhole number.

Step 2 Move the decimal point in the dividend to the right the same numberof places. Add zeros if necessary.

Step 3 Place the decimal point in the quotient directly above the decimalpoint of the dividend.

Step 4 Divide as you would with whole numbers.

Step by Step: To Divide by a Decimal

Let’s look at an example of the use of our division rule.

Example 6

Shift the decimal points two placesto the right to make the divisor awhole number.

Note that we add a 0 to carry thedivision to the hundredths place. Inthis case, we want to find thequotient to the nearest tenth.

Let’s look at some applications of our work in dividing by decimals.

Do you see the rule suggested by this example? We multiplied the numerator and the de-nominator (the dividend and the divisor) by 10. We made the divisor a whole number with-out altering the actual digits involved. All we did was shift the decimal point in the divisorand dividend the same number of places. This leads us to the following rule.

NOTE Of course, multiplyingby any whole-number power of10 greater than 1 is just amatter of shifting the decimalpoint to the right.

C H E C K Y O U R S E L F 6

Divide, rounding the quotient to the nearest tenth.

3.4 � 1.24

^

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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.

5.78

207 5 7032 3 729 0 5

3 3 203 3 20

0

41.5 �239.8 70

C H E C K Y O U R S E L F 7

A developer wants to subdivide a 12.6-acre piece of land into 0.45-acre lots. Howmany lots are possible?

NOTE Notice that we mustadd a zero to the dividend tocomplete the division process.

Andrea’s hourly rate of pay was $5.78.

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. If86.7 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 finalreading.

36,235 Final reading

� 34,563 Initial reading

1672 Miles covered

Next, divide the miles traveled by the number of gallons used. This will give us the milesper gallon.

1 9.28

867805 0780 324 7 017 3 47 3 606 9 36

4 24

Round 19.28 to 19.3 mi/gal.

86.7 �1672.0 00

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C H E C K Y O U R S E L F 8

John starts his trip with an odometer reading of 15,436 and ends with a reading of16,238. 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 pointto 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

305 35 0

3030

0

(b) Divide.

3.785

30078 570 08 508 00

500500

0

100 �378.500

10 �35.30The 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.

Here the dividend is 378.5, whereas the quotient is3.785. The decimal point is now shifted two placesto the left. In this case the divisor, 100, has twozeros.

Example 9 suggests the following rule.

Move the decimal point to the left the same number of places as there arezeros in the power of 10.

Rules and Properties: To Divide a Decimal by a Power of 10

C H E C K Y O U R S E L F 9

Perform each of the following divisions.

(a) 52.6 � 10 (b) 267.9 � 100

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Dividing a Decimal by a Power of 10

Divide.

(a) Shift one place to the left.

(b) Shift two places to the left.

(c) Shift three places to the left.

(d) The decimal after the 85 is implied.

(e) Shift four places to the left.

� 0.023572

� 0 0235.72235.72 � 104

� 0.085

� 0 085.85 � 1000

� 0.03975

� 0 039.7539.75 � 1000

� 0.5753

� 0 57.5357.53 � 100

� 2.73

� 2 7.327.3 � 10

Example 10

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

To convert from millimeters (mm) to meters (m), we divide by 1000. How many metersdoes 3450 mm equal?

3450 mm � 3 450. m Shift three places to the left to divide by 1000.

� 3.450 m

C H E C K Y O U R S E L F 1 0

Divide.

(a) 3.84 � 10 (b) 27.3 � 1000

NOTE As you can see, we mayhave to add zeros to correctlyplace the decimal point.

REMEMBER: 104 is a 1followed by four zeros.

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Recall that the order of operations is always used to simplify a mathematical expressionwith several operations. You should recall the order of operations as the following.

1. Perform any operations enclosed in parentheses.2. Apply any exponents.3. Do any multiplication and division, moving from left to right4. Do any addition and subtraction, moving from left to right.

Rules and Properties: The Order of Operations

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

Example 12

C H E C K Y O U R S E L F 1 2

Simplify each expression.

(a) 6.35 � (0.2 � 8.5)2 � 3.7 (b) 2.52 � (3.57 � 2.14) � 3.2 � 1.5

C H E C K Y O U R S E L F A N S W E R S

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

C H E C K Y O U R S E L F 1 1

A shipment of 1000 notebooks cost a stationery store $658. What was the cost per

notebook to the nearest cent?

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Exercises

Divide.

