3.2: Adding and Subtracting Decimals

Adding Decimals

Addition of decimal numbers is quite similar to addition of whole numbers. For example, suppose that we are asked to add 2.34 and 5.25. We could change these decimal numbers to mixed fractions and add.

\[ \begin{aligned} 2.34 + 5.25 & = 2 \frac{34}{100} + 5 \frac{25}{100} \\ & = 7 \frac{59}{100} \end{aligned}\nonumber \]

However, we can also line the decimal numbers on their decimal points and add vertically, as follows.

\[ \begin{array}{r} 2.34 \\ + 5.25 \\ \hline 7.59 \end{array}\nonumber \]

Note that this alignment procedure produces the same result, “seven and fifty nine hundredths.” This motivates the following procedure for adding decimal numbers.

Adding Decimals

To add decimal numbers, proceed as follows:

1. Place the numbers to be added in vertical format, aligning the decimal points.
2. Add the numbers as if they were whole numbers.
3. Place the decimal point in the answer in the same column as the decimal points above it.

Example 1

Add 3.125 and 4.814.

Solution

Place the numbers in vertical format, aligning on their decimal points. Add, then place the decimal point in the answer in the same column as the decimal points that appear above the answer.

\[ \begin{array}{r} 3.125 \\ +4.814 \\ \hline 7.939 \end{array}\nonumber \]

Thus, 3.125 + 4.814 = 7.939.

Exercise

Add: 2.864 + 3.029

Answer

5.893

Example 2

Jane has $4.35 in her purse. Jim has $5.62 in his wallet. If they sum their money, what is the total?

Solution

Arrange the numbers in vertical format, aligning decimal points, then add.

\[ \begin{array}{r} \$ 4.35 \\ + \$ 5.62 \\ \hline \$ 9.97 \end{array}\nonumber \]

Exercise

Alice has $8.63 in her purse and Joanna has $2.29. If they combine sum their money, what is the total?

Answer

$10.91

Before looking at another example, let’s recall an important observation.

Important Observation

Adding zeros to the end of the fractional part of a decimal number does not change its value. Similarly, deleting trailing zeros from the end of a decimal number does not change its value.

For example, we could add two zeros to the end of the fractional part of 7.25 to obtain 7.2500. The numbers 7.25 and 7.2500 are identical as the following argument shows:

\[ \begin{aligned} 7.2500 & = 7 \frac{2500}{10000} \\ & = 7 \frac{25}{100} \\ & = 7.25 \end{aligned}\nonumber \]

Example 3

Add 7.5 and 12.23.

Solution

Arrange the numbers in vertical format, aligning their decimal points in a column. Note that we add a trailing zero to improve columnar alignment.

\[ \begin{array}{r} 7.50 \\ +12.23 \\ \hline 19.73 \end{array}\nonumber \]

Hence, 7.5 + 12.23 = 19.73.

Exercise

Add: 9.7 + 15.86

Answer

25.56

Example 4

Find the sum: 12.2+8.352 + 22.44.

Solution

Arrange the numbers in vertical format, aligning their decimal points in a column. Note that we add trailing zeros to improve the columnar alignment.

\[ \begin{array}{r} 12.200 \\ 8.352 \\ + 22.440 \\ \hline 42.992 \end{array}\nonumber \]

Hence, 12.2+8.352 + 22.44 = 42.992.

Exercise

Add: 12.9+4.286 + 33.97

Answer

51.156

Subtracting Decimals

Subtraction of decimal numbers proceeds in much the same way as addition of decimal numbers.

Subtracting Decimals

To subtract decimal numbers, proceed as follows:

1. Place the numbers to be subtracted in vertical format, aligning the decimal points.
2. Subtract the numbers as if they were whole numbers.
3. Place the decimal point in the answer in the same column as the decimal points above it.

Example 5

Subtract 12.23 from 33.57.

Solution

Arrange the numbers in vertical format, aligning their decimal points in a column, then subtract. Note that we subtract 12.23 from 33.57.

\[ \begin{array}{r} 33.57 \\ -12.23 \\ \hline 21.34 \end{array}\nonumber \]

Hence, 33.57 − 12.23 = 21.34.

Exercise

Subtract: 58.76 − 38.95

Answer

19.81

As with addition, we add trailing zeros to the fractional part of the decimal numbers to help columnar alignment.

Example 6

Find the difference: 13.3 − 8.572.

Solution

Arrange the numbers in vertical format, aligning their decimal points in a column. Note that we add trailing zeros to the fractional part of 13.3 to improve columnar alignment.

\[ \begin{array}{r} 13.300 \\ -8.572 \\ \hline 4.728 \end{array}\nonumber \]

Hence, 13.3 − 8.572 = 4.728.

