# Division of Whole Numbers by Fractions

## Understand how to divide a whole number by a fraction.

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Division of Whole Numbers by Fractions
Source: https://www.flickr.com/photos/39908901@N06/11386507493/

Vanessa is making a game. She has a 40 inch long roll of paper and wants to divide this piece of paper into \begin{align*}\frac{1}{2}\end{align*} inch strips. How many strips of paper can Vanessa make?

In this concept, you will learn how to divide a whole number by a fraction.

### Dividing Whole Numbers by Fractions

Sometimes you will need to divide a whole number by a fraction. Here is an example.

\begin{align*}1 \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

At first glance, you might think that the answer would be one-half. This problem is not asking for \begin{align*}\frac{1}{2}\end{align*} of 1. It is asking for how many \begin{align*}\frac{1}{2}\end{align*}s are in 1. Here is a diagram of 1 whole.

Divide the whole into \begin{align*}\frac{1}{2}\end{align*}s.

There are 2 one-half sections. The answer is 2.

You learned that when you divide a fraction by a whole number, you instead multiply by the reciprocal of the divisor. The same applies to dividing a whole number by a fraction. Change the operation to multiplication and change the divisor to its reciprocal.

Here is the division problem again.

\begin{align*}1 \div \frac{1}{2} = \end{align*}

First, change the operation to multiplication and \begin{align*}\frac{1}{2}\end{align*} to its reciprocal.

\begin{align*}1 \div \frac{1}{2} = 1 \times \frac{2}{1} \end{align*}

Then, multiply. Remember that any whole number can be written as a fraction, \begin{align*}n=\frac{n}{1}\end{align*}. In this example, the identity property of multiplication states that a number multiplied by 1 will be the number itself. \begin{align*}\frac{2}{1}\end{align*} is also equal to 2.

\begin{align*}1 \times \frac{2}{1} = 1 \times 2 = 2\end{align*}

The answer is the same as when you used the pictures.

### Examples

#### Example 1

Earlier, you were given a problem about Vanessa and her game.

Vanessa wants to cut \begin{align*}\frac{1}{2}\end{align*} inch strips of paper from a roll of paper that is 40 inches long. Divide 40 by \begin{align*}\frac{1}{2}\end{align*} to find the total number of strips of paper she can make.

First, write a division problem.

\begin{align*}40 \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

Then, change this into a multiplication problem. Multiply by the reciprocal of the divisor.

\begin{align*}40 \div \frac{1}{2} = 40 \times \frac{2}{1} \text{strips of paper}\end{align*}

Next, multiply the fractions. Remember that \begin{align*}\frac{2}{1}\end{align*} is also equal to 2.

\begin{align*}40 \times \frac{2}{1} = 40 \times 2 = 80\end{align*}

Vanessa can make 80 strips from a 40 inch long roll of paper.

#### Example 2

Divide by fraction.

\begin{align*}25 \div \frac{2}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}

First, change the operation to multiplication and \begin{align*}\frac{2}{5}\end{align*} to its reciprocal.

\begin{align*}25 \div \frac{2}{5} = 25 \times \frac{5}{2} \end{align*}

Then, multiply. Remember that 25 can also be written as \begin{align*}\frac{25}{1}\end{align*}.

\begin{align*}\frac{25}{1} \times \frac{5}{2}=\frac{125}{2} \end{align*}

Next, simplify the fraction. Convert the improper fraction to a mixed number.

\begin{align*}\frac{125}{2} = 62 \frac{1}{2}\end{align*}

The answer is \begin{align*}62 \frac{1}{2}\end{align*}.

#### Example 3

Divide by fraction: \begin{align*}4 \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}.

First, change the expression. Multiply by the inverse of the divisor.

\begin{align*}4\div \frac{1}{2} = 4 \times \frac {2}{1}\end{align*}

Then, multiply.

\begin{align*}4 \times 2 = 8\end{align*}

#### Example 4

Divide by fraction: \begin{align*}6 \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}.

First, change the expression. Multiply by the inverse of the divisor.

\begin{align*}6 \div \frac{1}{3} = 6 \times \frac{3}{1}\end{align*}

Then, multiply.

\begin{align*}6 \times 3 = 18\end{align*}

#### Example 5

Divide by fraction: \begin{align*}12 \div \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}.

First, change the expression. Multiply by the inverse of the divisor.

\begin{align*}12 \div \frac{3}{4} =12 \times \frac{4}{3}\end{align*}

Then, multiply.

\begin{align*}12 \times \frac{4}{3} =\frac{\cancel{12}^4}{1} \times \frac{4}{\cancel{3}^1}= 16\end{align*}

### Review

Divide the following whole numbers and fractions.

1. \begin{align*}8 \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}18 \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}28 \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}14 \div \frac{1}{7} = \underline{\;\;\;\;\;\;\;}\end{align*}
5. \begin{align*}16 \div \frac{2}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
6. \begin{align*}22 \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
7. \begin{align*}24 \div \frac{2}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}
8. \begin{align*}36 \div \frac{2}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
9. \begin{align*}40 \div \frac{3}{10} = \underline{\;\;\;\;\;\;\;}\end{align*}
10. \begin{align*}60 \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
11. \begin{align*}12 \div \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
12. \begin{align*}48 \div \frac{2}{12} = \underline{\;\;\;\;\;\;\;}\end{align*}
13. \begin{align*}18 \div \frac{1}{6} = \underline{\;\;\;\;\;\;\;}\end{align*}
14. \begin{align*}30 \div \frac{2}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}
15. \begin{align*}45 \div \frac{5}{9} = \underline{\;\;\;\;\;\;\;}\end{align*}

To see the Review answers, open this PDF file and look for section 7.9.

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### Vocabulary Language: English

TermDefinition
Inverse Operation Inverse operations are operations that "undo" each other. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.
reciprocal The reciprocal of a number is the number you can multiply it by to get one. The reciprocal of 2 is 1/2. It is also called the multiplicative inverse, or just inverse.