4.2
Adding and Subtracting Decimals
4.2
OBJECTIVES
1. Add two or more decimals
2. Use addition of decimals to solve application
problems
3. Subtract one decimal from another
4. Use subtraction of decimals to solve application
problems
Working with decimals rather than common fractions makes the basic operations much
easier. Let’s start by looking at addition. One method for adding decimals is to write the
decimals as common fractions, add, and then change the sum back to a decimal.
0.34 0.52 52
86
34
0.86
100
100
100
It is much more efficient to leave the numbers in decimal form and perform the addition in
the same way as we did with whole numbers. You can use the following rule.
Step by Step: To Add Decimals
Step 1 Write the numbers being added in column form with their decimal
points 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.
Example 1 illustrates the use of this rule.
Example 1
Adding Decimals
Add 0.13, 0.42, and 0.31.
© 2001 McGraw-Hill Companies
NOTE Placing the decimal
points in a vertical line ensures
that we are adding digits of the
same place value.
0.13
0.42
0.31
0.86
CHECK YOURSELF 1
Add 0.23, 0.15, and 0.41.
In adding decimals, you can use the carrying process just as you did in adding whole
numbers. Consider the following.
303
304
CHAPTER 4
DECIMALS
Example 2
Adding Decimals Involving Carrying
Add 0.35, 1.58, and 0.67.
12
Carries
0.35
1.58
0.67
2.60
In the hundredths column:
5 8 7 20
Write 0 and carry 2 to the tenths column.
In the tenths column:
2 3 5 6 16
Write 6 and carry 1 to the ones column.
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.
CHECK YOURSELF 2
Add 23.546, 0.489, 2.312, and 6.135.
In adding decimals, the numbers may not have the same number of decimal places. Just
fill in as many zeros as needed so that all of the numbers added have the same number of
decimal places.
Recall that adding zeros to the right does not change the value of a decimal. 0.53 is the
same 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.
0.53
4.
2.7
3.234
Note that for a whole number, the decimal
is understood to be to its right. So 4 = 4.
Now fill in the missing zeros, and add as before.
0.530
4.000
2.700
3.234
10.464
Now all the numbers being added
have three decimal places.
CHECK YOURSELF 3
Add 6, 2.583, 4.7, and 2.54.
© 2001 McGraw-Hill Companies
NOTE Be sure that the decimal
points are in a vertical line.
ADDING AND SUBTRACTING DECIMALS
SECTION 4.2
305
Many applied problems require working with decimals. For instance, filling up at a gas
station 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?
NOTE Because we want a total
amount, addition is used for
the solution.
12.3
14.2
10.7
13.8
51.0 gal
CHECK YOURSELF 4
The Higueras kept track of the gasoline they purchased on a recent trip. If they
bought 12.4, 13.6, 9.7, 11.8, and 8.3 gal, how much gas did they buy on the trip?
Every day you deal with amounts of money. Because our system of money is a decimal
system, most problems involving money also involve operations with decimals.
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 the
total of his deposits for the month?
3.24
15.73
50.00
28.79
124.38
$222.14
© 2001 McGraw-Hill Companies
$
Simply add the amounts of money
deposited as decimals. Note that
we write $50 as $50.00.
The total of deposits for May
CHECK YOURSELF 5
Your textbooks for the fall term cost $63.50, $78.95, $43.15, $82, and $85.85. What
was the total cost of textbooks for the term?
In Chapter 1, we defined perimeter as the distance around the outside of a straight-edged
shape. Finding the perimeter often requires that we add decimal numbers.
CHAPTER 4
DECIMALS
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 the
land, measured in kilometers (km). How much fence does she need to buy?
0.45 km
0.36 km
0.16 km
0.62 km
0.26 km
0.61 km
Figure 1
The perimeter is the sum of the lengths of the sides, so we add those lengths to find the total
fencing 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.
CHECK YOURSELF 6
Manuel intends to build a walkway around the perimeter of his garden (Figure 2).
What will the total length of the walkway be?
5.6 m
2.3 m
1.2 m
6.4 m
8.8 m
2.8 m
1.2 m
5.1 m
Figure 2
Much of what we have said about adding decimals is also true of subtraction. To subtract
decimals, we use the following rule:
Step by Step: To Subtract Decimals
Step 1 Write the numbers being subtracted in column form with their decimal
points 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.
