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.

**Follow the below steps:**

(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.

Follow the below steps:**(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.**