Exercise

Subtract: 15.2 − 8.756

Answer

6.444

Adding and Subtracting Signed Decimal Numbers

We use the same rules for addition of signed decimal numbers as we did for the addition of integers.

Adding Two Decimals with Like Signs

To add two decimals with like signs, proceed as follows:

1. Add the magnitudes of the decimal numbers.
2. Prefix the common sign.

Example 7

Simplify: −3.2+(−18.95).

Solution

To add like signs, first add the magnitudes.

\[ \begin{array}{r} 3.20 \\ +18.95 \\ \hline 22.15 \end{array}\nonumber \]

Prefix the common sign. Hence, −3.2+(−18.95) = −22.15

Exercise

Simplify: −5.7 + (−83.85)

Answer

−89.55

We use the same rule as we did for integers when adding decimals with unlike signs.

Adding Two Decimals with Unlike Signs

To add two decimals with unlike signs, proceed as follows:

1. Subtract the smaller magnitude from the larger magnitude.
2. Prefix the sign of the decimal number with the larger magnitude.

Example 8

Simplify: −3 + 2.24.

Solution

To add unlike signs, first subtract the smaller magnitude from the larger magnitude.

\[ \begin{array}{r} 3.00 \\ -2.24 \\ \hline 0.76 \end{array}\nonumber \]

Prefix the sign of the decimal number with the larger magnitude. Hence, −3+2.24 = −0.76.

Exercise

Simplify: −8 + 5.74

Answer

−2.26

Subtraction still means add the opposite.

Example 9

Simplify: −8.567 − (−12.3).

Solution

Subtraction must first be changed to addition by adding the opposite.

\[−8.567 − (−12.3) = −8.567 + 12.3\nonumber \]

We have unlike signs. First, subtract the smaller magnitude from the larger magnitude.

\[ \begin{array}{r} 12.300 \\ − 8.567 \\ \hline 3.733 \end{array}\nonumber \]

Prefix the sign of the decimal number with the larger magnitude. Hence:

\[ \begin{aligned} −8.567 − (−12.3) & = −8.567 + 12.3 \\ & = 3.733 \end{aligned}\nonumber \]

Exercise

Simplify: −2.384 − (−15.2)

Answer

12.816

Order of operations demands that we simplify expressions contained in parentheses first.

Example 10

Simplify: −11.2 − (−8.45 + 2.7).

Solution

We need to add inside the parentheses first. Because we have unlike signs, subtract the smaller magnitude from the larger magnitude.

\[ \begin{array}{r} 8.45 \\ − 2.70 \\ \hline 5.75 \end{array}\nonumber \]

Prefix the sign of the number with the larger magnitude. Therefore,

\[−11.2 − (−8.45 + 2.7) = −11.2 − (−5.75)\nonumber \]

Subtraction means add the opposite.

\[−11.2 − (−5.75) = −11.2+5.75\nonumber \]

Again, we have unlike signs. Subtract the smaller magnitude from the larger magnitude.

\[ \begin{array}{r} 11.20 \\ − 5.75 \\ \hline 5.45 \end{array}\nonumber \]

Prefix the sign of the number with the large magnitude.

\[ −11.2+5.75 = −5.45\nonumber \]

Exercise

Simplify: −12.8 − (−7.44 + 3.7)

Answer

−9.06

Writing Mathematics

The solution to the previous example should be written as follows:

\[ \begin{aligned} −11.2 − (−8.45 + 2.7) & = −11.2 − (−5.75) \\ & = −11.2+5.75 \\ & = −5.45 \end{aligned}\nonumber \]

Any scratch work, such as the computations in vertical format in the previous example, should be done in the margin or on a scratch pad.

Exercises

Add the decimals.

1. \(31.9 + 84.7\)

3. \(4 + 97.18\)

9. \(52.671 + 5.97\)

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Subtract the decimals.

13. \(9 − 2.261\)

17. \(55.672 − 3.3\)

19. \(60.575 − 6\)

23. \(8.1 − 2.12\)

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Add or subtract the decimals, as indicated.

29. \(−34.7+(−56.214)\)

31. \(8.4+(−6.757)\)

33. \(−50.4+7.6\)

37. \(0.19 − 0.7\)

47. \(2.001 − 4.202\)

59. \(9.365 + (−5)\)

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Simplify the given expression.

69. \(−19.1 − (1.51 − (−17.35))\)

71. \(11.55 + (6.3 − (−1.9))\)

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Answers

1. 116.6

3. 101.18

9. 58.641

13. 6.739

17. 52.372

19. 54.575

23. 5.98

29. −90.914

31. 1.643

33. −42.8

37. −0.51

47. −2.201

59. 4.365

69. −37.96

71. 19.75