# How to subtract fractions with different denominator

In this post we will learn subtraction of two or more fractions with different denominator.

Here we will discuss two methods of subtraction, one is a traditional method and other is a shortcut.

To understand this post you should have basic knowledge about fraction and LCM concept.

## Subtracting Fractions with different denominators

Here are the two methods discussed in the post
(a) LCM method of subtraction
(b) Cross Multiplication Method

### Subtracting Fractions using LCM method

In order to subtract two or more fractions, it is important to have fractions with same denominator.

Here we will take help of LCM to convert fractions into common denominator.

(a) Find LCM of denominators.

(b) Multiply fractions to make a denominator = LCM value

(c) Now all the fractions have same denominator.
Simply subtract the numerator and retain the denominator.

For Example
Subtract \mathtt{\frac{7}{6} \ \ \&\ \ \frac{4}{8}}

Solution
Here both the fraction have different denominator.

(a) Find LCM of denominator.

LCM (6, 8) = 24

(b) Multiply the fractions to make denominator 24

Fraction 7/6
Multiply numerator and denominator by 4

\mathtt{\Longrightarrow \frac{7\ \times \ 4}{6\times \ 4} \ =\ \frac{28}{24}}

Fraction 4/8
Multiply numerator and denominator by 3

\mathtt{\Longrightarrow \frac{4\ \times \ 3}{8\ \times \ 3} \ =\ \frac{12}{24}}

(c) Now both the fractions have same denominator.
Simply subtract the numerator and retain the denominator.

\mathtt{\Longrightarrow \ \frac{28}{24} \ -\ \frac{12}{24} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{28\ -\ 12}{24}}\\\ \\ \mathtt{\Longrightarrow \ \frac{14}{24}}

The fraction can be further simplified.
Divide numerator and denominator by 2.

\mathtt{\Longrightarrow \ \frac{14\div 2}{24\div 2}}\\\ \\ \mathtt{\Longrightarrow \frac{7}{12}}

Hence, 7/12 is the solution.

### Fraction Subtraction using Cross Multiplication

This is a shortcut method of fraction subtraction.

The method works best when the digits are small, so that fast multiplication can be done.

Let a/b and c/d are the fractions to subtract.

In order to subtract the numbers, follow the below steps:

(a) For numerator cross multiply the fractions.

(b) For denominator, multiply the existing denominators.

Example of Cross Multiplication Fraction Subtraction
Subtract \mathtt{\frac{2}{3} \ \&\ \frac{1}{4} \ }

Solution
(a) For numerator, do the cross multiplication of fraction

(b) For denominator, simply multiply the given denominators.

So the fraction becomes 5/12.
Hence, 5/12 is the solution.

### Examples of fraction subtraction with different denominators

Example 01
Subtract the fractions \mathtt{\ \frac{4}{3} \ \&\ \frac{2}{7}}

Solution
(a) For numerator, do cross multiplication.

(b) For denominator, multiply the existing denominators.

(c) The fraction formed is 22/21

Hence, 22/21 is the solution.

Examples 02
Subtract the fractions; \mathtt{\frac{11}{5} \ \&\ \frac{1}{6}}

Solution
We will do the subtraction using cross multiplication method.

(a) For numerator, do the cross multiplication.

(b) For denominator, multiply the existing denominator.

So the fraction becomes 61/30.

Hence, 61/30 is the solution.

Example 03
Subtract the fraction \mathtt{\frac{15}{20} \ \&\ \frac{5}{25}}

Solution
We will solve the subtraction using LCM method.

(a) Find the LCM of denominators

LCM ( 20, 25 ) = 100

(b) Multiply the denominator such that denominator becomes 100

Fraction 15/20
Multiply numerator and denominator 5

\mathtt{\Longrightarrow \frac{15\ \times \ 5}{20\ \times \ 5} \ =\ \frac{75}{100}}

Fraction 5/25
Multiply numerator and denominator by 4

\mathtt{\Longrightarrow \frac{5\ \times \ 4}{25\ \times \ 4} \ =\ \frac{20}{100}}

(c) Now both fractions have same denominator.
Subtract the numerator and retain the denominator.

\mathtt{\Longrightarrow \ \frac{75}{100} \ -\ \frac{20}{100} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{75\ -\ 20}{100}}\\\ \\ \mathtt{\Longrightarrow \ \frac{55}{100}}

The fraction can be simplified further.
Divide numerator and denominator by 5.

\mathtt{\Longrightarrow \ \frac{55\div 5}{100\div 5}}\\\ \\ \mathtt{\Longrightarrow \frac{11}{20}}

Hence, 11/20 is the solution

Example 04
Subtract the fraction; \mathtt{\frac{\ 5}{6} \ \ \&\ \ \frac{2}{9}}

Solution
Subtracting the fraction using LCM method.

(a) Find LCM of denominator

LCM (6, 9) = 18

(b) Multiply the fraction such that the denominator becomes 18

Fraction 5/6
Multiply numerator and denominator by 3

\mathtt{\Longrightarrow \frac{5\ \times \ 3}{6\ \times \ 3} \ =\ \frac{15}{18}}

Fraction 2/9
Multiply numerator and denominator by 2

\mathtt{\Longrightarrow \frac{2\ \times \ 2}{9\ \times \ 2} \ =\ \frac{4}{18}}

(c) Now both the fraction have same denominator.
Subtract the numerator and retain the denominator.

\mathtt{\Longrightarrow \ \frac{15}{18} \ -\ \frac{4}{18} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{15\ -\ 4}{18}}\\\ \\ \mathtt{\Longrightarrow \ \frac{11}{18}}

Hence, 11/18 is the solution.

Example 05
Subtract \mathtt{\frac{\ 3}{10} \ \ \&\ \ \frac{3}{11}}

Let’s solve the fraction using cross multiplication method.

(a) For numerator, do the cross multiplication.

(b) For denominator, multiply the existing denominators.

So the fraction becomes 3/110.
Hence, fraction 3/110 is the solution.