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Addition and Subtraction of Decimals

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to add and subtract decimals. By the end of the module students should understand the method used for adding and subtracting decimals, be able to add and subtract decimals and be able to use the calculator to add and subtract decimals.

Section Overview

  • The Logic Behind the Method
  • The Method of Adding and Subtracting Decimals
  • Calculators

The Logic Behind the Method

Consider the sum of 4.37 and 3.22. Changing each decimal to a fraction, we have

4 37 100 + 3 22 100 4 37 100 + 3 22 100 size 12{4 { {"37"} over {"100"} } +3 { {"22"} over {"100"} } " Performing the addition, we get"} {} Performing the addition, we get

4 . 37 + 3 . 22 = 4 37 100 + 3 22 100 = 4 100 + 37 100 + 3 100 + 22 100 = 437 100 + 322 100 = 437 + 322 100 = 759 100 = 7 59 100 = seven and fifty-nine hundredths = 7.59 4 . 37 + 3 . 22 = 4 37 100 + 3 22 100 = 4 100 + 37 100 + 3 100 + 22 100 = 437 100 + 322 100 = 437 + 322 100 = 759 100 = 7 59 100 = seven and fifty-nine hundredths = 7.59

Thus, 4.37 + 3.22 = 7.594.37 + 3.22 = 7.59 size 12{4 "." "37 "+" 3" "." "22 "=" 7" "." "59"} {}.

The Method of Adding and Subtracting Decimals

When writing the previous addition, we could have written the numbers in col­umns.

4.37 + 3.22 ̲ 7.59 4.37 + 3.22 ̲ 7.59

This agrees with our previous result. From this observation, we can suggest a method for adding and subtracting decimal numbers.

Method of Adding and Subtracting Decimals

To add or subtract decimals:

  1. Align the numbers vertically so that the decimal points line up under each other and the corresponding decimal positions are in the same column.
  2. Add or subtract the numbers as if they were whole numbers.
  3. Place a decimal point in the resulting sum or difference directly under the other decimal points.

Sample Set A

Find the following sums and differences.

Example 1

9.813+2.1409.813+2.140 size 12{9 "." "813"+2 "." "140"} {}

9.813 The decimal points are aligned in the same column. + 2.140 ̲ 11.953 9.813 The decimal points are aligned in the same column. + 2.140 ̲ 11.953

Example 2

841.0056+47.016+19.058841.0056+47.016+19.058 size 12{"841" "." "0056"+"47" "." "016"+"19" "." "058"} {}

841.0056 47.016   + 19.058   ̲ 841.0056 47.016   + 19.058   ̲

To insure that the columns align properly, we can write a 0 in the position at the end of the numbers 47.016 and 19.058 without changing their values.

Vertical addition. 841.0056 plus 47.0160 plus 19.0580 equals 907.0796. A 1 needed to be carried in the hundredths, the tens, and the hundreds columns.

Example 3

1.314 - 0.58 1.314-0.58

1.314 - 0.58   ̲ Write a 0 in the thousandths position. 1.314 - 0.58   ̲ Write a 0 in the thousandths position.

Vertical subtraction. 1.314 minus 0.580 equals 0.734. The ones, and hundredths digits need to be borrowed from once, and the tenths needs to be borrowed from twice to perform the subtraction.

Example 4

16.01 - 7.053 16.01-7.053

16.01   - 7.053 ̲ Write a 0 in the thousandths position. 16.01   - 7.053 ̲ Write a 0 in the thousandths position.

Vertical subtraction. 16.101 minus 7.053 equals 8.957. The ones, tenths, hundredths, and thousandths digits need to be borrowed from once to perform the subtraction.

Example 5

Find the sum of 6.88106 and 3.5219 and round it to three decimal places.

6.88106 + 3.5219   ̲ Write a 0 in the ten thousandths position. 6.88106 + 3.5219   ̲ Write a 0 in the ten thousandths position.

Vertical addition. 6.88106 plus 3.52190 equals 10.40296. A 1 needs to be carried over the tenths and ones digits to perform the addition.