1. 16.68 � 6 2. 43.92 � 8

3. 1.92 � 4 4. 5.52 � 6

5. 5.48 � 8 6. 2.76 � 8

7. 13.89 � 6 8. 21.92 � 5

9. 185.6 � 32 10. 165.6 � 36

11. 79.9 � 34 12. 179.3 � 55

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 0.658 � 2.8 26. 0.882 � 0.36

0.52 �3.23180.046 �1.587

0.038 �0.81320.27 �1.8495

6.4 �3.6165.2 �11.622

2.9 �13.343.8 �7.22

0.8 �10.840.6 �11.07

76 �26.2252 �13.78

4.5

Name

Section Date

ANSWERS

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3.

4.

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Divide by moving the decimal point.

27. 5.8 � 10 28. 5.1 � 10

29. 4.568 � 100 30. 3.817 � 100

31. 24.39 � 1000 32. 8.41 � 100

33. 6.9 � 1000 34. 7.2 � 1000

35. 7.8 � 102 36. 3.6 � 103

37. 45.2 � 105 38. 57.3 � 104

Divide and round the quotient to the indicated decimal place.

39. 23.8 � 9 tenth 40. 5.27 � 8 hundredth

41. 38.48 � 46 hundredth 42. 3.36 � 36 thousandth

43. 125.4 � 52 tenth 44. 2.563 � 54 thousandth

45. hundredth 46. tenth

47. tenth 48. hundredth

49. hundredth 50. thousandth

Solve the following applications.

51. Cost of CDs. Marv paid $40.41 for three CDs on sale. What was the cost per CD?

52. Contributions. Seven employees of an office donated $172.06 during a charitydrive. What was the average donation?

53. Book purchases. A shipment of 72 paperback books cost a store $190.25. What wasthe average cost per book to the nearest cent?

64.2 �16.33.12 �4.75

5.8 �164.5 �8.415

0.6 �7.6950.7 �1.642

ANSWERS

27.

28.

29.

30.

31.

32.

33.

34.

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36.

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54. Cost. A restaurant bought 50 glasses at a cost of $39.90. What was the cost per glassto the nearest cent?

55. Cost. The cost of a case of 48 items is $28.20. What is the cost of an individual itemto the nearest cent?

56. Office supplies. An office bought 18 hand-held calculators for $284. What was thecost per calculator to the nearest cent?

57. Monthly payments. Al purchased a new refrigerator that cost $736.12 with interestincluded. He paid $100 as a down payment and agreed to pay the remainder in18 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 youmake a down payment of $50 and agree to pay the balance in 12 monthly payments,what will be the amount of each monthly payment?

59. Mileage. In five readings, Lucia’s gas mileage was 32.3, 31.6, 29.5, 27.3, and33.4 miles per gallon (mi/gal). What was her average gas mileage to the nearest tenthof a mile per gallon?

60. Pollution. Pollution index readings were 53.3, 47.8, 41.9, 55.8, 43.7, 41.7, and 52.3for 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 makelabels 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 perhour?

63. Cost per pound. A roast weighing 5.3 pounds (lb) sold for $14.89. Find the cost perpound to the nearest cent.

64. Weight. One nail weighs 0.025 ounce (oz). How many nails are there in 1 lb? (1 lb is16 oz.)

65. Mileage. A family drove 1390 miles (mi), stopping for gas three times. If theypurchased 15.5, 16.2, and 10.8 gallons (gal) of gas, find the number of miles pergallon (the mileage) to the nearest tenth.

66. Mileage. On a trip an odometer changed from 36,213 to 38,319. If 136 gal of gaswere used, find the number of miles per gallon (to the nearest tenth).

ANSWERS

54.

55.

56.

57.

58.

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61.

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67. Conversion. To convert from millimeters (mm) to inches, we can divide by 25.4. Iffilm is 35 mm wide, find the width to the nearest hundredth of an inch.

68. Conversion. To convert from centimeters (cm) to inches, we can divide by 2.54. Therainfall in Paris was 11.8 cm during 1 week. What was that rainfall to the nearesthundredth of an inch?

69. Construction. A road-paving project will cost $23,500. If the cost is to be shared by100 families, how much will each family pay?

70. Conversion. To convert from milligrams (mg) to grams (g), we divide by 1000. Atablet is 250 mg. What is its weight in grams?

71. Conversion. To convert from milliliters (mL) to liters (L), we divide by 1000. If abottle of wine holds 750 mL, what is its volume in liters?

