# Dividing Fractions – Definition with Examples

## What Do We Mean by Dividing Fractions?

By dividing fractions we mean the division involves at least one fraction. For example,

1. $\frac{3}{4} \div \frac{1}{2}$
2. $7 \div \frac{1}{2}$
3. $\frac{3}{4} \div 6$

The quotient of such multiplication could be a fraction or a whole number.

Let’s consider some examples to see what it means to divide fractions through visual models.

• Whole Number ÷ Fraction
• Fraction ÷ Whole
• Fraction ÷ Fraction

Visualising Fraction Division that Involves Remainder

Example 1:

Example 2:

## Steps to Divide Fractions

• Fraction by Fraction

To divide a fraction by another fraction, multiply the dividend fraction by the reciprocal of the divisor fraction. The reciprocal of a fraction can be found by interchanging its numerator and denominator. For example, $\frac{3}{4}$ is reciprocal of $\frac{4}{3}$ and vice-versa.

There is a quick way to remember the steps to divide fractions: Keep >> Change >> Flip.

Follow these steps to divide a fraction by fraction (either proper or improper):

STEP 1:

Keep the dividend the same.

Change the division sign to multiply

Flip the divisor by writing its reciprocal.

STEP 2: Multiply the fractions

STEP 3: Simplify the result, if needed.

Quick Tip: In case the dividend and/or divisor are mixed numbers, we first convert them into improper fractions and then follow the steps mentioned above.

• Fraction by Whole Number

Recall that the reciprocal of a whole number ‘a’ is the unit fraction $\frac{1}{a}$ because we can write a whole in fraction form with denominator equal to 1, that is, $\frac{a}{1}$.
Let’s understand how to divide when divisor is a whole number through an example:

Why do we flip the divisor after changing the division sign?

When we divide one number by another, let’s say, a is the dividend and b is the divisor, then we can write it as $\frac{a}{b}$. Also, we can write $\frac{a}{b}$ as the product of a and $\frac{1}{b}$. That is,

$a \div b = \frac{a}{b} = a \times \frac{1}{b}$

We can also understand it like this:

Division is the inverse operation of multiplication. So, dividing by a number is the same as multiplying by its reciprocal.

## Properties of Dividing Fractions

The properties of division with whole numbers hold true for fractions as well. Let’s check!

1. Property 1: When a fraction is divided by 1, the quotient is the fraction itself. $\frac{3}{4} \div 1 = \frac{3}{4}$
2. Property 2: When zero is divided by a non-zero fraction, then the quotient is always 0.
$0 \div \frac{3}{4} = 0$
3. Property 3: When a non-zero fraction is divided by itself, then the quotient is 1.
$\frac{3}{4} \div \frac{3}{4} = 1$
4. Property 4: Division by 0 is not possible and the result is not defined.
$\frac{3}{4}\div 0 =$ Not Defined

Solved Examples

Example 1: Divide: ​​$\frac{1}{5} \div \frac{1}{10}$.

Solution:

Keep $\frac{1}{5}$ as it is; change ÷ to × and flip $\frac{1}{10}$ and write it as $\frac{10}{1}$.

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

Example 2: Divide: ​​$1 \frac{2}{3} \div \frac{5}{7}$

Solution: Change 123 to improper fraction and then follow the steps to divide fractions.

$1\frac{2}{3} \div \frac{5}{7} = \frac{5}{3} \div \frac{5}{7} = \frac{5}{3} \times \frac{7}{5} = \frac{7}{3} = 2\frac{1}{3}$

Example 3: Max is painting toy cars. He has $2\frac{1}{4}$ L of paint. If each car requires $\frac{7}{3}$ L of paint, how many cars can Max paint?

Solution:

Quantity of paint with Max $= 2\frac{1}{4}$ L $= \frac{9}{4}$ L

Quantity of paint required to paint 1 car $= \frac{3}{8}$ L

Numbers of toy cars that Max can paint $= \frac{9}{4} \div \frac{3}{8}$

$= \frac{9}{4} \times \frac{8}{3}$

$= \frac{72}{12} = 6$

So, Max can paint 6 toys with  $2\frac{1}{4}$ L of paint.

Example 4: Melvin sang a medley of songs for 10 minutes. If one song was $2\frac{1}{2}$ minutes long, how many songs did Melvin sing?

Solution:

Duration for which Melvin sang $= 10$ minutes

Length of one song $= 2\frac{1}{2}$ minutes

Numbers of songs that Melvin sing $= 10 \div 2\frac{1}{2}$

$= 10 \div \frac{5}{2}$

$= 10 \times \frac{2}{5}$

$= \frac{20}{5} = 4$

So, Melvin sang 4 songs in 10 minutes.

## Practice Problems

1

### Divide: ​​$\frac{7}{8} \div \frac{3}{9}$

$\frac{21}{72}$
$\frac{24}{63}$
$\frac{21}{8}$
$\frac{8}{21}$
CorrectIncorrect
Correct answer is: $\frac{21}{8}$
$\frac{7}{8}\div\frac{3}{9} = \frac{7}{8}\times\frac{9}{3} = \frac{63}{24} = \frac{21}{8}$
2

### Divide: ​​$\frac{4}{8}\div 9$

9
$\frac{4}{89}$
$\frac{72}{4}$
$\frac{1}{18}$
CorrectIncorrect
Correct answer is: $\frac{1}{18}$
$\frac{4}{8}\div 9 = \frac{4}{8}\times\frac{1}{9} = \frac{4}{72} = \frac{1}{18}$
3

### Divide: $1\frac{1}{7}\div \frac{3}{5}$

$\frac{24}{35}$
$\frac{40}{21}$
$\frac{21}{40}$
$\frac{34}{35}$
CorrectIncorrect
Correct answer is: $\frac{40}{21}$
$1\frac{1}{7}\div\frac{3}{5} = \frac{8}{7}\div\frac{3}{5} = \frac{8}{7}\times\frac{5}{3} = \frac{40}{21}$
4

### Divide: $\frac{3}{7}\div\frac{3}{7}$

1
0
$\frac{9}{49}$
$\frac{49}{9}$
CorrectIncorrect