# 2.13: Decimal Division

**At Grade**Created by: CK-12

**Practice**Decimal Division

In this concept, you will learn how to divide decimals with and without rounding.

### Dividing Decimals

Division is useful anytime you want to split up an object or a group of objects. It is also useful when you want to reverse the multiplication done on a number.

Remember that the answer to a division problem is called the **quotient**. The number being divided is called the **dividend** and the number you are dividing by is called the **divisor**.

Here is where those terms are in a division problem.

\begin{align*}\text{dividend} \div \text{divisor} = \text{quotient}\end{align*}

\begin{align*}\overset{\qquad \ \ \text{quotient}}{\text{divisor} \overline{ ) {\text{dividend }}}}\\ \end{align*}

Dividing decimal numbers is very similar to dividing whole numbers, but with a couple of extra steps. In order to be able to do long division like normal, you will first want to change your divisor into a whole number by moving its decimal point. Then, in order to keep the expression equal, you will need to move the decimal point of the dividend in the same way.

Here are the steps for dividing decimal numbers.

- Move the decimal point of your divisor to the right as many places as necessary to turn it into a whole number.
- Move the decimal point of your dividend the same number of places. Add zeros if necessary.
- Temporarily ignore any decimal point in the dividend and divide using long division just as if you were dividing whole numbers.
- In your answer, insert a decimal point directly above the decimal point in your dividend.

Here is an example.

Divide \begin{align*}4.6 \div 2.3\end{align*}.

First, note that 4.6 is your dividend and 2.3 is your divisor.

Now, move the decimal point of your divisor, 2.3, to the right in order to make it a whole number. You will move the decimal point 1 place to the right to turn 2.3 into 23.

Next, move the decimal point of your dividend, 4.6, the same number of places. Move the decimal point 1 place to the right to turn 4.6 into 46.

Now, divide as usual using long division.

\begin{align*}\overset{\ \ \ \ \ 2}{23\overline{ ) {46\;}}}\\ \end{align*}

Last, decide where the decimal point should go in your answer. Because the dividend no longer has a decimal point, your answer will not have a decimal point.

The answer is \begin{align*}4.6 \div 2.3=2\end{align*}.

Remember that you can always check your answer to a division problem by multiplying. You can verify that \begin{align*}2.3 \times 2=4.6\end{align*}.

Here is an example where you will have to add zeros to the dividend.

Divide \begin{align*}9 \div 2.25\end{align*}.

First, note that 9 is your dividend and 2.25 is your divisor.

Now, move the decimal point of your divisor, 2.25, to the right in order to make it a whole number. You will move the decimal point 2 places to the right to turn 2.25 into 225.

Next, move the decimal point of your dividend, 9, the same number of places. Even though there is no written decimal point with the number 9, remember that 9 is the same as \begin{align*}9.000000 \ldots\end{align*}. When you move the decimal point you will be adding zeros. Move the decimal point 2 places to the right to turn 9 into 900.

Now, divide as usual using long division.

\begin{align*}\overset{\ \ \ \ \ \ \ \ \ 4}{225 \overline{ ) {900 \;}}}\\ \end{align*}

Last, decide where the decimal point should go in your answer. Because the dividend no longer has a decimal point, your answer will not have a decimal point.

The answer is \begin{align*}9 \div 2.25=4\end{align*}.

Sometimes you will want to round numbers before or after dividing them. Rounding is useful when you only need an approximate answer instead of an exact answer. Make sure to read the question carefully in order to figure out if you should round before or after doing the division.

Here is an example.

Round each number to the nearest tenth and then divide.

\begin{align*}67.521 \div 2.243\end{align*}

First, round each number to the nearest tenth. Remember that the tenths place is the first digit to the right of the decimal point.

- 67.521 rounds to 67.5
- 2.243 rounds to 2.2

Now, divide the rounded numbers. Note that 67.5 is your dividend and 2.2 is your divisor.

Next, move the decimal point of your divisor, 2.2, to the right in order to make it a whole number. You will move the decimal point 1 place to the right to turn 2.2 into 22.

Then, move the decimal point of your dividend, 67.5, the same number of places. Move the decimal point 1 place to the right to turn 67.5 into 675.

