Multiplication of Integers by a Fraction and a Mixed Number

In this article, we will learn how to multiply an integer with a fraction and a mixed number without any problem!

When it comes to multiplication exercises, there is no need to find a common denominator and all we have to do is convert the integers and mixed numbers into equivalent fractions.

Convert whole numbers and mixed numbers into equivalent fractions, with only the numerator and denominator and rewrite the exercise.

Multiply the numerators -> numerator by numerator by numerator

Multiply the denominators -> denominator by denominator by denominator

Product of the numerators -> will be the new numerator.

Product of the denominators -> will be the new denominator.

To convert an integer into an equivalent fraction, we will write the given number in the numerator and in the denominator we will write $1$.

Convert $3$ into an imaginary fraction.

Solution:

We will write $3$ in the numerator and $1$ in the denominator.

We obtain: $3 \over 1$

We will do this for any given number.

Examples:

$\frac{72}{1}=72$

$\frac{1}{1}=1$

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

We will multiply the whole number by the denominator and add the numerator to the obtained result.

The final result we obtain will appear in the new numerator.

The denominator will remain the same.

Convert the mixed number $4 \frac{2}{3}$ into an equivalent fraction.

Solution:

We will multiply the whole number $4$ by the denominator $3$ and then add $2$.

We obtain: $4\times 3+2=14$

$14$ will be the new numerator.

The denominator will remain the same: $3$.

We obtain that:

$4 \frac{2}{3}=\frac {14}{3}$

Now that we know how to convert whole and mixed numbers into equivalent fractions, we can continue.

$4\times \frac {1}{3}\times 5\frac {1}{2}$

Solution:

The first step

Convert whole numbers and mixed numbers into equivalent fractions and rewrite the exercise:

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

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

Note that, since this is a multiplication exercise, we can apply the substitution property.

It doesn't matter in which position we place the equivalent fractions we received, the result will not change.

Rewrite the exercise:

$\frac{1}{3}\times\frac{11}{2}\times\frac{4}{1}=$

Now let's go to the second step:

Multiply both the numerators and the denominators separately.

We obtain:

$\frac{1\times 11\times 4}{3\times 2\times 1}=\frac {44}{6}$

We can convert the result we obtained $\frac {44}{6}$ into a mixed number $2 \frac {2}{6}$.

$7 \frac {2}{9}\times 2\times \frac {2}{5}=$

Solution:

First, we convert all whole and mixed numbers into equivalent fractions. We obtain:

$7 \frac {2}{9}=\frac {7\times 9+2}{2}=\frac {65}{9}$

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

Rewrite the exercise only with equivalent fractions:

$\frac {65}{9}\times \frac {2}{1}\times \frac {2}{5}=$

Multiply both the numerators and the denominators separately and you get:

$\frac {65}{9}\times \frac {2}{1}\times \frac {2}{5}=\frac {260}{45}$

We can convert the final result we obtained $\frac {260}{45}$ into a mixed number $5 \frac {35}{45}$

Note: we can reduce the fraction even further and obtain: $5\frac{35}{45}=5​​\frac{7}{9}$

Think of it this way:

The number is composed of integers and another fraction. We don't touch the integers and leave $5$.

Now remains the fraction $\frac {35}{45}$ which is identical in value to the fraction $\frac {7}{9}$

And therefore $\frac {535}{45}$ and $\frac {57}{9}$ are identical in value.

How to convert an equivalent fraction into a mixed number?

Convert the equivalent fraction $\frac {241}{62}$ into a mixed number.

Solution:

To convert an equivalent fraction into a mixed number, we will divide the numerator by the denominator and refer only to the whole number we receive (ignore the remainder).

$241:62=3…….$

This will be the number of wholes.

Then, we subtract from the given numerator the result of multiplying the whole number by the denominator to see how much is left to "complete" it.

That is:

$241-(3\times 62)=55$

The result we receive will be written in the numerator.

The denominator will remain the same.

We obtain: $3 \frac {55}{62}$

You can always test yourself and see if you return to the same equivalent fraction.