4.2 adding and subtracting decimals - mcgraw hill …€¦ · adding and subtracting decimals...

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Adding and Subtracting Decimals4.2

4.2 OBJECTIVES

1. Add two or more decimals2. Use addition of decimals to solve application

problems3. Subtract one decimal from another4. Use subtraction of decimals to solve application

problems

303

Working with decimals rather than common fractions makes the basic operations mucheasier. Let’s start by looking at addition. One method for adding decimals is to write thedecimals as common fractions, add, and then change the sum back to a decimal.

It is much more efficient to leave the numbers in decimal form and perform the addition inthe same way as we did with whole numbers. You can use the following rule.

0.34 � 0.52 �34

100�

52

100�

86

100� 0.86

Step 1 Write the numbers being added in column form with their decimalpoints aligned (in line) vertically.

Step 2 Add just as you would with whole numbers.Step 3 Place the decimal point of the sum in line with the decimal points of

the addends.

Step by Step: To Add Decimals

Example 1 illustrates the use of this rule.

Example 1

Adding Decimals

Add 0.13, 0.42, and 0.31.

0.130.42

� 0.310.86

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

Add 0.23, 0.15, and 0.41.

NOTE Placing the decimalpoints in a vertical line ensuresthat we are adding digits of thesame place value.

In adding decimals, you can use the carrying process just as you did in adding wholenumbers. Consider the following.

304 CHAPTER 4 DECIMALS

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Note: The carrying process works with decimals, just as it did with whole numbers,because each place value is again one-tenth the value of the place to its left.


Add 23.546, 0.489, 2.312, and 6.135.

In adding decimals, the numbers may not have the same number of decimal places. Justfill in as many zeros as needed so that all of the numbers added have the same number ofdecimal places.

Recall that adding zeros to the right does not change the value of a decimal. 0.53 is thesame as 0.530.

Let’s see how this is used in Example 3.

Example 3

Adding Decimals

Add 0.53, 4, 2.7, and 3.234.

0.534.2.7

� 3.234

Now fill in the missing zeros, and add as before.

0.5304.0002.700

� 3.23410.464

Note that for a whole number, the decimalis understood to be to its right. So 4 = 4.

Now all the numbers being addedhave three decimal places.


Add 6, 2.583, 4.7, and 2.54.

NOTE Be sure that the decimalpoints are in a vertical line.

Adding Decimals Involving Carrying

Add 0.35, 1.58, and 0.67.

1 2

0.351.58

� 0.672.60

Carries In the hundredths column:5 � 8 � 7 � 20Write 0 and carry 2 to the tenths column.

In the tenths column:2 � 3 � 5 � 6 � 16Write 6 and carry 1 to the ones column.

Example 2

ADDING AND SUBTRACTING DECIMALS SECTION 4.2 305©

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Every day you deal with amounts of money. Because our system of money is a decimalsystem, most problems involving money also involve operations with decimals.


The Higueras kept track of the gasoline they purchased on a recent trip. If theybought 12.4, 13.6, 9.7, 11.8, and 8.3 gal, how much gas did they buy on the trip?

Example 5

An Application of the Addition of Decimals

Andre makes deposits of $3.24, $15.73, $50, $28.79, and $124.38 during May. What is thetotal of his deposits for the month?

$ 3.2415.7350.0028.79

� 124.38$222.14

Simply add the amounts of moneydeposited as decimals. Note thatwe write $50 as $50.00.

The total of deposits for May


Your textbooks for the fall term cost $63.50, $78.95, $43.15, $82, and $85.85. Whatwas the total cost of textbooks for the term?

In Chapter 1, we defined perimeter as the distance around the outside of a straight-edgedshape. Finding the perimeter often requires that we add decimal numbers.

NOTE Because we want a totalamount, addition is used forthe solution.

Many applied problems require working with decimals. For instance, filling up at a gasstation means reading decimal amounts.

Example 4

An Application of the Addition of Decimals

On a trip the Chang family kept track of their gas purchases. If they bought 12.3, 14.2, 10.7,and 13.8 gallons (gal), how much gas did they use on the trip?

12.314.210.7

� 13.851.0 gal


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Much of what we have said about adding decimals is also true of subtraction. To subtractdecimals, we use the following rule:


Manuel intends to build a walkway around the perimeter of his garden (Figure 2).What will the total length of the walkway be?

Figure 2

5.6 m

1.2 m2.3 m

6.4 m

1.2 m2.8 m

5.1 m

8.8 m

Step 1 Write the numbers being subtracted in column form with their decimalpoints aligned vertically.

