# 7.10: Division of Fractions by Whole Numbers

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

**Practice**Division of Fractions by Whole Numbers

Have you ever made your own game? Take a look at this dilemma.

As part of her project, Julie has decided to create a game about the rainforest. She will play the game with her classmates as part of her presentation, so the students can learn some information in a new way. Besides, Julie loves games! To create the game, Julie is going to create question cards on strips of paper. She wants each strip of paper to be \begin{align*}\frac{3}{4}''\end{align*}. That way she will have enough room to write the questions but the strips won’t be too wide either. Julie takes the big chart paper and cuts off a piece 20” wide. She is sure that she will have enough paper to cut her question strips. There are 25 students in Julie’s class. She wants each student to have one question to answer. Given the size of the chart paper and the size Julie wants each strip to be, does she have enough paper?

**Julie isn’t sure. She needs your help. To figure out the problem, Julie will need to divide fractions. You can help her. Pay attention in this Concept and you will learn all that you need to know about dividing fractions.**

### Guidance

You have learned a couple of things about dividing fractions. The first is that to divide fractions we are actually use the inverse operation, multiplication. The second is that the second fraction is going to become its reciprocal or opposite. These are a few basic notes, but we haven’t applied them to actually dividing yet. Let’s begin.

**How do we divide a fraction by a whole number?**

To divide a fraction by a whole number we have to think about what we are actually being asked to do. **We are being asked to take a part of something and split it up into more parts.**

\begin{align*}\frac{1}{2} \div 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

This problem is asking us to take one-half and divide into three parts. Here is a picture of what this would look like.

This is one half. If we were going to divide one-half into three parts, how much would be in each part?

Here we divided the one-half into three sections. But we couldn’t just do that with one part of the whole so we divided the other half into three sections too.

**Each part is** \begin{align*}\frac{1}{6}\end{align*} **of the whole.**

**How can we do this without drawing a lot of pictures?**

That is where multiplying by the reciprocal comes in handy.

\begin{align*}\frac{1}{2} \div 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

**First, change the division to multiplication.** **Next, invert the second fraction, which is a whole number 3, make it to the fraction \begin{align*}\frac{3}{1}\end{align*} then make that into its reciprocal \begin{align*}\frac{1}{3}\end{align*}.** **Now, we can find the product.**

\begin{align*}\frac{1}{2} \div 3 = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\end{align*}

**Notice that the answer is the same as when we divided using the pictures!!**

Practice solving these on your own. Remember to simplify the quotient (the answer) if you can.

#### Example A

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

**Solution: \begin{align*} \frac{1}{8}\end{align*}**

#### Example B

\begin{align*}\frac{3}{4} \div 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

**Solution: \begin{align*} \frac{1}{4}\end{align*}**

#### Example C

\begin{align*}\frac{4}{5} \div 2 = \underline{\;\;\;\;\;\;\;}\end{align*}

**Solution: \begin{align*} \frac{2}{5}\end{align*}**

Now let's help Julie figure out how to make her game.

**To figure this out, we can set up a division problem. We are dividing the 20” into as many \begin{align*}\frac{3}{4}''\end{align*} strips as possible.**

\begin{align*}20 \div \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

**Our first step is to change the operation to multiplication and to multiply 20 by the reciprocal of three-fourths.**

\begin{align*}20 \div \frac{3}{4} = \frac{20}{1} \times \frac{4}{3}\end{align*}

**Notice that we also made 20 into a fraction over one. Now we are ready to multiply and simplify.**

\begin{align*}20 \div \frac{3}{4} = \frac{20}{1} \times \frac{4}{3} = \frac{80}{3} = 26\frac{2}{3}\end{align*}

**Julie can cut 26 strips of paper from her large sheet. She will have enough strips for each student to have a question. There is also \begin{align*}\frac{2}{3}\end{align*} of another strip left over.**

### 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*}\frac{6}{8} \div 4 = \underline{\;\;\;\;\;\;\;}\end{align*}

**Answer**

To begin, we have to rewrite this problem as a multiplication problem.

\begin{align*} \frac{6}{8} \times \frac{1}{4} = \frac{3}{16}\end{align*}

**This is our answer.**

### Video Review

Here are videos for review.

Khan Academy Dividing Fractions Example

James Sousa Dividing Fractions

James Sousa Example of Dividing Fractions

### Practice

Directions: Divide each fraction and whole number.

1. \begin{align*}6 \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

2. \begin{align*}8 \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

3. \begin{align*}9 \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

4. \begin{align*}10 \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

5. \begin{align*}5 \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

6. \begin{align*}7 \div \frac{1}{8} = \underline{\;\;\;\;\;\;\;}\end{align*}

7. \begin{align*}4 \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

8. \begin{align*}7 \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

9. \begin{align*}12 \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

10. \begin{align*}11 \div \frac{1}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}

11. \begin{align*}\frac{1}{2} \div 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

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

13. \begin{align*}\frac{1}{9} \div 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

14. \begin{align*}\frac{2}{3} \div 4 = \underline{\;\;\;\;\;\;\;}\end{align*}

15. \begin{align*}\frac{4}{7} \div 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

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

17. \begin{align*}\frac{3}{7} \div 4 = \underline{\;\;\;\;\;\;\;}\end{align*}

18. \begin{align*}\frac{1}{5} \div 6 = \underline{\;\;\;\;\;\;\;}\end{align*}

19. \begin{align*}\frac{8}{9} \div 2 = \underline{\;\;\;\;\;\;\;}\end{align*}

20. \begin{align*}\frac{6}{7} \div 4 = \underline{\;\;\;\;\;\;\;}\end{align*}

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 fraction and a whole number.

## Concept Nodes:

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.