We need to round the sum to the thousandths position. Since the digit in the position immediately to the right is 9, and 9>5, we get

10.403

Example 6

Wendy has $643.12 in her checking account. She writes a check for $16.92. How much is her new account balance?

To find the new account balance, we need to find the difference between 643.12 and 16.92. We will subtract 16.92 from 643.12.

Vertical subtraction. 643.12 minus 16.92 equals 626.20.

After writing a check for $16.92, Wendy now has a balance of $626.20 in her checking account.

Practice Set A

Find the following sums and differences.

Exercise 1

3.187 + 2.9923.187 + 2.992 size 12{3 "." "187 "+" 2" "." "992"} {}

Solution

6.179

Exercise 2

14.987 - 5.34114.987 - 5.341 size 12{"14" "." "987 - 5" "." "341"} {}

Solution

9.646

Exercise 3

0.5261 + 1.07830.5261 + 1.0783 size 12{0 "." "5261 "+" 1" "." "0783"} {}

Solution

1.6044

Exercise 4

1.06 - 1.05351.06 - 1.0535 size 12{1 "." "06 - 1" "." "0535"} {}

Solution

0.0065

Exercise 5

16,521.07 + 9,256.1516,521.07 + 9,256.15 size 12{"16,521" "." "07 "+" 9,256" "." "15"} {}

Solution

25,777.22

Exercise 6

Find the sum of 11.6128 and 14.07353, and round it to two decimal places.

Solution

25.69

Calculators

The calculator can be useful for finding sums and differences of decimal numbers. However, calculators with an eight-digit display cannot be used when working with decimal numbers that contain more than eight digits, or when the sum results in more than eight digits. In practice, an eight-place decimal will seldom be encoun­tered. There are some inexpensive calculators that can handle 13 decimal places.

Sample Set B

Use a calculator to find each sum or difference.

Example 7

42.0638 + 126.551

Table 1
    Display Reads
Type 42.0638 42.0638
Press + 42.0638
Type 126.551 126.551
Press = 168.6148

The sum is 168.6148.

Example 8

Find the difference between 305.0627 and 14.29667.

Table 2
    Display Reads
Type 305.0627 305.0627
Press 305.0627
Type 14.29667 14.29667
Press = 290.76603

The difference is 290.76603

Example 9

51.07 + 3,891.001786

Since 3,891.001786 contains more than eight digits, we will be unable to use an eight-digit display calculator to perform this addition. We can, however, find the sum by hand.

51.070000 3891.001786 ̲ 3942.071786 51.070000 3891.001786 ̲ 3942.071786

The sum is 3,942.071786.

Practice Set B

Use a calculator to perform each operation.

Exercise 7

4.286 + 8.974.286 + 8.97 size 12{4 "." "286 "+" 8" "." "97"} {}

Solution

13.256

Exercise 8

452.0092- 392.558452.0092- 392.558 size 12{"452" "." "0092 - 392" "." "558"} {}

Solution

59.4512

Exercise 9

Find the sum of 0.095 and 0.001862

Solution

0.096862

Exercise 10

Find the difference between 0.5 and 0.025

Solution

0.475

Exercise 11

Find the sum of 2,776.00019 and 2,009.00012.

Solution

Since each number contains more than eight digits, using some calculators may not be helpful. Adding these by “hand technology,” we get 4,785.00031

Exercises

For the following 15 problems, perform each addition or subtraction. Use a calculator to check each result.

