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Division of Decimals by Decimals

Making the divisor a whole number

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Division of Decimals by Decimals
License: CC BY-NC 3.0

Liam is almost done building a bookcase. He just needs to add the shelves. Each shelf must be 26.5 inches long. He has a piece of wood that is 90.1 inches long. How any shelves can he make from that piece of wood?

In this concept, you will learn how to divide a decimal by a decimal.

Dividing Decimals by Decimals

You can divide a decimal number by another decimal number, but placing the decimal back into the quotient can become tricky. Simplify the process by change the divisor to a whole number.

Here is a division problem.

\begin{align*}10.4 \div 2.6 = \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

The divisor is 2.6. To change a decimal number to a whole number, multiply the decimal number by a power of ten to move the decimal point. Multiply 2.6 by 10 to move the decimal point one space to the right.

\begin{align*}2.6 \times 10 = 26\end{align*}

Dividing by 26 is easier than dividing by 2.6. However, dividing 10.4 by 26 is not the same as dividing 10.4 by 2.6. If you change the divisor, you must also change the dividend.

Remember that a division problem can also be written as a fraction. The dividend is placed in the numerator and the divisor is placed in the denominator. In this problem, 10.4 is the dividend and is placed in the numerator. 2.6 is the divisor and is placed in the denominator.

\begin{align*}10.4 \div 2.6 = \frac{10.4}{2.6}\end{align*}

If you change the denominator, you must also change the numerator the same way in order for the fraction to have the same value. Multiply the numerator and denominator by 10.

\begin{align*}10.4 \div 2.6 = \frac{10.4 \times 10}{2.6 \times 10} = \frac{104}{26}\end{align*}

Now write the new fraction as a division problem and divide to find the quotient.

\begin{align*}104 \div 26 \quad \text{ or } \quad \overset{}{26 \overline{ ) {104 }}}\end{align*}

\begin{align*}\begin{array}{rcl} && \overset{\quad \ 4}{26 \overline{ ) {104 }}}\\ && \ \ \underline{-104}\\ && \qquad 0 \end{array}\end{align*}

\begin{align*}10.4 \div 2.6\end{align*} is the same as \begin{align*}104 \div 26\end{align*}. The quotient of 10.4 divided by 2.6 is 4.

Here is another division problem. This time the divisor has two decimal places.

\begin{align*}\overset{}{0.45 \overline{ ) {1.44 }}}\end{align*}

First, change the divisor to a whole number by multiplying it by a power of ten. Multiply 0.45 by 100. Then, do the same thing to the dividend.

\begin{align*}\begin{array}{rcl} 0.45 \times 100 & = & 45\\ 1.44 \times 100 & = & 144 \end{array}\end{align*}

Here is the new division problem. Divide to find the quotient.

\begin{align*}\begin{array}{rcl} && \overset{\quad \ 3}{45 \overline{ ) {144 }}}\\ && \ \ \underline{-135\;}\\ && \qquad \ 9 \end{array}\end{align*}

Use zero placeholders to continue dividing.

\begin{align*}\begin{array}{rcl} && \overset{\quad \ 3.2}{45 \overline{ ) {144.{\color{red}0}}}}\\ && \ \ \underline{-135\;}\\ && \qquad \ 9{\color{red}0}\\ && \quad \ \ \underline{-90}\\ && \qquad \ \ \ 0 \end{array}\end{align*}

The quotient of 1.44 divided by 0.45 is 3.2.

Notice the pattern. You move the decimal the same number of spaces in the dividend as you do in the divisor.

\begin{align*}\begin{array}{rcl} 2.6 \rightarrow 2\underrightarrow{6} & & 0.45 \rightarrow \underrightarrow{45}\\ 10.4 \rightarrow 10\underrightarrow{4} & & 1.35 \rightarrow 1\underrightarrow{35} \end{array}\end{align*}

To multiply a decimal number by a decimal number, change the divisor to a whole number and move the decimal point the same number of spaces in the dividend. Then, divide to find the quotient.

Examples

Example 1

Earlier, you were given a problem about Liam and his bookcase.

He wants to know how many 26.5 inch shelves he can make from a piece of wood that is 90.1 inches long. Divide to find the answer.

\begin{align*}\overset{}{26.5 \overline{ ) {90.1}}}\end{align*}

First, change the divisor to a whole number.

\begin{align*}26.5 \rightarrow 26\underrightarrow{5}.\end{align*}

Then, do the same thing to the dividend.

\begin{align*}90.1 \rightarrow 90\underrightarrow{1}.\end{align*}

Next, write the new division problem and divide to find the quotient.

\begin{align*}\begin{array}{rcl} && \overset{\quad \ 3.4}{265 \overline{ ) {901.{\color{red}0}}}}\\ && \ \ \ \ \underline{-795\;}\\ && \qquad 106{\color{red}0}\\ && \quad \ \underline{-1060}\\ && \qquad \ \ \ 0 \end{array}\end{align*}

Liam can make at least 3 shelves from the piece of wood.