Step 2 Subtract just as you would with whole numbers.Step 3 Place the decimal point of the difference in line with the decimal

points of the numbers being subtracted.

Step by Step: To Subtract Decimals

Our next example illustrates the use of this rule.

Example 6

An Application Involving the Addition of Decimals

Rachel is going to put a fence around the perimeter of her farm. Figure 1 is a picture of theland, measured in kilometers (km). How much fence does she need to buy?

Figure 1

The perimeter is the sum of the lengths of the sides, so we add those lengths to find the totalfencing needed.

0.16 � 0.36 � 0.45 � 0.62 � 0.61 � 0.26 � 2.46

Rachel needs 2.46 km of fence for the perimeter of her farm.

0.62 km

0.61 km

0.26 km

0.16 km

0.36 km

0.45 km


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In subtracting decimals, as in adding, we can add zeros to the right of the decimal pointso that both decimals have the same number of decimal places.

Example 7

Subtracting a Decimal

Subtract 1.23 from 3.58.

3.58� 1.23

2.35

Subtract in the hundredths, the tenths,and then the ones columns.


Subtract 35.35 � 13.89.

Example 9


(a) Subtract 2.36 from 7.5.

4 1

� 2.365.14

(b) Subtract 3.657 from 9.

8 99111

� 3.6575.343

9.000

07.5 We have added a 0 to 7.5. Next,borrow 1 tenth from the 5 tenthsin the minuend.

In this case, move left to the ones place to begin the borrowingprocess.


Subtract 5 � 2.345.

NOTE When you aresubtracting, align the decimalpoints, then add zeros to theright to align the digits.

NOTE 9 has been rewritten as9.000.

Subtraction of a Decimal That Involves Borrowing

Subtract 1.86 from 6.54.

5141

� 1.864.68

6.54


Subtract 9.87 � 5.45.

Because each place value is one-tenth the value of the place to its left, borrowing, whenyou are subtracting decimals, works just as it did in subtracting whole numbers.

Example 8

Here, borrow in the tenths and onesplaces to do the subtraction.


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An Application of the Subtraction of a Decimal Number

At the grocery store, Sally buys a roast that is marked $12.37. She pays for her purchasewith a $20 bill. How much change does she get?

$20.00� 12.37

$ 7.63

Sally will receive $7.63 in change after her purchase.

Example 11

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

A stereo system that normally sells for $549.50 is discounted (or marked down) to$499.95 for a sale. What is the savings?

NOTE Sally’s change will be thedifference between the price ofthe roast and the $20 paid. Wemust use subtraction for thesolution.

Add zeros to write $20 as $20.00.Then subtract as before.

We can apply the subtraction methods of Examples 7 to 9 in solving applicationsinvolving decimals.

Example 10

An Application of the Subtraction of a Decimal Number

Jonathan was 98.3 centimeters (cm) tall on his sixth birthday. On his seventh birthday hewas 104.2 cm. How much did he grow during the year?

104.2 cm� 98.3 cm

5.9 cm

Jonathan grew 5.9 cm during the year.

NOTE We want to find thedifference between the twomeasurements, so we subtract.


A car’s highway mileage before a tune-up was 28.8 miles per gallon (mi/gal). Afterthe tune-up it measured 30.1 mi/gal. What was the increase in mileage?

The same method can be used in working with money.

Keeping your checkbook requires addition and subtraction of decimal numbers.


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To keep a running balance, we add the deposits and subtract the checks.

Beginning balance $234.15

Check # 301 23.88Balance 210.27

Check # 302 38.98Balance 171.29

Check # 303 114.66Balance 56.63

Deposit 175.75Balance 232.38

Check # 304 212.55Ending balance 19.83


For the following check register, add the deposits and subtract the checks to findthe balance.


Check # 401 19.75(a) Balance _______

Check # 402 56.88(b) Balance _______

Check # 403 117.59(c) Balance _______

Deposit 224.67(d) Balance _______

Check # 404 411.48(e) Ending balance _______

Example 12

An Application Involving the Addition and Subtraction of Decimals

For the following check register, find the running balance.


