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# Quotients of Fractions

## Understand the process of how to find a quotient between two fractions.

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Quotients of Fractions
Credit: Efraimstochter
Source: https://pixabay.com/en/waffle-waffle-irons-waffle-bake-203024/

Corey is planning on making waffles. The recipe says to use 34\begin{align*}\frac{3}{4}\end{align*} cups of flour, but Corey can only find the 13\begin{align*}\frac{1}{3}\end{align*}-cup measuring cup. How many 13\begin{align*}\frac{1}{3}\end{align*}-cups does Corey need to make his waffles?

In this concept, you will learn how to divide a fraction by a fraction.

### Dividing Fractions

When dividing whole numbers and fractions, you first change the operation to multiplication and then change the divisor to its reciprocal. The same rule applies to dividing a fraction by another fraction. Here is a division problem.

12÷13=\begin{align*}\frac{1}{2} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

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

12÷13=12×31\begin{align*}\frac{1}{2} \div \frac{1}{3} = \frac{1}{2} \times \frac{3}{1}\end{align*}

Then, multiply the fractions.

12×31=32\begin{align*}\frac{1}{2} \times \frac{3}{1} = \frac{3}{2}\end{align*}

Next, simply the fraction. Convert the improper fraction a mixed number.

32=112\begin{align*}\frac{3}{2} = 1\frac{1}{2}\end{align*}

The quotient is 112\begin{align*}1\frac{1}{2}\end{align*}.

As long as you apply the rules, the problem is very straightforward and simple to figure out. Here is another one.

89÷13\begin{align*}\frac{8}{9} \div \frac{1}{3}\end{align*}

First, change the operation and change 13\begin{align*}\frac{1}{3}\end{align*} to its reciprocal.

89÷13=89×31\begin{align*}\frac{8}{9} \div \frac{1}{3} = \frac{8}{9} \times \frac{3}{1} \end{align*}

Then, multiply the fractions.

89×31=249\begin{align*} \frac{8}{9} \times \frac{3}{1} = \frac{24}{9} \end{align*}

Next, simplify the fraction. Convert the improper fraction to a mixed number.

249=259\begin{align*}\frac{24}{9} = 2\frac{5}{9}\end{align*}

The quotient is 259\begin{align*}2\frac{5}{9}\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about Corey and his waffles.

Corey needs to measure out 34\begin{align*}\frac{3}{4}\end{align*} cups of flour for his waffles, but can only find a 13\begin{align*}\frac{1}{3}\end{align*} measuring cup. Divide 34\begin{align*}\frac{3}{4}\end{align*} by 13\begin{align*}\frac{1}{3}\end{align*} to find how many 13\begin{align*}\frac{1}{3}\end{align*} cups Corey should use.

First, write an expression.

34÷13\begin{align*} \frac{3}{4} \div \frac{1}{3}\end{align*}

Then, change the operation to multiplication and change the divisor to its reciprocal.

34÷13=34×31\begin{align*} \frac{3}{4} \div \frac{1}{3} = \frac{3}{4} \times \frac{3}{1}\end{align*}

Next, multiply the fractions.

34×31=94\begin{align*} \frac{3}{4} \times \frac{3}{1} = \frac {9}{4}\end{align*}

Finally, convert the improper fraction to a mixed number.

94=214\begin{align*}\frac {9}{4} = 2\frac {1}{4}\end{align*}

Corey can use a little more than 2 of the 13\begin{align*}\frac{1}{3}\end{align*} measuring cup to make his waffles.

#### Example 2

Divide the fractions: 49÷12\begin{align*} \frac{4}{9} \div \frac{1}{2}\end{align*}= _____. Answer in simplest form.

First, change the operation to multiplication and 12\begin{align*}\frac{1}{2}\end{align*} to its reciprocal.

34÷13\begin{align*} \frac{3}{4} \div \frac{1}{3}\end{align*}

Next, multiply the fractions.

49×21=89\begin{align*}\frac{4}{9} \times \frac{2}{1}= \frac {8}{9}\end{align*}

The quotient is 89\begin{align*} \frac{8}{9}\end{align*}.

#### Example 3

Divide the fractions: 14÷34=\begin{align*}\frac{1}{4} \div \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, change the expression. Multiply by the inverse of the divisor.

14÷34=14×43\begin{align*}\frac{1}{4} \div \frac{3}{4} = \frac {1}{4} \times \frac {4}{3}\end{align*}

Then, multiply.

141×413=13\begin{align*} \frac {1}{\cancel{4}^1} \times \frac {\cancel{4}^1}{3} = \frac {1}{3} \end{align*}

The quotient is 13\begin{align*} \frac{1}{3}\end{align*}.

#### Example 4

Divide the fractions: 78÷14=\begin{align*}\frac{7}{8} \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, change the expression. Multiply by the inverse of the divisor.

78÷14=78×41\begin{align*}\frac{7}{8} \div \frac{1}{4} = \frac{7}{8} \times \frac{4}{1} \end{align*}

Then, multiply.

782×411=72\begin{align*} \frac{7}{\cancel{8}^2} \times \frac{\cancel{4}^1}{1} = \frac {7}{2}\end{align*}

Next, convert the improper fraction to a mixed number.

72=312\begin{align*}\frac{7}{2} = 3 \frac {1}{2}\end{align*}

The quotient is 312\begin{align*}3 \frac{1}{2}\end{align*}

#### Example 5

Divide the fractions: 14÷13=\begin{align*}\frac{1}{4} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, change the expression. Multiply by the inverse of the divisor.

14÷13=14×31\begin{align*}\frac{1}{4} \div \frac{1}{3} = \frac {1}{4} \times \frac{3}{1}\end{align*}

Then, multiply.

14×31=34\begin{align*}\frac {1}{4} \times \frac{3}{1} = \frac {3}{4}\end{align*}

The quotient is 34\begin{align*} \frac{3}{4}\end{align*}.

### Review

Divide the fractions. Answer in simplest form.

1. 12÷13=\begin{align*}\frac{1}{2} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
2. 14÷15=\begin{align*}\frac{1}{4} \div \frac{1}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}
3. 25÷12=\begin{align*}\frac{2}{5} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
4. 47÷13=\begin{align*}\frac{4}{7} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
5. 68÷12=\begin{align*}\frac{6}{8} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
6. 49÷13=\begin{align*}\frac{4}{9} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
7. 56÷12=\begin{align*}\frac{5}{6} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
8. 610÷12=\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*}

To see the Review answers, open this PDF file and look for section 7.10.

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Color Highlighted Text Notes

### Vocabulary Language: English

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.