3.2: Adding and Subtracting Decimals
- Page ID
- 137911
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:
- Place the numbers to be added in vertical format, aligning the decimal points.
- Add the numbers as if they were whole numbers.
- 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:
- Place the numbers to be subtracted in vertical format, aligning the decimal points.
- Subtract the numbers as if they were whole numbers.
- 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:
- Add the magnitudes of the decimal numbers.
- 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:
- Subtract the smaller magnitude from the larger magnitude.
- 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. \(52.671 + 5.97\)
Subtract the decimals.
13. \(9 − 2.261\)
17. \(55.672 − 3.3\)
19. \(60.575 − 6\)
23. \(8.1 − 2.12\)
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. \(9.365 + (−5)\)
Simplify the given expression.
69. \(−19.1 − (1.51 − (−17.35))\)
71. \(11.55 + (6.3 − (−1.9))\)
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