Check # 301 23.88Balance _______

Check # 302 38.98Balance _______

Check # 303 114.66Balance _______

Deposit 175.75Balance _______

Check # 304 212.55Ending balance _______


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

1. 0.79 2. 32.482 3. 6.000 4. 55.8 gal 5. $353.452.5834.700

� 2.54015.823

6. 33.4 m 7. 4.42 8. 21.46 9. 2.655 10. 1.3 mi/gal

11. $49.55 12. (a) $378.25; (b) $321.37; (c) $203.78; (d) $428.45; (e) $16.97

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Exercises

Add.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 0.43 � 0.8 � 0.561 14. 1.25 � 0.7 � 0.259

15. 5 � 23.7 � 8.7 � 9.85 16. 28.3 � 6 � 8.76 � 3.8

17. 25.83 � 1.7 � 3.92 18. 4.8 � 32.59 � 4.76

19. 42.731 � 1.058 � 103.24 20. 27.4 � 213.321 � 39.38

1.33615.6857

7.9� 0.85

25.35826.51.898

� 0.69

8.6252.45

� 12.6

13.587.239

� 1.5

23.845� 7.29

5.28� 19.455

0.502.99

� 24.8

0.624.23

� 12.5

2.485� 1.25

1.045� 0.23

2.59� 0.63

0.28� 0.79

4.2

Name

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

311

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In exercises 21 to 24, use decimal square shading to represent the addition process. Shadeeach square and the total.

21.

22.

23.

24.

ANSWERS

21.

22.

23.

24.

312

� �

.32(Shade)

.15(Shade) (Shade Total)

� �

.43 .05

� �

.21 .25

� �

.56 .11

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Subtract.

25. 26.

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

39. Subtract 2.87 from 6.84. 40. Subtract 3.69 from 10.57.

41. Subtract 7.75 from 9.4. 42. Subtract 5.82 from 12.

43. Subtract 0.24 from 5. 44. Subtract 8.7 from 16.32.

15� 8.85

12� 5.35

45.6�8.75

7.02� 4.7

8.59�5.6

3.82� 1.565

7.83� 5.2

35.8� 7.45

40.092� 21.595

17.134� 3.502

48.03� 19.95

23.81� 6.57

5.68� 2.65

0.85� 0.59

ANSWERS

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

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45. Add twenty-three hundredths, five tenths, and two hundred sixty-eight thousandths.

46. Add seven tenths, four hundred fifty-eight thousandths, and fifty-six hundredths.

47. Add five and three tenths, seventy-five hundredths, twenty and thirteen hundredths,and twelve and seven tenths.

48. Add thirty-eight and nine tenths, five and fifty-eight hundredths, seven, and fifteenand eight tenths.

Solve the following applications.

49. Gas purchase. On a 3-day trip, Dien bought 12.7, 15.9, and 13.8 gallons (gal) ofgas. How many gallons of gas did he buy?

50. Distance. Felix ran 2.7 miles (mi) on Monday, 1.9 mi on Wednesday, and 3.6 mi onFriday. How far did he run during the week?

51. Rainfall was recorded in centimeters (cm) during the winter months as indicated onthe bar graph.

(a) How much rain fell during those months?

(b) How much more rain fell in December than in February?

52. Total length. A metal fitting has three sections, with lengths 2.5, 1.775, and1.45 inches (in.). What is the total length of the fitting?

53. Total expenses. Nicole had the following expenses on a business trip: gas, $45.69;food, $123; lodging, $95.60; and parking and tolls, $8.65. What were her totalexpenses during the trip?

Jan.Dec. Feb.

3.2

5.384.79

Rai

nfal

l (cm

)

ANSWERS

45.

46.

47.

48.

49.

50.

51.

52.

53.

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54. Textbook costs. Hok Sum’s textbooks for one quarter cost $29.95, $47, $52.85,$33.35, and $10. What was his total cost for textbooks?

55. Checking. Jordan wrote checks of $50, $11.38, $112.57, and $9.73 during a singleweek. What was the total amount of the checks he wrote?

56. The deposit slip shown indicates the amounts that made up a deposit Peter Rabbitmade. What was the total amount of his deposit?

57. Perimeter. Lupe is putting a fence around her yard. Her yard is rectangular andmeasures 8.16 yards (yd) long and 12.68 yd wide. How much fence should Lupepurchase?

58. Perimeter. Find the perimeter of the figure given below.

59. Fencing. The figure below gives the distance in miles (mi) of the boundary sectionsaround a ranch. How much fencing is needed for the property?

60. Discounts. A television set selling for $399.50 is discounted (or marked down) to$365.75. What is the savings?

1.903 mi 2.321 mi

2.887 mi

2.417 mi

2.007 mi

10.5 ft

6.3 ft

3.2 ft

7.4 ft5.8 ft

(OR TOTAL FROM OTHER SIDE)

SUB TOTAL

$

* LESS CASH RECEIVED

√ CASH 75.35

58.00

7.89

100.00

.

.

.

