# Dividing Fractions Using the Inverse Operation

## Introduction to Dividing Fractions

Dividing fractions may seem complex at first, but with the right technique, it becomes a straightforward process. By employing the inverse operation—a clever strategy that transforms division into multiplication—students can conquer fraction division with ease.

## Understanding Dividing Fractions – Explanation

Division is typically one of the four basic operations in arithmetic, but when it comes to fractions, we use a method called the inverse operation.

The inverse operation for dividing fractions involves flipping the second fraction (known as finding the reciprocal) and changing the division sign to a multiplication sign.

The reciprocal of a fraction is simply swapping its numerator and denominator. So, for instance, the reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$. By multiplying the first fraction by this reciprocal, we achieve the same result as division.

### Fraction Reciprocal – Examples

$\frac{4}{5}$ $\frac{5}{4}$

$\frac{1}{6}$ $6$

$8$ $\frac{1}{8}$

$\frac{1}{2}$ $2$

$\frac{2}{3}$ $\frac{3}{2}$

What is the reciprocal of $\frac{2}{5}$?
How would you divide $\frac{2}{5}$ by $\frac{1}{3}$ using the inverse operation?

## Dividing Fractions Using the Inverse Operation – Example

Let's look at a simple example to demonstrate how this method works in action.

Divide $\frac{3}{5}$ by $\frac{2}{7}$.

First, we identify the reciprocal of the second fraction, $\frac{2}{7}$. The reciprocal is $\frac{7}{2}$.

Now, we multiply the first fraction by this reciprocal:

$\frac{3}{5} \times \frac{7}{2} = \frac{21}{10}$

The result simplifies to $2 \frac{1}{10}$, which is our final answer.

## Dividing Fractions Using the Inverse Operation – Practice

Now, let's solve another problem together.

Divide $\frac{4}{9}$ by $\frac{2}{3}$ and simplify your answer.

Let's put your understanding to the test with some independent practice.8

Divide $\frac{7}{8}$ by $\frac{5}{6}$ and simplify your answer.
Divide $\frac{3}{7}$ by $\frac{1}{2}$ and simplify your answer.
Divide $\frac{5}{8}$ by $\frac{3}{4}$ and simplify your answer.
Divide $\frac{2}{5}$ by $\frac{3}{10}$ and simplify your answer.
Divide $\frac{7}{12}$ by $\frac{7}{9}$ and simplify your answer.
Divide $\frac{1}{6}$ by $\frac{1}{3}$ and simplify your answer.

## Dividing Fractions Using the Inverse Operation – Summary

Key Learnings from this Text:

• To divide fractions, we use the inverse operation which involves finding the reciprocal of the second fraction and then multiplying.
• The reciprocal of a fraction is obtained by switching its numerator and denominator.
• After multiplying, always simplify the result to its lowest terms.

Continue practicing with a variety of problems to strengthen your understanding of dividing fractions using the inverse operation. Explore other content, such as interactive practice problems and printable worksheets, to enhance your learning experience.

## Dividing Fractions Using the Inverse Operation – Frequently Asked Questions

What is the inverse operation in dividing fractions?
How do you find the reciprocal of a fraction?
Do you need to simplify after dividing fractions using the inverse operation?
How can you divide mixed numbers using the inverse operation?
Can we use the inverse operation for dividing fractions with whole numbers?
Why is it necessary to find the reciprocal when dividing fractions?
Is the reciprocal of a whole number still a whole number?
What happens when you multiply a fraction by its reciprocal?
Can you divide fractions using the inverse operation without simplifying?
Are there any exceptions to using the inverse operation when dividing fractions?

## Dividing Fractions Using the Inverse Operation exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the learning text Dividing Fractions Using the Inverse Operation.
• ### Find the reciprocal.

Hints

To find the reciprocal of a fraction swap the numerator and denominator, i.e. the reciprocal of $\frac{6}{7}$ is $\frac{7}{6}$.

The reciprocal of a unit fraction is a whole number. For example, the reciprocal of $\frac{1}{6}$ is $6$.

The reciprocal of a whole number is a unit fraction. For example, the reciprocal of $2$ is $\frac{1}{2}$.

Solution

The reciprocal of $\frac{2}{3}$ is $\bf{\frac{3}{2}}$.

The reciprocal of $\frac{4}{5}$ is $\bf{\frac{5}{4}}$.

