# 7.11: Division of Whole Numbers by Fractions

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

**Practice**Division of Whole Numbers by Fractions

Remember Julie and her game? Julie has 40 inches of paper and she wants to divide this piece of paper in one - half inch strips. How can she do it? In the last Concept, you divided a fraction by a whole number, but in this problem, you are going to work the other way around.

To help Julie figure out how to divide this piece of paper into one -half inch strips, you will need to divide a whole number by a fraction.

**Pay close attention and you will learn all that you to know in this Concept.**

### Guidance

We can also divide a whole number by a fraction. When we divide a whole number by a fraction we are taking a whole and dividing it into new wholes.

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

Now at first glance, you would think that this answer would be one-half, but it isn’t. We aren’t asking for \begin{align*}\frac{1}{2}\end{align*} of one we are asking for 1 divided by one-half. Let’s look at a picture.

Now we are going to divide one whole by one-half.

Now we have two one-half sections. **Our answer is two.**

**We can test this out by using the rule that we learned in the last Concept.**

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

**Our answer is the same as when we used the pictures.**

It’s time for you to try a few of these on your own. Find each quotient.

#### Example A

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

**Solution:\begin{align*}8\end{align*}**

#### Example B

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

**Solution:\begin{align*}18\end{align*}**

#### Example C

\begin{align*}12 \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

**Solution:\begin{align*}48\end{align*}**

Now back to Julie and the ribbon. Here is the original problem once again.

Remember Julie and her game? Julie has 40 inches of paper and she wants to divide this piece of paper in one - half inch strips. How can she do it? In the last Concept, you divide a fraction by a whole number, but in this problem, you are going to work the other way around.

To help Julie figure out how to divide this piece of paper into one -half inch strips, you will need to divide a whole number by a fraction.

To figure this out, we first can write an equation. Julie wants to divide 40" of paper into one - half inch strips.

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

Next, we can change this into a multiplication problem.

\begin{align*}40 \times \frac{2}{1} = 80\end{align*} strips of paper.

**This is our answer.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Inverse Operation
- opposite operation. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.

- Reciprocal
- the inverse of a fraction. We flip a fraction’s numerator and denominator to write a reciprocal. The product of a fraction and its reciprocal is one.

### Guided Practice

Here is one for you to try on your own.

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

**Answer**

First, we have to convert this problem to a multiplication problem.

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

Next, we convert this improper fraction to a mixed number.

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

### Video Review

Here are videos for review.

Khan Academy Dividing Fractions Example

James Sousa Dividing Fractions

James Sousa Example of Dividing Fractions

James Sousa Another Example of Dividing Fractions

### Practice

Directions: 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*}

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Term | Definition |
---|---|

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. |

### Image Attributions

Here you'll learn to divide a whole number by a fraction.