3–50/310

DEPOSITS MAY NOT BE AVAILABLE FOR IMMEDIATE WITHDRAWALDATE

SIGN HERE FOR CASH RECEIVED (IF REQUIRED) *

Briarpatch National Bank

DEPOSIT TICKET

Peter Rabbit123 East Derbunny St.

ANSWERS

54.

55.

56.

57.

58.

59.

60.

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61. Tubing radii. The outer radius of a piece of tubing is 2.8325 inches (in.). The innerradius is 2.775 in. What is the thickness of the wall of the tubing?

62. Temperature. If normal body temperature is 98.6°F and a person is running atemperature of 101.3°F, how much is that temperature above normal?

63. Amount of change. You pay your hotel bill of $84.58 with two $50 traveler’schecks. What change will you receive?

64. Perimeter. Given the following figure, find dimension a.

65. Credit cards. You make charges of $37.25, $8.78, and $53.45 on a credit card. If youmake a payment of $73.50, how much do you still owe?

66. Distance. At the start of a trip, Laura notes that her odometer (mileage indicator)reads 15,785.3 miles (mi). At the end of the trip it reads 16,479.8 mi. How far did shedrive?

67. Rainfall. Rainfall for the first 3 months of 2001 was recorded at 2.73, 1.41, and1.48 inches (in.). If the normal rainfall for that period is 6.51 in., by how much wasthe 2001 amount above or below normal?

68. Checkbook balance. For the following check register, find the running balance.


Check # 501 $425.69

Balance

Check # 502 $ 56.34

Balance

Check # 503 $ 41.89

Balance

Deposit $123.91

Balance

Check # 504 $356.98

Ending balance

0.65in. 0.375

in.

a

2.000 in.

ANSWERS

61.

62.

63.

64.

65.

66.

67.

68.

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Check # 601 $199.29

Balance

Service charge $ 18.00

Balance

Check # 602 $ 85.78

Balance

Deposit $250.45

Balance

Check # 603 $201.24

Ending balance



Check # 678 $175.63

Balance

Check # 679 $ 56.92

Balance

Deposit $121.12

Balance

Check # 680 $345.99

Ending balance



Check # 821 $ 234.99

Balance

Check # 822 $ 555.77

Balance

Deposit $ 126.77

Balance

Check # 823 $ 53.89

Ending balance

Estimation can be a useful tool when working with decimal fractions. To estimate a sum,one approach is to round the addends to the nearest whole number and add for your estimate.

ANSWERS

69.

70.

71.

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For instance, to estimate the sum below:

Round

19.8 203.5 4

24.2 24� 10.4 � 10

58

Use estimation to solve the following applications.

72. Alem’s restaurant bill is pictured below. Estimate his total by rounding to the nearestdollar.

73. Car maintenance. Your bill for a car tune-up includes $7.80 for oil, $5.90 for a filter,$3.40 for spark plugs, $4.10 for points, and $28.70 for labor. Estimate your total cost.

74. Payroll. The payroll at a car repair shop for 1 week was $456.73, utilities were$123.89, advertising was $212.05, and payments to distributors were $415.78.Estimate the amount spent in 1 week.

75. Expenses. On a recent business trip your expenses were $343.78 for airfare, $412.78for lodging, $148.89 for food, and $102.15 for other items. Estimate your totalexpenses.

76. Following are charges on a credit card account:$8.97, $32.75, $15.95, $67.32, $215.78, $74.95, $83.90, and $257.28

(a) Estimate the total bill for the charges by rounding each number to the nearestdollar and adding the results.

(b) Estimate the total bill by adding the charges and then rounding to the nearestdollar.

(c) What are the advantages and disadvantages of the methods in (a) and (b)?

ANSWERS

72.

73.

74.

75.

76.

318

Add for the estimate.

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77. Find the next number in the following sequence: 3.125, 3.375, 3.625, . . .

Recall that a magic square is one in which the sum of every row, column, and diagonal isthe same. Complete the magic squares below.

78. 79.

80. Find the next two numbers in each of the following sequences:

(a) 0.75 0.62 0.5 0.39

(b) 1.0 1.5 0.9 3.5 0.8

81. (a) Determine the average amount of rainfall (to the nearest hundredth of an inch) inyour town or city for each of the past 24 months.

(b) Determine the difference in rainfall amounts per month for each month from1 year to the next.

2.4 7.2

10.8

4.8

1.6 1.2

1

0.8

ANSWERS

77.

78.

79.

80.

81.

