6.4: Addition and Subtraction of Decimals
- understand the method used for adding and subtracting decimals
- be able to add and subtract decimals
- be able to use the calculator to add and subtract decimals
The Logic Behind the Method
Consider the sum of 4.37 and 3.22. Changing each decimal to a fraction, we have
\(4 \dfrac{37}{100} + 3 \dfrac{22}{100}\) Performing the addition, we get
\(\begin{array} {rcl} {4.37 + 3.22 = 4 \dfrac{37}{100} + 3 \dfrac{22}{100}} & = & {\dfrac{4 \cdot 100 + 37}{100} + \dfrac{3 \cdot 100 + 22}{100}} \\ {} & = & {\dfrac{437}{100} + \dfrac{322}{100}} \\ {} & = & {\dfrac{437 + 322}{100}} \\ {} & = & {\dfrac{759}{100}} \\ {} & = & {7 \dfrac{59}{100}} \\ {} & = & {\text{seven and fifty-nine hundredths}} \\ {} & = & {7.59} \end{array}\)
Thus, \(4.37 + 3.22 = 7.59\).
The Method of Adding and Subtracting Decimals
When writing the previous addition, we could have written the numbers in columns.
\(\begin{array} {r} {4.37} \\ {\underline{+3.22}} \\ {7.59} \end{array}\)
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:
Align the numbers vertically so that the decimal points line up under each other and the corresponding decimal positions are in the same column.
Add or subtract the numbers as if they were whole numbers.
Place a decimal point in the resulting sum or difference directly under the other decimal points.
Find the following sums and differences.
\(9.813 + 2.140\)
Solution
\(\begin{array} {r} {9.813} \\ {\underline{+2.140}} \\ {11.953} \end{array}\) The decimal points are aligned in the same column.
\(841.0056 + 47.016 + 19.058\)
Solution
\(\begin{array} {r} {841.0056} \\ {47.016\ \ } \\ {\underline{+19.058\ \ }} \end{array}\)
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.
\(1.314 - 0.58\)
Solution
\(\begin{array} {r} {1.314} \\ {\underline{-0.58\ \ }} \end{array}\) Write a 0 in the thousandths position.
\(16.01 - 7.053\)
Solution
\(\begin{array} {r} {16.01\ \ } \\ {\underline{-7.053}} \end{array}\) Write a 0 in the thousandths position.
Find the sum of 6.88106 and 3.5219 and round it to three decimal places.
Solution
\(\begin{array} {r} {6.88106} \\ {\underline{+3.5219\ \ }} \end{array}\) Write a 0 in the ten thousandths position.
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
Wendy has $643.12 in her checking account. She writes a check for $16.92. How much is her new account balance?
Solution
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.
After writing a check for $16.92, Wendy now has a balance of $626.20 in her checking account.
Pracitce Set A
Find the following sums and differences.
\(3.187 + 2.992\)
- Answer
-
6.179
Pracitce Set A
\(14.987 - 5.341\)
- Answer
-
9.646
Pracitce Set A
\(0.5261 + 1.0783\)
- Answer
-
1.6044
Pracitce Set A
\(1.06 - 1.0535\)
- Answer
-
0.0065
Pracitce Set A
\(16,521.07 + 9,256.15\)
- Answer
-
25,777.22
Pracitce Set A
Find the sum of 11.6128 and 14.07353, and round it to two decimal places.
- Answer
-
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.
Use a calculator to find each sum or difference.
42.0638 + 126.551
Solution
| Display Reads | ||
| Type | 42.0638 | 42.0638 |
| Press | + | 42.0638 |
| Type | 126.551 | 126.551 |
| Press | = | 168.6148 |
The sum is 168.6148.
Find the difference between 305.0627 and 14.29667.
Solution
| Display Reads | ||
| Type | 305.0627 | 305.0627 |
| Press | - | 305.0627 |
| Type | 14.29667 | 14.29667 |
| Press | = | 290.76603 |
The difference is 290.76603
51.07 + 3,891.001786
Solution
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.
\(\begin{array} {r} {51.070000} \\ {\underline{3891.001786}} \\ {3942.071786} \end{array}\)
The sum is 3,942.071786.
