Division of Fractions

Fraction division is a pleasant and simple topic, especially if you already know how to solve fraction multiplications.
In this article, you will learn how to operate to divide fractions and discover how easy it is to do it with this method.

We will solve fraction divisions in the following way:
First step
Let's look at the exercise.

• In case there is any mixed number - we will convert it into a fraction
• In case there is any whole number - we will convert it into a fraction

Second step
We will convert the division into multiplication
Also, we will swap between the numerator and the denominator in the second fraction.
Third step
We will solve by multiplying numerator by numerator and denominator by denominator.

How are fractions multiplied?
Numerator by numerator and denominator by denominator.

How is a mixed number converted to a fraction?
Multiply the denominator by the whole number and add the numerator.
Write the result in the numerator. The denominator remains unchanged.

How is a whole number converted to a fraction?
Write the whole number in the numerator
and in the denominator write $1$.

Let's practice and see all the possible cases we might encounter:

Here is an exercise:

$\frac{4}{5}:\frac{3}{2}=$

Solution:
We will change the division operation to a multiplication and swap the numerator and denominator in the second fraction.

We will obtain:

$\frac{2}{3} \times \frac{4}{5}=$

We will solve the exercise as it is done in fraction multiplication - Numerator by numerator and denominator by denominator.

We will obtain:
$8 \over 15$

Now let's move on to a slightly more complex example:

$\frac{4}{6}:5\frac{1}{2}=$

Solution:
Upon examining the exercise, we will realize that there is a mixed number -> $5\frac{1}{2}$
We will convert it to a fraction:
$\frac{11}{2}$

Pay attention! After obtaining another fraction, the exercise looks as follows:
$\frac{4}{6}:\frac{11}{2}=$

Clearly, it remains a division exercise, now we must operate as is done to solve a division exercise:
We will change the division operation to multiplication and swap between the numerator and the denominator in the second fraction.
We will obtain:

$\frac{4}{6} \times \frac{11}{2}=$

Upon solving we will obtain:

$​​\frac{4}{33}=\frac{8}{66}$

Here is an exercise:
$3:\frac{1}{2}=$

Solution:
First, we will convert the whole number to a fraction: $3=\frac{3}{1}$
Let's copy the new exercise:
$\frac{3}{1}:\frac{1}{2}=$
We will operate according to the rules of fraction division, we will obtain:
$\frac{2}{1} \times \frac{3}{1}=$
$\frac{6}{1}=6$

Sometimes you will come across word problems from which you must extract the appropriate division exercise.

A seamstress has $30$ meters of fabric. The seamstress uses $1\frac{3}{4}$ m. to sew each shirt.

How many shirts can the seamstress make with the amount of fabric she has?
The solution may include a number that is not an integer.

Solution:
To find out how many shirts the seamstress can make with $30$ meters of fabric, we must divide $30$ (the amount of fabric she has) by the amount of fabric she uses for each shirt -> $1\frac{3}{4}$
The division exercise will be: $30:1\frac{3}{4}=$

We will convert the mixed number to a fraction:  $1\frac{3}{4}=\frac{7}{4}$
And the whole number $30$ to a fraction: $30=\frac{30}{1}$
Let's rewrite the exercise:
$\frac{30}{1}:\frac{7}{4}=$
Let's convert to multiplication and swap between the numerator and the denominator in the second fraction.
We will obtain:
$\frac{4}{7} \times \frac{30}{1}=$
Upon solving, we will obtain:
$\frac{120}{7}=17\frac{1}{7}$
The seamstress can make $17\frac{1}{7}$ shirts with the fabric she has.