Now, divide as usual using long division. The decimal point in your answer will go directly above the decimal point in the dividend.

\begin{align*}\begin{array}{rcl} && \overset{\quad \ \ 30.68181 \ldots}{22 \overline{ ) {675.000000 \;}}}\\ && \ \ \underline{- 66\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\\ && \qquad 150 \\ && \quad \ \underline{ - 132\;\;\;\;\;\;}\\ && \qquad \ \ 180 \\ && \quad \ \ \ \underline{ - 176\;\;\;\;\;\;}\\ && \qquad \quad \ \ 40 \\ && \qquad \ \ \ \underline{ - 22}\\ && \qquad \quad \ \ 180 \\ && \qquad \ \ \ \underline{ - 176\;\;\;\;\;\;}\\ && \qquad \qquad \ \ 40 \\ && \qquad \quad \ \ \ \underline{- 22 \;\;\;\;\;\;}\\ && \qquad \qquad \ \ 180 \end{array}\end{align*}

Notice that this time there is a repeating pattern in the long division. This means the answer is a repeating decimal. The \begin{align*}818181 \ldots\end{align*} pattern will repeat over and over forever. In this case, you have two options.

- You can use a bar above the repeating portion of the quotient to indicate that it repeats: \begin{align*}30.6 \overline{81}\end{align*}.
- You can round your answer: 30.682.

In this case, because the original dividend and divisor were already rounded and your answer is already only an approximate answer, it makes sense to round.

The answer is \begin{align*}67.521 \div 2.243\end{align*} is approximately 30.682.

### Examples

#### Example 1

Earlier, you were given a problem about Lucy and her cookies.

She's baking up 21 cookies to share with her friends and she has 39.9 ounces of dough. She wants to figure out how much each ball of cookie dough should weigh so that she will have 21 equally sized cookies.

To figure this out, Lucy needs to divide \begin{align*}39.9 \div 21\end{align*}.

First, she should note that 39.9 is her dividend and 21 is her divisor.

Her divisor is already a whole number, so she will not need to move any decimal points.

Next, she should divide as usual using long division. The decimal point in her answer will go directly above the decimal point in the dividend.

\begin{align*}\begin{array}{rcl} && \overset{ \quad 1.9}{21 \overline{ ) {39.9}}}\\ && \ \ \underline{ - 21 \;\;\;}\\ && \quad \ \ 189 \\ && \ \ \ \underline{ - 189 } \\ && \quad \ \ \ \ \ 0 \end{array}\end{align*}

The answer is \begin{align*}39.9 \div 21=1.9\end{align*}. Lucy should make each ball of cookie dough 1.9 ounces.

#### Example 2

The Landry’s living room is a rectangle with an area of 97.92 square meters and a length of 13.6 meters. What is the width of the living room?

First, remember that the area of a rectangle is given by length times width. This means that if you have the area and divide by the length, you will get the width.

\begin{align*}\begin{array}{rcl} w &=& A \div l \\ w &=& 97.92 \div 13.6 \end{array}\end{align*}

Note that for this problem, 97.92 is your dividend and 13.6 is your divisor.

Now, move the decimal point of your divisor, 13.6, to the right in order to make it a whole number. You will move the decimal point 1 place to the right to turn 13.6 into 136.

Next, move the decimal point of your dividend, 97.92, the same number of places. Move the decimal point 1 place to the right to turn 97.92 into 979.2.

Now, divide as usual using long division. The decimal point in your answer will go directly above the decimal point in the dividend.

\begin{align*}\begin{array}{rcl} && \overset{ \qquad \ \ 7.2 }{136 \overline{ ) {979.2 \;}}}\\ && \quad \ \underline{ - 952 \;\;\;}\\ && \qquad \ \ \ 272 \\ && \qquad \underline{ -272 } \\ && \qquad \quad \ \ 0 \end{array}\end{align*}The answer is the width is 7.2 meters.

#### Example 3

Divide \begin{align*}3.96 \div 1.2\end{align*}.

First, note that 3.96 is your dividend and 1.2 is your divisor.

