Dividing Whole Numbers by Fractions and Mixed Numbers

🏆Practice integer division between a fraction and a mixed number

Whole number division by a fraction and a mixed number

Dividing whole numbers by a fraction:

1)    Convert the whole number to a fraction
2)    Convert to a multiplication problem, remembering to swap the numerator and denominator of the second fraction
3)    Solve by multiplying fractions

Whole number division by a mixed number:

1)    Convert the whole number to a fraction
2)    Convert the mixed number to an improper fraction
3)    Convert the division problem to a multiplication problem, remembering to swap the numerator and denominator of the second fraction
4)    Solve by multiplying fractions

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Test yourself on integer division between a fraction and a mixed number!

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\( \frac{1}{3}:3= \)

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Division of a whole number by a fraction and a mixed number

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

How do we solve whole number division by a fraction?

The first step –

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 11.

For example:

Convert the number 88 to a fraction:
In the numerator, write 88 and in the denominator, write 11.
We get:
818 \over 1


Convert the number 11 to a fraction:
In the numerator, we write 11 (because this is our whole number) and in the denominator, we also write 11 because that is the rule.
We get:
111 \over 1

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The second step -

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 –
41:23=\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:
4132=\frac{4}{1}*\frac{3}{2}=

The third step –

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

For example:
4132=122\frac{4}{1}*\frac{3}{2} = \frac{12}{2}

122=6\frac{12}{2} = 6


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

5:45=5:\frac{4}{5}=

Solution:

Let's solve it step by step. First, we will convert the whole number 55 into an improper fraction.
In the numerator, we write 55 and in the denominator 11. We get:

51:45=\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:

5154=\frac{5}{1}\cdot\frac{5}{4}=

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

254=614\frac{25}{4}= 6\frac{1}{4}

Do you know what the answer is?

How do we solve the division of a whole number by a mixed number?

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.

The first step:

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)

Check your understanding

The second step:

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 5265 \frac{2}{6} to a fraction.

Solution:

It seems that we can simplify 262 \over 6 to 131 \over 3
Therefore, we rewrite the mixed number as 5135 \frac{1}{3} and convert it to an improper fraction.
We multiply the whole number 55 by the denominator 33 and add the numerator 11
53+1=165*3+1=16

The number we obtained (1616) will be written in the numerator and the denominator will remain the same.
We get:
16316 \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.

The third step:

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 –
7:213=7:2\frac{1}{3}=

Solution:

Let's convert the whole number and the mixed number to improper fractions.
The whole number 77 is converted to 717\over1
The mixed number 2132 \frac{1}{3}, which cannot be simplified, is converted to an improper fraction.
Multiply the whole number 22 by the denominator 33 and add 11 to get 737 \over 3 and write:

71:73=\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:

7137=\frac{7}{1}\cdot\frac{3}{7}=

We will solve by multiplying fractions:

217=3\frac{21}{7} = 3

Do you think you will be able to solve it?
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