Dividing Whole Numbers by Fractions and Mixed Numbers

Whole number division exercises with fractions and mixed numbers are easy and not intimidating if you just follow the steps.

Let's turn the whole number into an improper fraction.
How do we do that?
In the numerator, we write the number itself (the whole number) and in the denominator, we always write $1$.

For example:

Convert the number $8$ to a fraction:
In the numerator, write $8$ and in the denominator, write $1$.
We get:
$8 \over 1$

Convert the number $1$ to a fraction:
In the numerator, we write $1$ (because this is our whole number) and in the denominator, we also write $1$ because that is the rule.
We get:
$1 \over 1$

After converting the whole number to an improper fraction and having only fractions in the exercise, we replace the division operation with multiplication and switch the positions of the numerator and the denominator in the second fraction.

For example:

Let's perform the second step in the exercise –
$\frac{4}{1}:\frac{2}{3}=$

We will turn the division operation into multiplication and do not forget to switch the positions of the numerator and the denominator in the second fraction. We get:
$\frac{4}{1}*\frac{3}{2}=$

We solve by multiplying fractions – numerator times numerator and denominator times denominator.

For example:
$\frac{4}{1}*\frac{3}{2} = \frac{12}{2}$

$\frac{12}{2} = 6$

And now let's practice!
Here is the exercise:

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

Solution:

Let's solve it step by step. First, we will convert the whole number $5$ into an improper fraction.
In the numerator, we write $5$ and in the denominator $1$. We get:

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

Now we move on to the second step and turn the exercise into a multiplication exercise without forgetting to switch the positions of the numerator and the denominator in the second fraction. We get:

$\frac{5}{1}\cdot\frac{5}{4}=$

We will solve by multiplying fractions – numerator times numerator and denominator times denominator, and we get:

$\frac{25}{4}= 6\frac{1}{4}$

First, let's recall the difference between an improper fraction and a mixed number:
Improper fraction – a fraction that consists only of a numerator and a denominator.
Mixed number – a number that consists of both whole numbers and a fraction.

Convert the whole number to an improper fraction.
In the numerator, write the whole number and in the denominator, write the number 1 (just as we learned at the beginning of the article)

Convert the mixed number to an improper fraction

How do you convert a mixed number to an improper fraction?

• The denominator will remain the same.
• To find the numerator - multiply the whole number by the denominator and then add the numerator to it. The number you get is the new number that will appear in the numerator.

Important tip!
Before converting the mixed fraction to an improper fraction, check if it can be simplified and only then convert it to an improper fraction.
For example:
Convert the mixed number $5 \frac{2}{6}$ to a fraction.

Solution:

It seems that we can simplify $2 \over 6$ to $1 \over 3$
Therefore, we rewrite the mixed number as $5 \frac{1}{3}$ and convert it to an improper fraction.
We multiply the whole number $5$ by the denominator $3$ and add the numerator $1$
$5*3+1=16$

The number we obtained ($16$) will be written in the numerator and the denominator will remain the same.
We get:
$16 \over 3$

Note – we kept the denominator of the fraction after the reduction (because we turned it into an improper fraction) and not the denominator of the original fraction.

After converting the whole number to an improper fraction and the mixed number to an improper fraction, and we only have fractions in the exercise, we will replace the division operation with a multiplication operation and switch the positions of the numerator and the denominator in the second fraction. (As we learned at the beginning of the article).

Let's practice!

Here is the exercise –
$7:2\frac{1}{3}=$

Solution:

Let's convert the whole number and the mixed number to improper fractions.
The whole number $7$ is converted to $7\over1$
The mixed number $2 \frac{1}{3}$, which cannot be simplified, is converted to an improper fraction.
Multiply the whole number $2$ by the denominator $3$ and add $1$ to get $7 \over 3$ and write:

$\frac{7}{1}:\frac{7}{3}=$

We will invert the numerator and the denominator in the second fraction and change the division to multiplication. We get:

$\frac{7}{1}\cdot\frac{3}{7}=$

We will solve by multiplying fractions:

$\frac{21}{7} = 3$