Vanessa is making a game. She has a 40 inch long roll of paper and wants to divide this piece of paper into

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 s are in 1. Here is a diagram of 1 whole.

Divide the whole into

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

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,

. In this example, the identity property of multiplication states that a number multiplied by 1 will be the number itself. 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 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*}

The answer is 8.

#### 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*}

The answer is 18.

#### 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*}

The answer is 16.

### Review

Divide the following whole numbers and fractions.

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

### Review (Answers)

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