Our next example illustrates the use of this rule.
© 2001 McGraw-Hill Companies
306
ADDING AND SUBTRACTING DECIMALS
SECTION 4.2
307
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.
CHECK YOURSELF 7
Subtract 9.87 5.45.
Because each place value is one-tenth the value of the place to its left, borrowing, when
you are subtracting decimals, works just as it did in subtracting whole numbers.
Example 8
Subtraction of a Decimal That Involves Borrowing
Subtract 1.86 from 6.54.
5141
6.54
1.86
4.68
Here, borrow in the tenths and ones
places to do the subtraction.
CHECK YOURSELF 8
Subtract 35.35 13.89.
In subtracting decimals, as in adding, we can add zeros to the right of the decimal point
so that both decimals have the same number of decimal places.
Example 9
Subtracting a Decimal
(a) Subtract 2.36 from 7.5.
41
NOTE When you are
© 2001 McGraw-Hill Companies
subtracting, align the decimal
points, then add zeros to the
right to align the digits.
7.5 0
2.36
5.14
We have added a 0 to 7.5. Next,
borrow 1 tenth from the 5 tenths
in the minuend.
(b) Subtract 3.657 from 9.
8 99
111
NOTE 9 has been rewritten as
9.000.
9.000
3.657
5.343
In this case, move left to the
ones place to begin the borrowing
process.
CHECK YOURSELF 9
Subtract 5 2.345.
308
CHAPTER 4
DECIMALS
We can apply the subtraction methods of Examples 7 to 9 in solving applications
involving 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 he
was 104.2 cm. How much did he grow during the year?
NOTE We want to find the
difference between the two
measurements, so we subtract.
104.2 cm
98.3 cm
5.9 cm
Jonathan grew 5.9 cm during the year.
CHECK YOURSELF 10
A car’s highway mileage before a tune-up was 28.8 miles per gallon (mi/gal). After
the tune-up it measured 30.1 mi/gal. What was the increase in mileage?
The same method can be used in working with money.
Example 11
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 purchase
with a $20 bill. How much change does she get?
difference between the price of
the roast and the $20 paid. We
must use subtraction for the
solution.
$20.00
12.37
$ 7.63
Add zeros to write $20 as $20.00.
Then subtract as before.
Sally will receive $7.63 in change after her purchase.
CHECK YOURSELF 11
A stereo system that normally sells for $549.50 is discounted (or marked down) to
$499.95 for a sale. What is the savings?
Keeping your checkbook requires addition and subtraction of decimal numbers.
© 2001 McGraw-Hill Companies
NOTE Sally’s change will be the
ADDING AND SUBTRACTING DECIMALS
SECTION 4.2
309
Example 12
An Application Involving the Addition and Subtraction of Decimals
For the following check register, find the running balance.
Beginning balance
Check # 301
Balance
Check # 302
Balance
Check # 303
Balance
Deposit
Balance
Check # 304
Ending balance
$234.15
23.88
_______
38.98
_______
114.66
_______
175.75
_______
212.55
_______
To keep a running balance, we add the deposits and subtract the checks.
Beginning balance
$234.15
Check # 301
Balance
23.88
210.27
Check # 302
Balance
38.98
171.29
Check # 303
Balance
114.66
56.63
Deposit
Balance
175.75
232.38
Check # 304
Ending balance
212.55
19.83
CHECK YOURSELF 12
For the following check register, add the deposits and subtract the checks to find
the balance.
© 2001 McGraw-Hill Companies
(a)
(b)
(c)
(d)
(e)
Beginning balance
Check # 401
Balance
Check # 402
Balance
Check # 403
Balance
Deposit
Balance
Check # 404
Ending balance
$398.00
19.75
_______
56.88
_______
117.59
_______
224.67
_______
411.48
_______
CHAPTER 4
DECIMALS
CHECK YOURSELF ANSWERS
1. 0.79
6. 33.4 m
11. $49.55
2. 32.482
3. 6.000
2.583
4.700
2.540
15.823
4. 55.8 gal
5. $353.45
7. 4.42
8. 21.46
9. 2.655
10. 1.3 mi/gal
12. (a) $378.25; (b) $321.37; (c) $203.78; (d) $428.45; (e) $16.97
© 2001 McGraw-Hill Companies
310
Name
4.2
Exercises
Section
Date
Add.
