In this post we will learn to add two or more fractions.

The process requires understanding of the concept of fraction and LCM. I strongly suggest to get basic understanding of these concepts before moving on with this lesson.

(a) If denominator is same for the given fraction, then simply add the numerator and retain the same denominator.

(b) If the fractions have different denominator then do the following:

(i) Take L.C.M (Least Common Multiple) of all the denominator.

(ii) Multiply each fraction to make denominator equal to LCM value

(iii) Now with all the fraction with same denominator, simply add the numerator and retain the denominator.

Let us understand the process with examples:

Example 01

Solution
Both the fractions have same denominator.
In this case, simply add the numerator and retain the denominator

\mathtt{\Longrightarrow \ \frac{7}{3} \ +\ \frac{4}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7\ +\ 4}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{11}{3}}

Hence 11/3 is the solution.

Note:
In fraction addition, the process is very simple when the denominator is same.
The complexity arise when we get fractions with different denominators.

### Why do we want fractions with same denominator?

Because we can’t manually add the fractions with different denominator.

Consider addition of fractions \mathtt{\frac{1}{2} \ \&\ \frac{1}{3}}

Fraction 1/2 means object is divided into two parts and one part is shaded.

Where as fraction 1/3 means object is divided into three equal parts and one part is shaded.

On observing the image you will find that the object is divided into different parts which make it difficult to add the fraction.

So we will take the LCM of denominators
LCM (2, 3) = 2 x 3 = 6

Now we will try to make all the fraction with denominator 6.

For Fraction (1/2)
Multiply numerator and denominator with 3

\mathtt{\Longrightarrow \frac{1\ \times \ 3}{2\ \times \ 3} \ =\ \frac{3}{6}}

For Fraction (1/3)
Multiply numerator and denominator by 2

\mathtt{\Longrightarrow \frac{1\ \times \ 2}{3\ \times \ 2} \ =\ \frac{2}{6}}

Now we have fraction 3/6 and 2/6, both have same denominator. Since both the object is divided into 6 parts, the addition of fraction is now possible.

When the denominator is same, simply add the numerators and retain the denominator.

\mathtt{\Longrightarrow \ \frac{3}{6} \ +\ \frac{2}{6}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3\ +\ 2}{6}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5}{6}}

Hence, 5/6 is the solution.

Example 02

Here the fractions have different denominator.

(a) Find the L.C.M of the denominator

LCM (5, 9) = 45

(b) Multiply each fractions to make denominator 45

For fraction (2/5)
Multiply numerator and denominator with 9

\mathtt{\Longrightarrow \frac{2\ \times \ 9}{5\ \times \ 9} \ =\ \frac{18}{45}}

Note:
You always have to multiply numerator and denominator with same number otherwise the process will go wrong.

For Fraction (6/9)
Multiply numerator and denominator with 5

\mathtt{\Longrightarrow \frac{6\ \times \ 5}{9\ \times \ 5} \ =\ \frac{30}{45}}

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

\mathtt{\Longrightarrow \ \frac{18}{45} \ +\ \frac{30}{45}}\\\ \\ \mathtt{\Longrightarrow \ \frac{18\ +\ 30}{45}}\\\ \\ \mathtt{\Longrightarrow \ \frac{48}{45}}

(d) Fraction 48/45 is the solution.
If you want, you can further simplify the fraction.

Both numbers 48 & 45 are divisible by 3.
So dividing numerator and denominator by 3, we get;

\mathtt{\Longrightarrow \frac{48\ \div \ 3}{45\ \div \ 3} \ =\ \frac{16}{15}}

Hence, 16/15 is the solution.

Example 03

The fractions have different denominators.

(a) Find the LCM of denominator

LCM (11, 15) = 165

(b) Multiply each fractions to make each denominator 165

For Fraction 9/11
Multiply numerator and denominator by 15

\mathtt{\Longrightarrow \frac{9\ \times \ 15}{11\ \times \ 15} \ =\ \frac{135}{165}}

For Fraction 7/15
Multiply numerator and denominator by 11

\mathtt{\Longrightarrow \frac{7\ \times \ 11}{15\ \times \ 11} \ =\ \frac{77}{165}}

(c) Now both the fractions have same denominator.
Add the numerators and retain the denominators.

\mathtt{\Longrightarrow \ \frac{135}{165} \ +\ \frac{77}{165}}\\\ \\ \mathtt{\Longrightarrow \ \frac{135\ +\ 77}{165}}\\\ \\ \mathtt{\Longrightarrow \ \frac{212}{165}}

Hence, 212/165 is the solution.