The reciprocal of $3$ is $\bf{\frac{1}{3}}$.

The reciprocal of $\frac{7}{2}$ is $\bf{\frac{2}{7}}$.

The reciprocal of $\frac{1}{4}$ is $\bf{4}$.

• ### Use the inverse operation to solve the division questions.

Hints

The inverse of multiplication is division.

Remember to use the reciprocal (swap the numerator and denominator). For example, the reciprocal of $\frac{3}{8}$ is $\frac{8}{3}$.

For example:

$\frac{2}{7} \div \frac{3}{4} = \frac{2}{7} \times \frac{4}{3} = \frac{8}{21}$

Solution

$\frac{2}{5} \div \frac{1}{4} = \frac{2}{5} \bf{\times \frac{4}{1}}$ $= \frac{8}{5}$

${}$

$\frac{5}{6} \div \frac{2}{7} = \frac{5}{6} \bf{\times \frac{7}{2}}$ $= \frac{35}{12}$

${}$

$\frac{3}{4} \div \frac{5}{3} = \frac{3}{4} \bf{\times \frac{3}{5}}$ $= \frac{9}{20}$

${}$

$\frac{4}{7} \div 5 = \frac{4}{7} \bf{\times \frac{1}{5}}$ $= \frac{4}{35}$

${}$

$8 \div \frac{3}{5} = 8 \bf{\times \frac{5}{3}}$ $=\frac{40}{3}$

• ### What is $\frac{2}{5} \div \frac{4}{9}$?

Hints

Multiply the first fraction by the reciprocal of the second fraction.

We can also say use the inverse operation - swap the numerator and denominator and multiply.

For example, $\frac{3}{8} \div \frac{5}{4} = \frac{3}{8} \times \frac{4}{5}$

There are two correct answers, one is simplified and the other is unsimplified.

For example, hint two has two possible answers.

$\frac{3}{8} \div \frac{5}{4} = \frac{3}{8} \times \frac{4}{5} = \bf{\frac{12}{40}}$ or $\bf{\frac{3}{10}}$

Solution

$\frac{2}{5} \div \frac{4}{9}$

$= \frac{2}{5} \times \frac{9}{4}$

$= \bf{\frac{18}{20}} = \bf{\frac{9}{10}}$

• ### Divide the fractions.

Hints

The reciprocal of a unit fraction is a whole number. For example, the reciprocal of $\frac{1}{6}$ is $6$.

The reciprocal of a whole number is a unit fraction. For example, the reciprocal of $2$ is $\frac{1}{2}$.

Remember when multiplying a fraction by a whole number we only multiply the numerator. For example, $\frac{2}{3} \times 4 = \frac{8}{3}$.

Solution

$\frac{3}{8} \div \frac{2}{5} = \frac{3}{8} \times \frac{5}{2} =$ $\bf{\frac{15}{16}}$

$\frac{3}{4} \div 5 = \frac{3}{4} \times \frac{1}{5} =$ $\bf{\frac{3}{20}}$

$6 \div \frac{7}{2} = 6 \times \frac{2}{7} =$ $\bf{\frac{12}{7}}$

$\frac{5}{9} \div \frac{1}{4} = \frac{5}{9} \times \frac{4}{1} =$ $\bf{\frac{20}{9}}$

• ### What is $\frac{2}{3} \div \frac{1}{5}$?

Hints

Multiply the first fraction by the reciprocal of the second fraction.

To find the reciprocal swap the numerator and denominator. For example, the reciprocal of $\frac{3}{8}$ is $\frac{8}{3}$.

An example:

$\frac{3}{4} \div \frac{2}{3} = \frac{3}{4} \times \frac{3}{2} = \frac{9}{8}$.

Solution

$\frac{2}{3} \div \frac{1}{5} = \frac{2}{3} \times \frac{5}{1} = \frac{10}{3}$

• ### What is $\frac{5}{6} \div \frac{5}{9}$?

Hints

Multiply the first fraction by the reciprocal of the second fraction.

A mixed number has a whole number part and a fractional part, i.e. $3 \frac{2}{5}$.
$\frac{5}{6} \div \frac{5}{9} = \frac{5}{6} \times \frac{9}{5} = \frac{45}{30} = \frac{3}{2} = 1\frac{1}{2}$