1.
0.28
0.79
2.
2.59
0.63
ANSWERS
1.
2.
3.
1.045
0.23
4.
2.485
1.25
3.
4.
5.
0.62
4.23
12.5
6.
0.50
2.99
24.8
5.
6.
7.
7.
5.28
19.455
8.
23.845
7.29
8.
9.
10.
9.
13.58
7.239
1.5
10.
8.625
2.45
12.6
11.
12.
13.
11.
25.3582
6.5
1.898
0.69
12.
1.336
15.6857
7.9
0.85
14.
15.
16.
17.
13. 0.43 0.8 0.561
14. 1.25 0.7 0.259
18.
© 2001 McGraw-Hill Companies
19.
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
20.
311
ANSWERS
21.
In exercises 21 to 24, use decimal square shading to represent the addition process. Shade
each square and the total.
22.
21.
23.
24.
.32
(Shade)
.15
(Shade)
(Shade Total)
22.
.21
.25
23.
.56
.11
.43
312
.05
© 2001 McGraw-Hill Companies
24.
ANSWERS
25.
Subtract.
25.
0.85
0.59
26.
5.68
2.65
26.
27.
28.
27.
23.81
6.57
28.
48.03
19.95
29.
30.
31.
29.
17.134
3.502
30.
40.092
21.595
32.
33.
34.
31.
35.8
7.45
32.
7.83
5.2
35.
36.
37.
33.
3.82
1.565
34.
8.59
5.6
38.
39.
40.
35.
7.02
4.7
36.
45.6
8.75
41.
42.
43.
© 2001 McGraw-Hill Companies
37.
12
5.35
38.
15
8.85
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.
44.
313
ANSWERS
45.
45. Add twenty-three hundredths, five tenths, and two hundred sixty-eight thousandths.
46.
47.
46. Add seven tenths, four hundred fifty-eight thousandths, and fifty-six hundredths.
48.
49.
47. Add five and three tenths, seventy-five hundredths, twenty and thirteen hundredths,
and twelve and seven tenths.
50.
51.
48. Add thirty-eight and nine tenths, five and fifty-eight hundredths, seven, and fifteen
52.
and eight tenths.
53.
Solve the following applications.
49. Gas purchase. On a 3-day trip, Dien bought 12.7, 15.9, and 13.8 gallons (gal) of
gas. 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 on
Friday. How far did he run during the week?
51. Rainfall was recorded in centimeters (cm) during the winter months as indicated on
the bar graph.
(a) How much rain fell during those months?
(b) How much more rain fell in December than in February?
5.38
Rainfall (cm)
4.79
Dec.
Jan.
Feb.
52. Total length. A metal fitting has three sections, with lengths 2.5, 1.775, and
1.45 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 total
expenses during the trip?
314
© 2001 McGraw-Hill Companies
3.2
ANSWERS
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?
54.
55.
56.
55. Checking. Jordan wrote checks of $50, $11.38, $112.57, and $9.73 during a single
57.
week. What was the total amount of the checks he wrote?
58.
56. The deposit slip shown indicates the amounts that made up a deposit Peter Rabbit
59.
made. What was the total amount of his deposit?
60.
DEPOSIT TICKET
Peter Rabbit
123 East Derbunny St.
√
CASH
75.35
3–50/310
58.00
7.89
DATE
DEPOSITS MAY NOT BE AVAILABLE FOR IMMEDIATE WITHDRAWAL
100.00
(OR TOTAL FROM OTHER SIDE)
SIGN HERE FOR CASH RECEIVED (IF REQUIRED) *
Briarpatch National Bank
SUB TOTAL
.
* LESS CASH
RECEIVED
.
$
.
57. Perimeter. Lupe is putting a fence around her yard. Her yard is rectangular and
measures 8.16 yards (yd) long and 12.68 yd wide. How much fence should Lupe
purchase?
58. Perimeter. Find the perimeter of the figure given below.
6.3 ft
10.5 ft
3.2 ft
7.4 ft
5.8 ft
59. Fencing. The figure below gives the distance in miles (mi) of the boundary sections
around a ranch. How much fencing is needed for the property?
