# Subtraction of fraction || How to subtract fractions?

In this post we will methods to subtract two or more fractions.

To understand the methods, you should have basic understanding of the concept of fractions and LCM.

## How to subtract fractions?

There are two types of subtraction methods:

(a) Subtraction when fractions have same denominator.

(b) Subtraction when fractions have different denominator.

We will learn both the methods, step by step.

### Subtracting Fractions with same denominator

In this case, simply subtract the numerator and retain the same denominator.

In the above image; A/C & B/C are the fractions with same denominator.

Note that in the subtraction, we simply subtracted the numerator (A – B) and maintained the same denominator C.

### Subtracting Fractions with different denominator

In this case, we have to manipulate the fractions so that all of them have common denominator.

(a) Find LCM of the denominators.

(b) Multiply the fractions such that denominator becomes equal to LCM value.

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

### Why its important to have same denominator in fraction subtraction?

Because with different denominator it becomes difficult to subtract fraction.

For example, consider the fraction 1/2 and 2/3.

Fraction 1/2 mean that the object is divided into two equal part and 1 part is shaded.

Fraction 2/3 mean that object is divided into three equal part and 2 part is shaded.

Since the objects are divided into different parts it is very difficult to do subtraction.

Now consider the fraction 2/4 and 1/4.

Fraction 2/4 mean that the object is divided into 4 parts and 1 part is shaded.

Similarly fraction 1/4 signifies that the object is divided into 4 equal part and 1 part is shaded.

Since the object is divided into same parts (i.e. same denominator), the subtraction process gets straightforward.

Notice the above image, here we have simply subtracted the numerator and left the denominator as it is.

### Subtraction of Fraction examples

Example 01
Subtract the fractions; \mathtt{\ \frac{6}{3} \ \ \&\ \ \frac{2}{3}}

Solution
The fractions have same denominator.
So, simply subtract the numerator and retain denominator.

\mathtt{\Longrightarrow \ \frac{6}{3} \ -\ \frac{2}{3} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{6\ -\ 2\ }{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4}{3}}

Hence, 4/3 is the solution.

Example 02
Subtract the fractions \mathtt{\frac{2}{9} \ \ \&\ \ \frac{1}{18}}

Solution
Here the fractions have different denominator.

(a) Find LCM of denominator

LCM (9, 18) = 18

(b) Multiply the fractions to make denominator 18

Fraction 2/9
Multiply numerator and denominator by 2

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

Fraction 1/18
Here denominator is already 18, so no need to multiply anything.

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

\mathtt{\Longrightarrow \ \frac{4}{18} \ -\ \frac{1}{18} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{4\ -\ 1}{18}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3}{18}}

The fraction can be further simplified.
Divide numerator and denominator by 3

\mathtt{\Longrightarrow \ \frac{3\div 3}{18\div 3}}\\\ \\ \mathtt{\Longrightarrow \frac{1}{6}}

Hence, 1/6 is the solution.

Example 03
Subtract the fractions, \mathtt{\frac{3}{2} \ \ \&\ \ \frac{1}{5}}

Solution
The fractions have different denominator.

(a) Find LCM of denominator

LCM (2, 5) = 10

(b) Multiply fractions to make denominator 10

Fraction 3/2
Multiply numerator and denominator by 5

\mathtt{\Longrightarrow \frac{3\ \times \ 5}{2\ \times \ 5} \ =\ \frac{15}{10}}

Fraction 1/5
Multiply numerator and denominator by 2

\mathtt{\Longrightarrow \frac{1\ \times \ 2}{5\ \times \ 2} \ =\ \frac{2}{10}}

(c) Now both the fractions have same denominator.
Simply subtract the numerator and leave the denominator as it is.

\mathtt{\Longrightarrow \ \frac{15}{10} \ -\ \frac{2}{10} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{15\ -\ 2}{10}}\\\ \\ \mathtt{\Longrightarrow \ \frac{13}{10}}

Hence, 13/10 is the solution.

Example 04
Subtract the fractions, \mathtt{\frac{19}{11} \ \ \&\ \ \frac{3}{7}}

(a) Find LCM of denominators

LCM (11, 7) = 77

(b) Multiply the fractions to make denominator 77

Fraction 19/11
Multiply numerator and denominator by 7

\mathtt{\Longrightarrow \frac{19\ \times \ 7}{11\times \ 7} \ =\ \frac{133}{77}}

Fraction 3/7
Multiply numerator and denominator by 11

\mathtt{\Longrightarrow \frac{3\ \times \ 11}{7\times \ 11} \ =\ \frac{33}{77}}

(c) Both the fraction now have same denominator.
Subtract the numerator and retain the denominator.

