# Introduction to Addition and Subtraction of Fractions

Fraction is defined as a numerical part of a whole number. Sometimes we want to add or subtract two fractions to find out the total number.

For example, Malini ate 1½ roties for lunch and 2½ roties for dinner. What is the total number of roties Malini ate? In the above example, it is necessary to add the fractions.

All fractions cannot be added or subtracted easily. The following methods are available to add or subtract fraction.

- Add or subtract fraction with same denominator
- Add or subtract fraction with different denominator
- Add or subtract mixed number fraction

**Adding or subtracting fractions with same denominator**

In this method, addition or subtraction of fraction is very easy. Because, the denominators of the two fractions are same.

Example: ^{2}/_{4} + ^{3}/_{4} = ?

In the above fractions, the denominators are same.

^{2}/_{4 }=

1 | 2 |

^{ }^{3}/_{4} =

1 | 2 | 3 |

In above fractions, the box is divided into four parts. For ^{2}/_{4 } and ^{3}/_{4 }, take 3 and 1 numerator parts respectively out of 4 equal parts.

Steps involved adding fractions, if denominators are same.

Step 1 : Check the given fractions, if denominators are same or not.

Step 2: If denominators are same, take the numerators of two fractions and add or subtract it.

Step 3: Give a final answer with denominator.

Example 1: ^{5}/_{8} + ^{2}/_{8} = ?

Step 1 : Denominators are same.

Step 2: Take the numerators 5 and 2 respectively and add it

^{ }= ^{(5+2)}/_{8}

Step 3: Give final answer.

= ^{7}/_{8}

Example 2: ^{5}/_{8} – ^{2}/_{8} = ?

Step 1 : Denominators are same.

Step 2: Take the numerators 5 and 2 respectively and subtract the smaller from the bigger numerator.

^{ }= ^{(5-2)}/_{8}

Step 3: Give final answer.

= ^{3}/_{8}

**Adding or subtracting fractions with different denominator**

In this method, the denominators are not the same in two fractions. So we need to make them same.

In this method, the numerators and denominators are different in the two fractions.

Let us add two fractions, ^{3}/_{8 }and ^{5}/_{12}.

In above fractions, numerators and denominators are different.

Solution:

Step 1: Take LCM for denominators of above fractions. i.e. 8 and 12 respectively.

(LCM is a smallest number which is used as common multiple of two numbers)

24 is a common multiple for 8 and 12.

Step 2: Convert denominators 8 and 12 into 24.

In ^{3}/_{8}, multiply numerator and denominator by 3 = ^{3}/_{8} x^{3}/_{3} = ^{9}/_{24}

In ^{5}/_{12}, multiply numerator and denominator by 2 = ^{5}/_{12}x ^{2}/_{2} = ^{10}/_{24}

(Note: We must do to the numerator what we do to the denominator)

Step 3: Now, the denominators of the two fractions are same.

^{9}/_{24} , ^{10}/_{24}

Step 4: Take the numerators 9 and 10 respectively and add it

= ^{(9+10)}/_{24}

Step 5 : Give a final answer

= ^{19}/_{24}

Example: ^{3}/_{8 }– ^{5}/_{12} =?

Solution: Denominators are different. So, take Least Common Multiple (LCM).

24 is a common multiple of 8 and 12.

^{3}/_{8} => ^{3}/_{8} x^{3}/_{3} = ^{9}/_{24}

^{5}/_{12} => ^{5}/_{12}x ^{2}/_{2} = ^{10}/_{24}

^{3}/_{8} – ^{5}/_{12} = ^{9}/_{24} – ^{10}/_{24}

= ^{9-10}/_{24 }(subtract the smaller numerator from the bigger)

Answer = ^{-1}/_{24 }

**Adding or subtracting mixed fractions**

A *mixed fraction* is defined as a fraction and a whole number combined into one “*mixed*” *number*.

Depends on the denominator, two following methods of available for add or subtract of mixed fraction.

- If same denominators are present in the mixed fraction.
- If different denominators are present in the mixed fraction.

**If same denominators are present in the mixed fraction**

In this method, mixed fraction consists of same denominators. The following steps are involved to add or subtract the mixed fraction.

Step 1 : Add the whole number of two fraction separately.

Step 2: Add the fractions separately.

Step 3 : Combine the whole number and the fraction both

Step 3: Convert improper fraction into mixed fraction.

Example: 3 ^{2}/_{5} + 1 ^{4}/_{5} = ?

Step 1 : Add the whole number of two fraction separately

= 3+1 = 4 — (1)

Step 2: Add the fractions separately

= ^{2}/_{5} + ^{4}/_{5} = ^{6}/_{5} —– (2)

Step 3 : Combine the both equation (1) and (2)

= 4 ^{6}/_{5} —— (3)

In 4 ^{6}/_{5}, ^{6}/_{5} is an improper fraction. So, convert into proper mixed fraction.

Equation (2) —–> ^{6}/_{5} = 1 ^{1}/_{5} —– (3)

Step 4: Now, add equation (1) and (3)

4 + 1 ^{1}/_{5} = 5 ^{1}/_{5}

Answer = 5 ^{1}/_{5}

**If different denominators are present in the mixed fraction**

In this method, mixed fraction consists of different denominators. The following steps are involved to add or subtract the mixed fraction.

Example: 6 ^{3}/_{4} + 3 ^{5}/_{8} = ?

Solution: In the above mixed fractions, the denominators are different. So we need to make them same.

Step 1: Take LCM for denominators of above fractions. i.e. 4 and 8 respectively.

(LCM is a smallest number which is used as common multiple of two numbers)

8 is a common multiple for 4 and 8.

Step 2: Convert denominators 4 and 8 into 8.

In ^{3}/_{4}, multiply numerator and denominator by 2 = ^{3}/_{4} x^{2}/_{2} = ^{6}/_{8}

In ^{5}/_{8}, multiply numerator and denominator by 1 = ^{5}/_{8}x ^{1}/_{1} = ^{5}/_{8}

(Note: We must do to the numerator what we do to the denominator)

Now, denominators are same in the mixed fraction.

6 ^{3}/_{4} + 3 ^{5}/_{8} = 6 ^{6}/_{8} + 3 ^{5}/_{8}

Step 3 : Add the whole number of two fraction separately

= 6+3 = 9 ———— (1)

Step 4: Add the fractions separately

= ^{6}/_{8} + ^{5}/_{8}= ^{11}/_{8} —– (2)

^{11}/_{8} is an improper fraction. So, convert into proper fraction.

^{11}/_{8} = 1 ^{3}/_{8 }————(3)

Step 5 : Combine the both equation (1) and (3)

9+ 1 ^{3}/_{8 }= 10 ^{3}/_{8}

Answer : 10 ^{3}/_{8}