Practice Set B
Use a calculator to perform each operation.
\(4.286 + 8.97\)
- Answer
-
13.256
Practice Set B
\(452.0092 - 392.558\)
- Answer
-
59.4512
Practice Set B
Find the sum of 0.095 and 0.001862
- Answer
-
0.096862
Practice Set B
Find the difference between 0.5 and 0.025
- Answer
-
0.475
Practice Set B
Find the sum of 2,776.00019 and 2,009.00012.
- Answer
-
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 \(\PageIndex{1}\)
\(1.84 + 7.11\)
- Answer
-
8.95
Exercise \(\PageIndex{2}\)
\(15.015 - 6.527\)
Exercise \(\PageIndex{3}\)
\(11.842 + 28.004\)
- Answer
-
39.846
Exercise \(\PageIndex{4}\)
\(3.16 - 2.52\)
Exercise \(\PageIndex{5}\)
\(3.55267 + 8.19664\)
- Answer
-
11.74931
Exercise \(\PageIndex{6}\)
\(0.9162 - 0.0872\)
Exercise \(\PageIndex{7}\)
\(65.512 - 8.3005\)
- Answer
-
57.2115
Exercise \(\PageIndex{8}\)
\(761.0808 - 53.198\)
Exercise \(\PageIndex{9}\)
\(4.305 + 2.119 - 3.817\)
- Answer
-
2.607
Exercise \(\PageIndex{10}\)
\(19.1161 + 27.8014 + 39.3161\)
Exercise \(\PageIndex{11}\)
\(0.41276 - 0.0018 - 0.00011\)
- Answer
-
0.41085
Exercise \(\PageIndex{12}\)
\(2.181 + 6.05 + 1.167 + 8.101\)
Exercise \(\PageIndex{13}\)
\(1.0031 + 6.013106 + 0.00018 + 0.0092 + 2.11\)
- Answer
-
9.135586
Exercise \(\PageIndex{14}\)
\(27 + 42 + 9.16 - 0.1761 + 81.6\)
Exercise \(\PageIndex{15}\)
\(10.28 + 11.111 + 0.86 + 5.1\)
- Answer
-
27.351
For the following 10 problems, solve as directed. A calculator may be useful.
Exercise \(\PageIndex{16}\)
Add 6.1121 and 4.916 and round to 2 decimal places.
Exercise \(\PageIndex{17}\)
Add 21.66418 and 18.00184 and round to 4 decimal places.
- Answer
-
39.6660
Exercise \(\PageIndex{18}\)
Subtract 5.2121 from 9.6341 and round to 1 decimal place.
Exercise \(\PageIndex{19}\)
Subtract 0.918 from 12.006 and round to 2 decimal places.
- Answer
-
11.09
Exercise \(\PageIndex{20}\)
Subtract 7.01884 from the sum of 13.11848 and 2.108 and round to 4 decimal places.
Exercise \(\PageIndex{21}\)
A checking account has a balance of $42.51. A check is written for $19.28. What is the new balance?
- Answer
-
$23.23
Exercise \(\PageIndex{22}\)
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 \(\PageIndex{23}\)
A person buys $4.29 worth of hamburger and pays for it with a $10 bill. How much change does this person get?
- Answer
-
$5.71
Exercise \(\PageIndex{24}\)
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 \(\PageIndex{25}\)
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?
- Answer
-
869.37
Exercises for Review
Exercise \(\PageIndex{26}\)
Find the difference between 11,206 and 10,884.
Exercise \(\PageIndex{27}\)
Find the product, \(820 \cdot 10,000\).
- Answer
-
8,200,000
Exercise \(\PageIndex{28}\)
Find the value of \(\sqrt{121} - \sqrt{25} + 8^2 + 16 \div 2^2\).
Exercise \(\PageIndex{29}\)
Find the value of \(8 \dfrac{1}{3} \cdot \dfrac{36}{75} \div 2 \dfrac{2}{5}\).
- Answer
-
\(\dfrac{20}{9} = \dfrac{5}{3}\) or \(2 \dfrac{2}{9}\)
Exercise \(\PageIndex{30}\)
Round 1.08196 to the nearest hundredth.