Now, move the decimal point of your divisor, 1.2, to the right in order to make it a whole number. You will move the decimal point 1 place to the right to turn 1.2 into 12.

Next, move the decimal point of your dividend, 3.96, the same number of places. Move the decimal point 1 place to the right to turn 3.96 into 39.6.

Now, divide as usual using long division. The decimal point in your answer will go directly above the decimal point in the dividend.

\begin{align*}\begin{array}{rcl} && \overset{ \quad \ 3.3 }{12 \overline{ ) {39.6 \;}}}\\ && \ \underline{\ -36\;\;\;}\\ && \qquad \ 36 \\ && \quad \ \ \underline{-36\;\;} \\ && \quad \ \ \ \ \ 0 \end{array}\end{align*}

The answer is \begin{align*}3.96 \div 1.2=3.3\end{align*}.

#### Example 4

Find the quotient of \begin{align*}24.288 \div 9.6\end{align*}.

First, note that 24.288 is your dividend and 9.6 is your divisor.

Now, move the decimal point of your divisor, 9.6, to the right in order to make it a whole number. You will move the decimal point 1 place to the right to turn 9.6 into 96.

Next, move the decimal point of your dividend, 24.288, the same number of places. Move the decimal point 1 place to the right to turn 24.288 into 242.88.

\begin{align*}\begin{array}{rcl} && \overset{\quad \quad 2.53 }{96 \overline{ ){242.88 \;}}}\\ && \ \ \ \underline{ - 192 }\\ && \qquad \ 508 \\ && \quad \ \ \underline{ -480 } \\ && \qquad \ \ \ 288 \\ && \quad \quad \underline{ -288 } \\ && \qquad \quad \ 0 \end{array}\end{align*}

The answer is \begin{align*}24.288 \div 9.6=2.53\end{align*}.

#### Example 5

Round to the nearest tenth and then divide.

\begin{align*}4.721 \div 2.465\end{align*}

First, round each number to the nearest tenth. Remember that the tenths place is the first digit to the right of the decimal point.

- 4.721 rounds to 4.7
- 2.465 rounds to 2.5

Now, divide the rounded numbers. Note that 4.7 is your dividend and 2.5 is your divisor.

Next, move the decimal point of your divisor, 2.5, to the right in order to make it a whole number. You will move the decimal point 1 place to the right to turn 2.5 into 25.

Then, move the decimal point of your dividend, 4.7, the same number of places. Move the decimal point 1 place to the right to turn 4.7 into 47.

\begin{align*}\begin{array}{rcl} && \overset{ \ \ \ 1.88}{25 \overline{ ) {47.00 \;}}}\\ && \ \ \underline{ - 25 }\\ && \quad \ \ \ 220 \\ && \quad \underline{ - 200 } \\ && \quad \quad \ 200 \\ && \quad \ \ \underline{-200 } \\ && \qquad \quad 0 \end{array}\end{align*}

The answer is \begin{align*}4.721 \div 2.465\end{align*} is approximately 1.88.

### Review

Find the quotient.

- \begin{align*}5.4\div4.5\end{align*}
- \begin{align*}8.71\div6.7\end{align*}
- \begin{align*}3.375\div2.25\end{align*}
- \begin{align*}11.2\div5.6\end{align*}
- \begin{align*}19.11\div1.3\end{align*}
- \begin{align*}28.992\div18.12\end{align*}
- \begin{align*}113.52\div12.9\end{align*}
- \begin{align*}31.93\div3.1\end{align*}
- \begin{align*}46.125\div6.15\end{align*}
- \begin{align*}84.28\div17.2\end{align*}

Find the quotient and then round to the nearest hundredth.

- \begin{align*}113.409\div25.21\end{align*}
- \begin{align*}81.862\div6.4\end{align*}
- \begin{align*}377.15\div1.54\end{align*}
- \begin{align*}9.17\div4.5\end{align*}
- \begin{align*}7.56\div2.12\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.13.

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 , 6 is the divisor and 152 is the dividend.Estimation

Estimation is the process of finding an approximate answer to a problem.Quotient

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

In this concept, you will learn how to divide decimals with and without rounding.

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