Fractions with different denominators are called Unlike Fractions.

Example:

$\frac{5}{13}$ , $\frac{7}{2}$ , $\frac{8}{3}$ , $\frac{11}{17}$ ........... are called as unlike fractions.

There are a number of basic operations that can be performed on unlike fractions. Some of them are:

• Subtracting Unlike Fractions
• Multiplying Unlike Fractions
• Comparing Unlike Fractions

## What are Unlike Fractions?

When the denominators of any fractions are found to be unequal or are not the same they are termed as unlike fractions.

Operations like addition, subtraction, multiplication and division cannot be completed directly on unlike fractions.

The usage of proper applications of fundamental rules could change their form so that they become like fractions.

Example: $\frac{7}{3}$ , $\frac{4}{5}$ ....

The process for addition of unlike fractions is as follows:

1. Find the least common denominator of all the fractions
2. Rewrite the fractions to have the denominators equal to the LCD obtained in step 1
3. Add the numerators of all the fractions keeping the denominator value equal to the LCD obtained in step 1
4. Express the fraction in lowest terms.

### Examples on Adding Unlike Fractions

Given below are some of the examples on adding unlike fractions.

Example 1:

Find the sum of $\frac{5}{3}$ and $\frac{7}{4}$

Solution:

The denominators of the given fractions are 3 and 4.

The least common denominator (LCD) of 3 and 4 is 12.

Now, rewrite the fractions to have the denominators equal to the LCD

$\frac{5}{3}$ = $\frac{(5\times4)}{(3\times4)}$ = $\frac{20}{12}$

$\frac{7}{4}$ = $\frac{(7\times3)}{(4\times3)}$ = $\frac{21}{12}$

$\frac{5}{3}$ + $\frac{7}{4}$ = $\frac{20}{12}$ + $\frac{21}{12}$

$\frac{5}{3}$ + $\frac{7}{4}$ = $\frac{(20 + 21)}{12}$ = $\frac{41}{12}$

Since 41 and 12 are co primes, $\frac{41}{12}$ is the answer.

Example 2:

Find the sum of $\frac{6}{5}$ , $\frac{7}{4}$ and $\frac{2}{3}$

Solution:

The denominators of the given fractions are 5, 4 and 3.

The least common denominator (LCD) of 5, 4 and 3 is 60.

Now, rewrite the fractions to have the denominators equal to the LCD

$\frac{6}{5}$ = $\frac{(6\times12)}{(5\times12)}$ = $\frac{72}{60}$

$\frac{7}{4}$ = $\frac{(7\times15)}{(4\times15)}$ = $\frac{105}{60}$

$\frac{2}{3}$ = $\frac{(2\times20)}{(3\times20)}$ = $\frac{40}{60}$

$\frac{6}{5}$ + $\frac{7}{4}$ + $\frac{2}{3}$ = $\frac{72}{60}$ + $\frac{105}{60}$ + $\frac{40}{60}$

Express the fractions in the lowest terms by canceling the common factors in the numerator and the denominator.

$\frac{215}{60}$ = $\frac{(5\times43)}{(5\times12)}$ = $\frac{43}{12}$

Since 43 and 12 are co primes, $\frac{43}{12}$ is the answer.

## Subtracting Unlike Fractions

The process for subtraction of unlike fractions is as follows:

1. Find the least common denominator of all the fractions
2. Rewrite the fractions to have the denominators equal to the LCD obtained in step 1
3. Add the numerators of all the fractions keeping the denominator value equal to the LCD obtained in step 1
4. Express the fraction in the lowest terms.

### Examples on Subtracting Unlike Fractions

Given below are some of the examples on subtracting unlike fractions.

Example 1:

Simplify $\frac{5}{3}$ - $\frac{7}{4}$

Solution:

The denominators of the given fractions are 3 and 4.

The least common denominator (LCD) of 3 and 4 is 12.

Now, rewrite the fractions to have the denominators equal to the LCD

$\frac{5}{3}$ = $\frac{(5\times4)}{(3\times4)}$ = $\frac{20}{12}$

$\frac{7}{4}$ = $\frac{(7\times3)}{(4\times3)}$ = $\frac{21}{12}$

$\frac{5}{3}$ - $\frac{7}{4}$ = $\frac{20}{12}$ - $\frac{21}{12}$

Now, subtract the numerators

$\frac{5}{3}$ - $\frac{7}{4}$ = $\frac{(20 - 21)}{12}$ = $\frac{(- 1)}{12}$

Example 2:

Simplify $\frac{6}{5}$ - $\frac{7}{4}$

Solution:

The denominators of the given fractions are 5 and 4.

The least common denominator (LCD) of 5 and 4 is 20.

Now, rewrite the fractions to have the denominators equal to the LCD

$\frac{6}{5}$ = $\frac{(6\times4)}{(5\times4)}$ = $\frac{24}{20}$

$\frac{7}{4}$ = $\frac{(7\times5)}{(4\times5)}$ = $\frac{35}{20}$

$\frac{6}{5}$ - $\frac{(7)}{(4)}$ = $\frac{24}{20}$ - $\frac{35}{20}$

Now, subtract the numerators

$\frac{6}{5}$ - $\frac{(7)}{(4)}$ = $\frac{(24 - 35)}{20}$ = $\frac{(- 11)}{20}$

Express the fractions in the lowest terms by cancelling the common factors in the numerator and the denominator. In this case, the answer can't be reduced further. Therefore, the answer is $\frac{(- 11)}{20}$

## Multiplying Unlike Fractions

The procedure for multiplying unlike fractions is very much similar to that of multiplying like fractions. It is explained below:

1. Multiply the numerators of the unlike fractions
2. Multiply the denominators of the unlike fractions
3. The product of the two unlike fractions will have the result of Step 1 as the numerator and the result of Step 2 as the denominator
4. Write the product fraction in it's lowest terms by canceling the common factors in both the numerator and the denominator.