72. Unit cost. A shipment of 100 calculators cost a store $593.88. Find the cost percalculator (to the nearest cent).

73. The blood alcohol content (BAC) of a person who has been drinking is determined bythe formula

A 125-lb person is driving and is stopped by a policewoman on suspicion of drivingunder the influence (DUI). The driver claims that in the past 2 hours he consumedonly six 12-oz bottles of 3.9% beer. If he undergoes a breathalyzer test, what will hisBAC be? Will this amount be under the legal limit for your state?

74. Four brands of soap are available in a local store.

BAC �oz of alcohol � % of alcohol � 0.075 of body wt.

(hours of drinking � 0.015)

ANSWERS

67.

68.

69.

70.

71.

72.

73.

74.

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Brand Ounces Total Price Unit Price

Squeaky Clean 5.5 $0.36Smell Fresh 7.5 0.41Feel Nice 4.5 0.31Look Bright 6.5 0.44

Compute the unit price, and decide which brand is the best buy.

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75. Sophie is a quality control expert. She inspects boxes of #2 pencils. Each pencilweighs 4.4 grams (g). The contents of a box of pencils weigh 66.6 g. If a box islabeled CONTENTS: 16 PENCILS, should Sophie approve the box as meetingspecifications? Explain your answer.

76. Write a plan to determine the number of miles per gallon (mpg) your car (or yourfamily car) gets. Use this plan to determine your car’s actual mpg.

77. Express the width and length of a $1 bill in centimeters (cm). Then express this samelength in millimeters (mm).

78. If the perimeter of a square is 19.2 cm, how long is each side?

79. If the perimeter of an equilateral triangle (all sides have equal length) is 16.8 cm, howlong is each side?

80. If the perimeter of a regular (all sides have equal length) pentagon is 23.5 in., howlong is each side?

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

P � 23.5 in.

P � 16.8 cm

P � 19.2 cm

ANSWERS

75.

76.

77.

78.

79.

80.

81.

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Answers1. 2.78 3. 0.48 5. 0.685 7. 2.315 9. 5.8 11. 2.3513. 0.265 15. 18.45 17. 1.9 19. 2.235 21. 6.85 23. 34.525. 0.235 27. 0.58 29. 0.04568 31. 0.02439 33. 0.006935. 0.078 37. 0.000452 39. 2.6 41. 0.84 43. 2.4 45. 2.3547. 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/gal67. 1.38 in. 69. $235 71. 0.75 L 73. 75.

77. 79. 5.6 cm 81. 11.8 83. 17.0291

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369

Using Your Calculator to Divide Decimals

It would be most surprising if you had reached this point without using your calculator todivide decimals. It is a good way to check your work, and a reasonable way to solveapplications. 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.

C H E C K Y O U R S E L F 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’sintroduce a twist.

Example 2

An Application Involving the Division of Decimals

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 thereis no way you can compute gas mileage to the nearest hundred-millionth mile. How do youdecide where to round off the answer that the calculator gives you? A good rule of thumbis to never report more digits than the least number of digits in any of the numbers that youare given in the problem. In this case, you were given a number with four digits and anotherwith 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 forgas mileage, how precise an answer would you give? The best answer to this questionwould be to give the nearest whole number of miles per gallon: 28 miles per gallon.

C H E C K Y O U R S E L F 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?

C H E C K Y O U R S E L F A N S W E R S

1. 31.7 2. 4.10 yd

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Calculator Exercises

Divide and check.

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

Divide and round to the indicated place.

7. 2.546 � 1.38 hundredth 8. 45.8 � 9.4 tenth

9. 0.5782 � 1.236 thousandth 10. 1.25 � 0.785 hundredth

11. 1.34 � 2.63 two decimal places 12. 12.364 � 4.361 three decimal places

Solve the following applications.

13. Salary. In 1 week, Tom earned $178.30 by working 36.25 hours (h). What was hishourly rate of pay to the nearest cent?

14. Area. An 80.5-acre piece of land is being subdivided into 0.35-acre lots. How manylots are possible in the subdivision?

If the circumference of a circle is known, the diameter can be determined by dividing thecircumference by p. In each of the following, determine the diameter for the givencircumference. Use 3.14 for p.

15. 3.925 cm 16. 8.3838 in.

17. 1.7584 cm 18. 13.0624 mm

Answers1. 3.45 3. 2.89 5. 0.245 7. 1.84 9. 0.468 11. 0.5113. $4.92 15. 1.25 cm 17. 0.56 cm