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Answers1. 1.07 3. 1.275 5. 17.35 7. 24.735 9. 22.319 11. 34.4462

22

13. 1.791 15. 17. 31.45 19. 147.029 21. 0.47

23. 0.67 25. 0.26 27. 29. 13.632 31. 28.35

33. 2.255 35. 2.32 37. 39. 3.97 41. 1.65 43. 4.76

1

45. 0.998 47. 49. 42.4 gal 51. (a) 13.37 cm; (b) 0.59 cm

53. $272.94 55. $183.68 57. 41.68 yd 59. 11.535 mi61. 0.0575 in. 63. $15.42 65. $25.98 67. 0.89 in. below normal69. End balance: $202.14 71. End balance: $627.35 73. $5075. $1008 77. 79. 81.

2.4 8.4 7.2

10.8 6 1.2

4.8 3.6 9.6

5.300.75

20.13� 12.70

38.88

12.00� 5.35

6.65

11 71

23.81� 6.57

17.24

5.0023.708.70

� 9.8547.25

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321

Using Your Calculator to Addand Subtract Decimals

Entering decimals in your calculator is similar to entering whole numbers. There is just onedifference: The decimal point key is used to place the decimal point as you enter thenumber.

•REMEMBER: The reason forthis book is to help you reviewthe basic skills of arithmetic. Weare using these calculatorsections to show you how thecalculator can be helpful as atool. Unless your instructor saysotherwise, you should be usingyour calculator only on theproblems in these specialsections.

Example 1


Enter 14.367 on your calculator.

Entering a Decimal Number into a Calculator

To enter 0.678, press

Display 0.678

876•

Example 2

Example 3


Enter 0.398 on your calculator.

NOTE You don’t have to pressthe 0 key for the digit 0 to theleft of the decimal point.

The process of adding and subtracting decimals on your calculator is the same as we sawearlier in the sections about adding and subtracting whole numbers.

Entering a Decimal Number into a Calculator

To enter 12.345, press

Display 12.345

543•21

Adding Decimals

To add 2.567 � 0.89 � 5, enter

2.567 0.89 5

Display 8.457

��

NOTE You don’t need to worryabout the fact that the decimalsdon’t have the same number ofplaces. If a whole number isinvolved, just enter that wholenumber. The decimal point keyis not necessary.


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Subtraction of decimals on the calculator is similar.


Add on your calculator

5.39 � 0.68 � 9.7


Subtract on your calculator

16.3 � 7.895


Use your calculator to find

52.8 � 36.9 � 15.87 � 9.36

Example 4


To subtract 4.2 � 2.875, enter

4.2 2.875

Display 1.325

��

Often both addition and subtraction are involved in a calculation. In this case, just enterthe decimals and the operation signs, � or �, as they appear in the problem.

Example 5

Adding and Subtracting Decimals

To find 23.7 � 5.2 � 3.87 � 2.341, enter

23.7 5.2 3.87 2.341

Display 20.029

��

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

1. 14.367 2. 0.398 3. 15.77 4. 8.405 5. 22.41

NOTE Again there aredifferences in the operation ofvarious calculators. Try thisproblem on yours to check thatits operation sequence iscorrect.

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Section Date

ANSWERS

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2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

323

Calculator Exercises

Solve the following exercises using your calculator.

1. 5.87 � 3.6 � 9.25 2. 3.456 � 10 � 2.8 � 5.62

3. 28.21 � 387.6 � 3935.21 4. 10,345.2 � 2308.35 � 153.58

5. 4.59 � 2.389 6. 19.375 � 14.2

7. 27.85 � 3.45 � 2.8 8. 8.8 � 4.59 � 2.325 � 8.5

9. 14 � 3.2 � 9.35 � 3.375 10. 8.7675 � 2.8 � 3.375 � 6

Solve the following applications using your calculator.

11. Checking balance. Your checking account has a balance of $532.89. You writechecks of $50, $27.54, and $134.75 and make a deposit of $50. What is your endingbalance?

12. Checking balance. Your checking account has a balance of $278.45. You makedeposits of $200 and $135.46. You write checks for $389.34, $249, and $53.21. Whatis your ending balance? Be careful with this problem. A negative balance means thatyour account is overdrawn.

13. Car costs. You buy a car for $9548. If you buy additional options for $85.75,$236, and $95.50 and make a down payment of $1500, how much do you owe onthe car?

14. Profit. A small store makes a profit of $934.20 in the first week of a given month,$1238.34 in the second week, and $853 in the third week. If the goal is a profit of$4000 for the month, what profit must the store make during the remainder of themonth?

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324

Answers1. 18.72 3. 4351.02 5. 2.201 7. 21.6 9. 4.475 11. $370.6013. $8465.25

4.2 adding and subtracting decimals - mcgraw hill …€¦ · adding and subtracting decimals...

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