\mathtt{\Longrightarrow \ \frac{133}{77} \ -\ \frac{33}{77} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{133\ -\ 33}{77}}\\\ \\ \mathtt{\Longrightarrow \ \frac{100}{77}}

Hence, 100/77 is the solution.

Example 05
Subtract the fraction, \mathtt{\frac{17}{20} \ \ \&\ \ \frac{11}{20}}

Solution
Both the subtraction have same denominator.
So, subtract the numerator and retain the denominator.

\mathtt{\Longrightarrow \ \frac{17}{20} \ -\ \frac{11}{20} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{17\ -\ 11}{20}}\\\ \\ \mathtt{\Longrightarrow \ \frac{6}{20}}

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

\mathtt{\Longrightarrow \ \frac{6\div 2}{20\div 2}}\\\ \\ \mathtt{\Longrightarrow \frac{3}{10}}

Hence, 3/10 is the right answer.

## Fraction Subtraction -Solved Problems

(01) Subtract the fractions; \mathtt{\ \frac{4}{3} \ \ \&\ \ \frac{2}{3}}

(a) 4/3
(b) 2/3
(c) 5/3

Option (b) is correct

Solution
The fractions have same denominator.
Simply subtract the numerator and retain the denominator.

\mathtt{\Longrightarrow \ \frac{4}{3} \ -\ \frac{2}{3} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{4\ -\ 2}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2}{3}}

Hence, 2/3 is the solution.

(02) Subtract the fraction; \mathtt{\frac{7}{4} \ -\ 1}

(a) 3/4
(b) 5/4
(c) 1/4

Option (a) is correct

Solution
The given numbers are 7/4 and 1.

1 can be written as 4/4.

Now we have same denominator fractions 7/4 and 4/4.

Subtract the numerator and retain the denominator.

\mathtt{\Longrightarrow \ \frac{7}{4} \ -\ \frac{4}{4} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{7\ -\ 4}{4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3}{4}}

Hence, 3/4 is the solution.

(03) Subtract the fractions, \mathtt{\frac{6}{10} \ \ \&\ \ \frac{2}{5}}

(a) 2/5
(b) 1/5
(c) 6/10
(d) 8/10

Option (b) is correct

Solution
Here the fractions have different denominator.

(a) Find LCM of the denominators

LCM (10, 5) = 10

(b) Multiply the fractions to get denominator 10.

Fraction 6/10
No need to do anything.

Fraction 2/5
Multiply numerator and denominator by 2

\mathtt{\Longrightarrow \frac{2\ \times \ 2}{5\times \ 2} \ =\ \frac{4}{10}}

(c) Now we have fractions with same denominator.
Subtract the numerator and leave the denominator as it is.

\mathtt{\Longrightarrow \ \frac{6}{10} \ -\ \frac{4}{10} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{6\ -\ 4}{10}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2}{10}}

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

\mathtt{\Longrightarrow \ \frac{2\div 2}{10\div 2}}\\\ \\ \mathtt{\Longrightarrow \frac{1}{5}}

Hence, 1/5 is the solution.

(04) Subtract the fraction, \mathtt{\frac{7}{25} \ \ \&\ \ \frac{9}{50}}

(a) 2/10
(b) 5/10
(c) 1/10
(d) 3/10

Option (c) is correct

Solution
The fractions have different denominator.

(a) LCM of denominator

LCM (25, 50) = 50

(b) Multiply the fraction to make denominator 50

Fraction 7/25
Multiply numerator and denominator by 2

\mathtt{\Longrightarrow \frac{7\ \times \ 2}{25\times \ 2} \ =\ \frac{14}{50}}

Fraction 9/50
No need to do anything.

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

\mathtt{\Longrightarrow \ \frac{14}{50} \ -\ \frac{9}{50} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{14\ -\ 9}{50}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5}{50}}

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

\mathtt{\Longrightarrow \ \frac{5\div 5}{50\div 5}}\\\ \\ \mathtt{\Longrightarrow \frac{1}{10}}

Hence, 1/10 is the solution.

(05) Subtract the fraction; \mathtt{\frac{9}{20} \ \ \&\ \ \frac{4}{20}}

(a) 1/5
(b) 1/3
(c) 1/7
(d) 1/4

Option (d) is correct

Both the fractions have same denominator.
Subtract the fraction and leave the denominator as it is.

\mathtt{\Longrightarrow \ \frac{9}{20} \ -\ \frac{4}{20} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{9\ -\ 4}{20}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5}{20}}

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

\mathtt{\Longrightarrow \ \frac{5\div 5}{20\div 5}}\\\ \\ \mathtt{\Longrightarrow \frac{1}{4}}