### Examples on Multiplying Unlike Fractions

Given below are some of the examples on multiplying unlike fractions.

Example 1:

Find the product of $\frac{2}{3}$ and $\frac{3}{4}$

Solution:

In the fractions $\frac{2}{3}$ and $\frac{3}{4}$

The product of the numerators is 2 x 3 = 6

The product of the denominators is 3 x 4 = 12

Now, place the product of the numerator over the product of the denominator

$\frac{2}{3}$ X $\frac{3}{4}$ = $\frac{6}{12}$

Reducing the product fraction into the simplest form, we get

$\frac{6}{12}$ = $\frac{(1\times6)}{(2\times6)}$ = $\frac{1}{2}$

Therefore, the product of $\frac{2}{3}$ and $\frac{3}{4}$ is $\frac{1}{2}$

Example 2:

Find the product of $\frac{3}{8}$ and $\frac{4}{3}$

Solution:

In the fractions $\frac{3}{8}$ and $\frac{4}{3}$

The product of the numerators is 3 x 4 = 12
The product of the denominator is 8 x 3 = 24

Now, place the product of the numerator over the product of the denominator

$\frac{3}{8}$ X $\frac{4}{3}$ = $\frac{12}{24}$

Reducing the product fraction into the simplest form, we get

$\frac{12}{24}$ = $\frac{12\times1}{12\times2}$ = $\frac{1}{2}$

Therefore, the product of $\frac{3}{8}$ and $\frac{4}{3}$ is $\frac{1}{2}$

## Comparing Unlike Fractions

The process for comparing unlike fractions is as follows:

• Multiply both the numerator and denominator of one fraction or both the fractions by the same number in order to make the denominators of both the fractions equal.
• Now, the denominator of both the fractions are equal.
• The fraction with the greatest numerator is greater than the one with the least denominator.

### Examples on Comparing Unlike Fractions

Given below are some of the examples on comparing unlike fractions.

Example:

Compare $\frac{7}{12}$ and $\frac{1}{3}$

Solution:

Multiply the numerator and denominator of $\frac{1}{3}$ by 4 in order to make the denominators of both the fractions equal.

$\frac{1}{3}$ = $\frac{(1\times4)}{(3\times4)}$ = $\frac{4}{12}$

Now, compare $\frac{7}{12}$ and $\frac{4}{12}$

The numerator of $\frac{7}{12}$ is greater than $\frac{4}{12}$. Therefore, $\frac{7}{12}$ > $\frac{4}{12}$. That is, $\frac{7}{12}$ > $\frac{1}{3}$.

## Examples of Unlike Fractions

Given below are some examples of unlike fractions.

Example 1:

Find the sum of $\frac{5}{7}$, $\frac{3}{4}$ and $\frac{3}{8}$

Solution:

The denominator of the given fractions are 7, 4 and 8.

The least common denominator (LCD) of 7, 4 and 8 is 56.

Now, rewrite the fractions to have the denominators equal to the LCD

$\frac{5}{7}$ = $\frac{(5\times8)}{(7\times8)}$ = $\frac{40}{56}$

$\frac{3}{4}$ = $\frac{(3\times14)}{(4\times14)}$ = $\frac{42}{56}$

$\frac{3}{8}$ =$\frac{(3\times7)}{(8\times7)}$ = $\frac{21}{56}$

$\frac{5}{7}$ + $\frac{3}{4}$ + $\frac{3}{8}$ = $\frac{40}{56}$ + $\frac{42}{56}$ + $\frac{21}{56}$

$\frac{5}{7}$ + $\frac{3}{4}$ + $\frac{3}{8}$ = $\frac{(40 + 42 + 21)}{56}$ = $\frac{123}{56}$

Express the fractions in the lowest terms by canceling the common factors in the numerator and the denominator

Since 123 and 56 are co primes, $\frac{123}{56}$ is the answer.

Example 2:

Simplify $\frac{7}{6}$ - $\frac{3}{4}$

Solution:

Given $\frac{7}{6}$ - $\frac{3}{4}$

The denominators of the given fractions are 6 and 4.

The least common denominator (LCD) of 6 and 4 is 24.

Now, rewrite the fractions to have the denominators equal to the LCD

Multiply by 4 for the fraction $\frac{7}{6}$ = $\frac{7\times4}{6\times4}$ = $\frac{28}{24}$

Multiply by 6 for the fraction $\frac{3}{4}$ = $\frac{3\times6}{4\times6}$ = $\frac{18}{24}$

$\frac{7}{6}$ - $\frac{3}{4}$ = $\frac{28}{24}$ - $\frac{18}{24}$

Now, subtract the numerators

$\frac{28}{24}$ - $\frac{18}{24}$ = $\frac{(28-18)}{24}$ = $\frac{10}{24}$ = $\frac{5}{12}$

Example 3:

Find the product of $\frac{9}{2}$ and $\frac{1}{6}$

Solution:

Given $\frac{9}{2}$ X $\frac{1}{6}$

In the fractions $\frac{9}{2}$ and $\frac{1}{6}$

The product of the numerators is 9 x 1 = 9
The product of the denominator is 2 x 6 = 12

Now, place the product of the numerator over the product of the denominator

$\frac{9}{2}$ X $\frac{1}{6}$ = $\frac{9}{12}$

Reducing the product fraction into the simplest form, we get

$\frac{9}{12}$ = $\frac{3\times3}{3\times4}$ = $\frac{3}{4}$

Therefore, the product of $\frac{9}{2}$ and $\frac{1}{6}$ is $\frac{3}{4}$