Multiplication and Division of Mixed Numbers

In this article, you will see how easy it is to multiply and divide mixed numbers.
You will understand the method, practice, and become a specialist in the topic!
Shall we start?

In multiplication and division exercises with mixed numbers, the first thing we should do is convert the mixed number into a fraction.

Mixed number – Number composed of a fraction and a whole number, for example: $3 \frac{1}{2}$
Fraction - Number composed of numerator and denominator, for example: $\frac{15}{5}$

• The denominator does not change.
• To find the numerator: Multiply the whole number by the denominator and then add the numerator. The result is the new number that will appear in the numerator.

For example:
Convert the mixed number $5 \frac{2}{3}$ to a fraction.

Solution:
We will multiply the whole number by the denominator and add the numerator
$5 \times 3+2=$
$15+2=17$
The obtained number ($17$) will be written in the numerator, while the denominator will not change.
This gives us:
$5 \frac{2}{3} = \frac{17}{3}$

Important recommendation!
Before converting the mixed number to a fraction, check if the fractional part can be reduced, and if so, convert it after performing the reduction.
The reduction will help you later in the exercises of multiplication and division of mixed numbers.

Given the following mixed number:
$6 \frac{25}{45}$
We can reduce it - We will reduce the numerator and the denominator by $5$ without touching the whole numbers. We will obtain:
$6 \frac{25}{45} = 6\frac{5}{9}$
It will be easier for us to operate with the reduced fraction.

After completing the first step and having converted all mixed numbers into fractions,
we will move on to the second step
Numerator by numerator and denominator by denominator.

$1\frac{2}{4} \cdot 5\frac{9}{18}$

Solution:

First, we will reduce the fractions as much as possible to make the following steps easier.
Let's rewrite the exercise:
$1\frac{1}{2} \cdot 5\frac{1}{2}$
Now we will convert the mixed numbers to fractions and rewrite the exercise:
$1\frac{1}{2} =\frac{3}{2}$

$5\frac{1}{2} =\frac{11}{2}$

$\frac{3}{2} \cdot \frac{11}{2}=$
Now we will multiply numerator by numerator and denominator by denominator, we will obtain:
$\frac{33}{4} = 8\frac{1}{4}$

$4\frac{1}{6} \cdot 7\frac{3}{6}$

Solution:
First, we will reduce what is possible and rewrite the exercise:
$4\frac{1}{6} \cdot 7\frac{1}{2}$
Now we will convert the mixed numbers to fractions and rewrite the exercise:
$\frac{25}{6} \cdot \frac{15}{2}=$

We will solve by multiplying numerator by numerator and denominator by denominator and we will obtain:
$\frac{25}{6} \cdot \frac{15}{2}=\frac{375}{12}$

We will simplify by 3 and obtain:
$\frac{375}{12}=\frac{125}{4}=31\frac{1}{4}$

After having reduced the fractions and having converted all the mixed numbers into fractions, all we have to do is:
Convert the division into multiplication
and change the location of the numerator and denominator in the second fraction -> that is, the fraction that is found after the sign.
Then we will solve by multiplying numerator by numerator and denominator by denominator.

Here we have a common exercise of division with mixed numbers:

$2\frac{3}{5}:3\frac{5}{15}=$

Solution:
The first thing we have to do is check if the fractions can be reduced.
In this exercise, we can only reduce the second fraction. We will reduce it and rewrite the exercise:

$2\frac{3}{5}:3\frac{1}{3}=$
The second thing we must do is convert the mixed numbers into fractions.
We will do it and rewrite the exercise:
$2\frac{3}{5}=\frac{13}{5}$
$3\frac{1}{3}=\frac{10}{3}$

$\frac{13}{5}:\frac{10}{3}=$
The third task awaiting us is to change the division operation to multiplication and swap the location of the numerator and the denominator in the second fraction -> that is, the fraction that is after the sign.
We will do it and obtain:
$\frac{13}{5} \cdot\frac{3}{10}=$
Now we will solve by multiplying numerator by numerator and denominator by denominator, we will obtain:
$\frac{13}{5} \cdot\frac{3}{10}=\frac{39}{50}$

Solve the exercise:

$5\frac{4}{12}:1\frac{5}{6}=$

Solution:
First, we will reduce what is possible and rewrite the exercise:
$5\frac{1}{3}:1\frac{5}{6}=$
Now we will convert the mixed numbers to fractions and rewrite the exercise:
$\frac{16}{3}:\frac{11}{6}=$
Now we will change the division operation to multiplication and swap the locations between the numerator and the denominator in the second fraction. We will obtain:
$\frac{16}{3}\cdot\frac{6}{11}=$
We will solve by multiplying numerator by numerator and denominator by denominator and we will obtain:
$\frac{16}{3}\cdot\frac{6}{11}=\frac{96}{33}$
We will reduce by $3$ and obtain:
$\frac{96}{33}=\frac{32}{11}=2\frac{10}{11}$