# 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*}

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*} 112.**

**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*} 259.**

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*} 13**

#### Example B

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

**Solution: \begin{align*}3 \frac{1}{2}\end{align*} 312**

#### Example C

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

**Solution: \begin{align*} \frac{3}{4}\end{align*} 34**

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*}

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*} 89.**

### 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*}

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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.