© 2001 McGraw-Hill Companies
1.903 mi
2.321 mi
2.007 mi
2.887 mi
2.417 mi
60. Discounts. A television set selling for $399.50 is discounted (or marked down) to
$365.75. What is the savings?
315
ANSWERS
61.
61. Tubing radii. The outer radius of a piece of tubing is 2.8325 inches (in.). The inner
radius is 2.775 in. What is the thickness of the wall of the tubing?
62.
63.
62. Temperature. If normal body temperature is 98.6°F and a person is running a
temperature of 101.3°F, how much is that temperature above normal?
64.
65.
63. Amount of change. You pay your hotel bill of $84.58 with two $50 traveler’s
66.
checks. What change will you receive?
67.
64. Perimeter. Given the following figure, find dimension a.
68.
0.65
in.
0.375
in.
a
2.000 in.
65. Credit cards. You make charges of $37.25, $8.78, and $53.45 on a credit card. If you
make 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 she
drive?
67. Rainfall. Rainfall for the first 3 months of 2001 was recorded at 2.73, 1.41, and
1.48 inches (in.). If the normal rainfall for that period is 6.51 in., by how much was
the 2001 amount above or below normal?
68. Checkbook balance. For the following check register, find the running balance.
Balance
Check # 502
$ 56.34
Balance
Check # 503
$ 41.89
Balance
Deposit
$123.91
Balance
Check # 504
$356.98
Ending balance
316
$896.74
$425.69
© 2001 McGraw-Hill Companies
Beginning balance
Check # 501
ANSWERS
69. Checkbook balance. For the following check register, find the running balance.
Beginning balance
Check # 601
$456.00
$199.29
Balance
Service charge
$ 18.00
Balance
Check # 602
$ 85.78
Balance
Deposit
$250.45
Balance
Check # 603
$201.24
69.
70.
71.
Ending balance
70. Checkbook balance. For the following check register, find the running balance.
Beginning balance
Check # 678
$589.21
$175.63
Balance
Check # 679
$ 56.92
Balance
Deposit
$121.12
Balance
Check # 680
$345.99
Ending balance
71. Checkbook balance. For the following check register, find the running balance.
© 2001 McGraw-Hill Companies
Beginning balance
Check # 821
$1345.23
$ 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.
317
ANSWERS
72.
For instance, to estimate the sum below:
73.
Round
19.8
3.5
24.2
10.4
74.
75.
20
4
24
10
58
Add for
the estimate.
76.
Use estimation to solve the following applications.
72. Alem’s restaurant bill is pictured below. Estimate his total by rounding to the nearest
dollar.
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, $412.78
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 nearest
dollar and adding the results.
(b) Estimate the total bill by adding the charges and then rounding to the nearest
dollar.
(c) What are the advantages and disadvantages of the methods in (a) and
(b)?
318
© 2001 McGraw-Hill Companies
for lodging, $148.89 for food, and $102.15 for other items. Estimate your total
expenses.
ANSWERS
77. Find the next number in the following sequence: 3.125, 3.375, 3.625, . . .
77.
78.
79.
Recall that a magic square is one in which the sum of every row, column, and diagonal is
the same. Complete the magic squares below.
80.
81.
78.
79.
1.6
1.2
2.4
1
7.2
10.8
0.8
4.8
80. Find the next two numbers in each of the following sequences:
(a) 0.75
(b) 1.0
0.62
1.5
0.5
0.9
0.39
3.5
0.8
81. (a) Determine the average amount of rainfall (to the nearest hundredth of an inch) in
your town or city for each of the past 24 months.
© 2001 McGraw-Hill Companies
(b) Determine the difference in rainfall amounts per month for each month from
1 year to the next.
319
Answers
1. 1.07
3. 1.275
5. 17.35
7. 24.735
9. 22.319
11. 34.4462
22
13. 1.791
23. 0.67
15.
17. 31.45
5.00
23.70
8.70
9.85
47.25
25. 0.26
27.
19. 147.029
29. 13.632
1 1 71
21. 0.47
31. 28.35
23.81
6.57
17.24
33. 2.255
35. 2.32
37.