Example 2

Divide the decimals.

\begin{align*}\overset{}{3.45 \overline{ ) {7.245 }}}\end{align*}

First, change 3.45 to a whole number by multiplying it by a power of ten. Multiply 3.45 by 100 or simply move the decimal point two spaces to the right.

\begin{align*}3.45 \times 100 = 345 \quad \text{ or } \quad 3\underrightarrow{45}.\end{align*}

Then, do the same thing to the dividend. Multiply 7.245 by 100 or move the decimal point two spaces to the right.

\begin{align*}7.245 \times 100 = 724.5 \quad \text{ or } \quad 7\underrightarrow{24}.5\end{align*}

Next, write the new division problem and divide to find the quotient.

\begin{align*}\begin{array}{rcl} && \overset{\qquad 2.1}{345 \overline{ ) {724.5}}}\\ && \quad \ \underline{-690\;}\\ && \qquad \ \ 345\\ && \quad \ \ \ \underline{-345}\\ && \qquad \quad \ \ 0 \end{array}\end{align*}

The quotient of 7.245 divided by 3.45 is 2.1.

Divide the decimals. Use zero placeholders if needed.

Example 3

\begin{align*}\overset{}{1.2 \overline{ ) {4.8}}}\end{align*}

First, change the divisor to a whole number.

\begin{align*}1.2 \times 10 = 12 \quad \text{ or } \quad 1\underrightarrow{2}.\end{align*}

Then, do the same thing to the dividend.

\begin{align*}4.8 \times 10 = 48 \quad \text{ or } \quad 4\underrightarrow{8}.\end{align*}

Next, write the new division problem and divide to find the quotient.

\begin{align*}\begin{array}{rcl} && \overset{\quad \ 4}{12 \overline{ ) {48 }}}\\ && \ \ \ \underline{-48}\\ && \qquad 0 \end{array}\end{align*}

The quotient of 4.8 divided by 1.2 is 4.

Example 4

\begin{align*}\overset{}{5.6 \overline{ ) {14.28}}}\end{align*}

First, change the divisor to a whole number.

\begin{align*}5.6 \times 10 = 56 \quad \text{ or } \quad 5\underrightarrow{6}.\end{align*}

Then, do the same thing to the dividend.

\begin{align*}14.28 \times 10 = 142.8 \quad \text{ or } \quad 14\underrightarrow{2}.8\end{align*}

Next, write the new division problem and divide to find the quotient.

\begin{align*}\begin{array}{rcl} && \overset{\quad \ \ 2.55}{56 \overline{ ) {142.8{\color{red}0}}}}\\ && \ \ \underline{\;-112\;}\\ && \qquad 308\\ && \quad \ \underline{-280}\\ && \qquad \ \ 28{\color{red}0}\\ && \quad \ \ \ \underline{-280}\\ && \qquad \quad \ \ 0 \end{array}\end{align*}

The quotient of 14.28 divided by 5.6 is 2.55.

Example 5

\begin{align*}\overset{}{6.98 \overline{ ) {13.96}}}\end{align*}

First, change the divisor to a whole number.

\begin{align*}6.98 \times 100 = 698 \quad \text{ or } \quad 6\underrightarrow{98}.\end{align*}

Then, do the same thing to the dividend.

\begin{align*}13.96 \times 100 = 1,396 \quad \text{ or } \quad 13\underrightarrow{96}.\end{align*}

Next, write the new division problem and divide to find the quotient.

\begin{align*}\begin{array}{rcl} && \overset{\quad \ \ \ 2}{698 \overline{ ) \ { 1,396 }}}\\ && \quad \ \underline{-\; 1,396}\\ && \qquad \quad 0 \end{array}\end{align*}The quotient of 13.96 divided by 6.98 is 2.

Review

Divide the following decimals. Use zero place holders if needed.

  1. \begin{align*}\overset{}{1.2 \overline{ ) {4.08 }}}\end{align*}
  2. \begin{align*}\overset{}{3.5 \overline{ ) {12.6 }}}\end{align*}
  3. \begin{align*}\overset{}{14.5 \overline{ ) {29 }}}\end{align*}
  4. \begin{align*}\overset{}{8.9 \overline{ ) {11.57 }}}\end{align*}
  5. \begin{align*}\overset{}{0.32 \overline{ ) {0.08 }}}\end{align*}
  6. \begin{align*}\overset{}{1.2 \overline{ ) {7.8 }}}\end{align*}
  7. \begin{align*}\overset{}{9.6 \overline{ ) {11.52 }}}\end{align*} 
  8. \begin{align*}\overset{}{14.5 \overline{ ) {33.35 }}}\end{align*}
  9. \begin{align*}\overset{}{12.5 \overline{ ) {7.5 }}}\end{align*}
  10. \begin{align*}\overset{}{2.5 \overline{ ) {13.29 }}}\end{align*}
  11. \begin{align*}\overset{}{1.2 \overline{ ) {7.2 }}}\end{align*} 
  12. \begin{align*}\overset{}{0.8 \overline{ ) {1.8 }}}\end{align*} 
  13. \begin{align*}\overset{}{4.6 \overline{ ) {10.58 }}}\end{align*}
  14. \begin{align*}\overset{}{1.6 \overline{ ) {0.3 }}}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.16. 

Resources

 

 

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Vocabulary

Dividend

In a division problem, the dividend is the number or expression that is being divided.

divisor

In a division problem, the divisor is the number or expression that is being divided into the dividend. For example: In the expression 152 \div 6, 6 is the divisor and 152 is the dividend.

Quotient

The quotient is the result after two amounts have been divided.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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