6.4: Addition and Subtraction of Decimals

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Learning Objectives

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.

Sample Set A

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.

Sample Set A

$841.0056 + 47.016 + 19.058$

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.

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

Sample Set A

$1.314 - 0.58$

Solution

$\begin{array} {r} {1.314} \\ {\underline{-0.58\ \ }} \end{array}$ Write a 0 in the thousandths position.

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

Sample Set A

$16.01 - 7.053$

Solution

$\begin{array} {r} {16.01\ \ } \\ {\underline{-7.053}} \end{array}$ Write a 0 in the thousandths position.

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

Sample Set A

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.

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

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

Sample Set A

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.

$Vertical subtraction. 643.12 minus 16.92 equals 626.20.$

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

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.

Sample Set B

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.

Sample Set B

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

Sample Set B

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.