Exercise 12

1.84+7.111.84+7.11 size 12{1 "." "84"+7 "." "11"} {}

Solution

8.95

Exercise 13

15.0156.52715.0156.527 size 12{"15" "." "015"-6 "." "527"} {}

Exercise 14

11.842+28.00411.842+28.004 size 12{"11" "." "842"+"28" "." "004"} {}

Solution

39.846

Exercise 15

3.162.523.162.52 size 12{3 "." "16"-2 "." "52"} {}

Exercise 16

3.55267+8.196643.55267+8.19664 size 12{3 "." "55267"+8 "." "19664"} {}

Solution

11.74931

Exercise 17

0.91620.08720.91620.0872 size 12{0 "." "9162"-0 "." "0872"} {}

Exercise 18

65.5128.300565.5128.3005 size 12{"65" "." "512"-8 "." "3005"} {}

Solution

57.2115

Exercise 19

761.080853.198761.080853.198 size 12{"761" "." "0808"-"53" "." "198"} {}

Exercise 20

4.305+2.1193.8174.305+2.1193.817 size 12{4 "." "305"+2 "." "119"-3 "." "817"} {}

Solution

2.607

Exercise 21

19.1161+27.8014+39.316119.1161+27.8014+39.3161 size 12{"19" "." "1161"+"27" "." "8014"+"39" "." "3161"} {}

Exercise 22

0.412760.00180.000110.412760.00180.00011 size 12{0 "." "41276"-0 "." "0018"-0 "." "00011"} {}

Solution

0.41085

Exercise 23

2.181+6.05+1.167+8.1012.181+6.05+1.167+8.101 size 12{2 "." "181"+6 "." "05"+1 "." "167"+8 "." "101"} {}

Exercise 24

1.0031+6.013106+0.00018+0.0092+2.11

Solution

9.135586

Exercise 25

27+42+9.160.1761+81.627+42+9.160.1761+81.6 size 12{"27"+"42"+9 "." "16"-0 "." "1761"+"81" "." 6} {}

Exercise 26

10.28+11.111+0.86+5.110.28+11.111+0.86+5.1 size 12{"10" "." "28"+"11" "." "111"+0 "." "86"+5 "." 1} {}

Solution

27.351

For the following 10 problems, solve as directed. A calculator may be useful.

Exercise 27

Add 6.1121 and 4.916 and round to 2 decimal places.

Exercise 28

Add 21.66418 and 18.00184 and round to 4 decimal places.

Solution

39.6660

Exercise 29

Subtract 5.2121 from 9.6341 and round to 1 decimal place.

Exercise 30

Subtract 0.918 from 12.006 and round to 2 decimal places.

Solution

11.09

Exercise 31

Subtract 7.01884 from the sum of 13.11848 and 2.108 and round to 4 decimal places.

Exercise 32

A checking account has a balance of $42.51. A check is written for $19.28. What is the new balance?

Solution

$23.23

Exercise 33

A checking account has a balance of $82.97. One check is written for $6.49 and another for $39.95. What is the new balance?

Exercise 34

A person buys $4.29 worth of hamburger and pays for it with a $10 bill. How much change does this person get?

Solution

$5.71

Exercise 35

A man buys $6.43 worth of stationary and pays for it with a $20 bill. After receiving his change, he realizes he forgot to buy a pen. If the total price of the pen is $2.12, and he buys it, how much of the $20 bill is left?

Exercise 36

A woman starts recording a movie on her video cassette recorder with the tape counter set at 21.93. The movie runs 847.44 tape counter units. What is the final tape counter reading?

Solution

869.37

Exercises for Review

Exercise 37

((Reference)) Find the difference between 11,206 and 10,884.

Exercise 38

((Reference)) Find the product, 820 10,000820 10,000 size 12{"820 " cdot " 10,000"} {}.

Solution

8,200,000

Exercise 39

((Reference)) Find the value of 12125+82+16÷2212125+82+16÷22 size 12{ sqrt {"121"} - sqrt {"25"} +8 rSup { size 8{2} } +"16" div 2 rSup { size 8{2} } } {} .

Exercise 40

((Reference)) Find the value of 8133675÷2258133675÷225 size 12{8 { {1} over {3} } cdot { {"36"} over {"75"} } div 2 { {2} over {5} } } {}.

Solution

209=53 or 229209=53 or 229 size 12{ { {"20"} over {9} } = { {5} over {3} } " or " { {2} over {9} } } {}

Exercise 41

((Reference)) Round 1.08196 to the nearest hundredth.

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