12.00
5.35
6.65
39. 3.97
41. 1.65
43. 4.76
1
5.30
49. 42.4 gal
51. (a) 13.37 cm; (b) 0.59 cm
0.75
20.13
12.70
38.88
$272.94
55. $183.68
57. 41.68 yd
59. 11.535 mi
0.0575 in.
63. $15.42
65. $25.98
67. 0.89 in. below normal
End balance: $202.14
71. End balance: $627.35
73. $50
$1008
77.
79.
81.
45. 0.998
2.4
8.4
7.2
10.8
6
1.2
4.8
3.6
9.6
© 2001 McGraw-Hill Companies
53.
61.
69.
75.
47.
320
Using Your Calculator to Add
and Subtract Decimals
REMEMBER: The reason for
this book is to help you review
the basic skills of arithmetic. We
are using these calculator
sections to show you how the
calculator can be helpful as a
tool. Unless your instructor says
otherwise, you should be using
your calculator only on the
problems in these special
sections.
Entering decimals in your calculator is similar to entering whole numbers. There is just one
difference: The decimal point key • is used to place the decimal point as you enter the
number.
Example 1
Entering a Decimal Number into a Calculator
To enter 12.345, press
1 2
• 3 4 5
Display
12.345
CHECK YOURSELF 1
Enter 14.367 on your calculator.
Example 2
Entering a Decimal Number into a Calculator
To enter 0.678, press
NOTE You don’t have to press
the 0 key for the digit 0 to the
left of the decimal point.
• 6 7 8
Display
0.678
CHECK YOURSELF 2
Enter 0.398 on your calculator.
© 2001 McGraw-Hill Companies
The process of adding and subtracting decimals on your calculator is the same as we saw
earlier in the sections about adding and subtracting whole numbers.
Example 3
Adding Decimals
NOTE You don’t need to worry
about the fact that the decimals
don’t have the same number of
places. If a whole number is
involved, just enter that whole
number. The decimal point key
is not necessary.
To add 2.567 0.89 5, enter
2.567 0.89 5 Display
8.457
321
322
CHAPTER 4
DECIMALS
CHECK YOURSELF 3
Add on your calculator
5.39 0.68 9.7
Subtraction of decimals on the calculator is similar.
Example 4
Subtracting a Decimal
To subtract 4.2 2.875, enter
4.2 2.875 Display
1.325
CHECK YOURSELF 4
Subtract on your calculator
16.3 7.895
Often both addition and subtraction are involved in a calculation. In this case, just enter
the decimals and the operation signs, or , as they appear in the problem.
Example 5
Adding and Subtracting Decimals
differences in the operation of
various calculators. Try this
problem on yours to check that
its operation sequence is
correct.
To find 23.7 5.2 3.87 2.341, enter
23.7 5.2 3.87 2.341 Display
20.029
CHECK YOURSELF 5
Use your calculator to find
52.8 36.9 15.87 9.36
CHECK YOURSELF ANSWERS
1. 14.367
2. 0.398
3. 15.77
4. 8.405
5. 22.41
© 2001 McGraw-Hill Companies
NOTE Again there are
Name
Calculator Exercises
Section
Date
Solve the following exercises using your calculator.
1. 5.87 3.6 9.25
2. 3.456 10 2.8 5.62
ANSWERS
1.
2.
3. 28.21 387.6 3935.21
4. 10,345.2 2308.35 153.58
3.
4.
5. 4.59 2.389
6. 19.375 14.2
5.
6.
7. 27.85 3.45 2.8
8. 8.8 4.59 2.325 8.5
7.
8.
9. 14 3.2 9.35 3.375
10. 8.7675 2.8 3.375 6
9.
10.
11.
Solve the following applications using your calculator.
12.
11. Checking balance. Your checking account has a balance of $532.89. You write
checks of $50, $27.54, and $134.75 and make a deposit of $50. What is your ending
balance?
13.
14.
12. Checking balance. Your checking account has a balance of $278.45. You make
deposits of $200 and $135.46. You write checks for $389.34, $249, and $53.21. What
is your ending balance? Be careful with this problem. A negative balance means that
your account is overdrawn.
© 2001 McGraw-Hill Companies
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 on
the 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 the
month?
323
Answers
5. 2.201
7. 21.6
9. 4.475
11. $370.60
© 2001 McGraw-Hill Companies
1. 18.72
3. 4351.02
13. $8465.25
324