# 7.12: Quotients of Fractions

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

**Practice**Quotients of Fractions

Are you usually hungry after school? Julie is, take a look.

After school, Julie arrives at home. She is starving after a busy day and looks around the kitchen for something to eat. She finds \begin{align*} \frac{1}{2}\end{align*} a pan of brownies. Julie divides the brownies into quarter sections. How many brownies are in each part?

Do you know how to figure this out?

Julie will need to divide using the following expression.

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

To solve this problem, you will need to understand how to divide fractions.

**This Concept is all about dividing a fraction by another fraction.**

### Guidance

In the past two Concepts, you have been dividing whole numbers by fractions and fractions by whole numbers. We can also use what we have learned when dividing a fraction by another fraction. Here is the rule.

Let’s apply these rules to dividing a fraction by another fraction.

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

**Start by applying the first rule and change the sign to multiplication. Then apply the second rule, the reciprocal of one-third is three over one.**

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

**Next, we multiply across and simplify.**

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

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

**As long as we apply the rules, the problem is very straightforward and simple to figure out. Let’s try another one.**

\begin{align*}\frac{8}{9} \div \frac{1}{3} = \frac{8}{9} \times \frac{3}{1} = \frac{24}{9} = 2\frac{5}{9}\end{align*}

We started with a fraction divided by a fraction, so we multiplied by the reciprocal. Our product was an improper fraction which we converted to a mixed number.

**Our quotient is \begin{align*}2\frac{5}{9}\end{align*}.**

It’s time for you to practice a few of these on your own. Be sure that your answer is in simplest form.

#### Example A

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

**Solution: \begin{align*} \frac{1}{3}\end{align*}**

#### Example B

\begin{align*}\frac{7}{8} \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

**Solution: \begin{align*}3 \frac{1}{2}\end{align*}**

#### Example C

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

**Solution: \begin{align*} \frac{3}{4}\end{align*}**

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

After school, Julie arrives at home. She is starving after a busy day and looks around the kitchen for something to eat. She finds \begin{align*} \frac{1}{2}\end{align*} a pan of brownies. Julie divides the brownies into quarter sections. How many brownies are in each part?

Do you know how to figure this out?

Julie will need to divide using the following expression.

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

First, Julie can change this problem to a multiplication problem by multiplying the first fraction by the reciprocal of the second fraction.

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

When we simplify, we get an answer of 2.

**Each section will have two brownies in it.**

### 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{4}{9} \div \frac{1}{2}\end{align*}

**Answer**

First, we change this problem into a multiplication problem.

\begin{align*} \frac{4}{9} \times \frac{2}{1}\end{align*}

Next, we multiply across.

**The answer is \begin{align*} \frac{8}{9}\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 each pair of fractions.

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

2. \begin{align*}\frac{1}{4} \div \frac{1}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}

3. \begin{align*}\frac{2}{5} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

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

5. \begin{align*}\frac{6}{8} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

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

7. \begin{align*}\frac{5}{6} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

8. \begin{align*}\frac{6}{10} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

9. \begin{align*}\frac{9}{18} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

10. \begin{align*}\frac{8}{9} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

11. \begin{align*}\frac{15}{16} \div \frac{1}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}

12. \begin{align*}\frac{8}{11} \div \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

13. \begin{align*}\frac{12}{16} \div \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

14. \begin{align*}\frac{20}{24} \div \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

15. \begin{align*}\frac{18}{20} \div \frac{4}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}

### Notes/Highlights Having trouble? Report an issue.

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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 fraction by a fraction.