Example 04
\mathtt{Add\ \frac{1}{13} \ ,\ \frac{3}{14} \ \&\ \frac{5}{8}}

(a) Find LCM of denominator

LCM (13, 14, 8) = 728

(b) Multiply fractions to make denominator 728

Fraction 1/13
Multiply 56 on both numerator and denominator

\mathtt{\Longrightarrow \frac{1\ \times \ 56}{13\ \times \ 56} \ =\ \frac{56}{728}}

Fraction 3/14
Multiply numerator and denominator by 52

\mathtt{\Longrightarrow \frac{3\ \times \ 52}{14\ \times \ 52} \ =\ \frac{156}{728}}

Fraction 5/8
Multiply numerator and denominator with 91

\mathtt{\Longrightarrow \frac{5\ \times \ 91}{8\ \times \ 91} \ =\ \frac{455}{728}}

(c) Now all the fractions have same denominator.
Add the numerators and retain the denominator.

\mathtt{\Longrightarrow \ \frac{56}{728} \ +\ \frac{156}{728} +\frac{455}{728}}\\\ \\ \mathtt{\Longrightarrow \ \frac{56\ +\ 156+455}{728}}\\\ \\ \mathtt{\Longrightarrow \ \frac{667}{728}}

Hence, 667/728 is the solution.

Example 05

(a) Find LCM of denominator

LCM (25, 23) = 25 x 23 = 575

(b) Multiply fractions to make denominator 575

Fraction 11/25
Multiply numerator and denominator with 23

\mathtt{\Longrightarrow \frac{11\ \times \ 23}{25\ \times \ 23} \ =\ \frac{253}{575}}

Fraction 4/23
Multiply numerator and denominator with 25
\mathtt{\Longrightarrow \frac{4\ \times \ 25}{23\ \times \ 25} \ =\ \frac{100}{575}}

(c) Now we have fractions with same denominator.
Add all the numerators while retaining the same denominator.

\mathtt{\Longrightarrow \ \frac{253}{575} \ +\ \frac{100}{575}}\\\ \\ \mathtt{\Longrightarrow \ \frac{253\ +\ 100}{575}}\\\ \\ \mathtt{\Longrightarrow \ \frac{353}{575}}

Hence, 353/575 is the solution.

## Fraction Addition – Solved Problems

\mathtt{\frac{1}{4} \ \&\ \frac{9}{4}}

(a) 5/6
(b) 5/2
(c) 3/2
(d) 7/6

Option (b) is correct

Explanation

Both the fraction have same denominator.

Simply add the numerator and retain the denominator.

\mathtt{\Longrightarrow \ \frac{1}{4} \ +\ \frac{9}{4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1\ +\ 9}{4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10}{4}}

Simplify the fraction further.
Divide numerator and denominator by 2.

\mathtt{\Longrightarrow \ \frac{10\div 2}{4\ \div 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5}{2}}

Hence, 5/2 is the solution.

(02) \mathtt{Add\ \frac{2}{7} \ \&\ \frac{6}{5}}

(a) 52/35
(b) 36/15
(c) 92/35

Option (a) is correct

Solution

Here the fractions have different denominator.

(a) Find LCM of denominators
LCM (7, 5) = 35

(b) Multiply the fraction to make denominator 35

Fraction 2/7
Multiply numerator and denominator with 5

\mathtt{\Longrightarrow \frac{2\ \times \ 5}{7\ \times \ 5} \ =\ \frac{10}{35}}

Fraction 6/5
Multiply numerator and denominator by 7

\mathtt{\Longrightarrow \frac{6\ \times \ 7}{5\ \times \ 7} \ =\ \frac{42}{35}}

(c) Both fractions have same denominator.
Add the numerator and retain the denominator.

\mathtt{\Longrightarrow \ \frac{10}{35} \ +\ \frac{42}{35}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10\ +\ 42}{35}}\\\ \\ \mathtt{\Longrightarrow \ \frac{52}{35}}

Hence, 52/35 is the solution.

(03) \mathtt{Add\ \frac{10}{11} \ \&\ \frac{16}{11}}

(a) 25/11
(b) 26/11
(c) 27/11

Option (b) is correct

Explanation
Both the fractions have same denominator, so simply add the numerator.

\mathtt{\Longrightarrow \ \frac{10}{11} \ +\ \frac{16}{11}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10\ +\ 16}{11}}\\\ \\ \mathtt{\Longrightarrow \ \frac{26}{11}}

Hence, 26/11 is the solution

(04) \mathtt{Add\ \frac{1}{2} \ ,\frac{1}{3} \ \&\ \frac{1}{5}}

(a) 28/30
(b) 29/30
(c) 30/30
(d) 31/30

Option (d) is correct

Explanation
Given are the fractions with different denominator.

(i) Find LCM of denominators
LCM (2, 3, 5) = 30

(ii) Multiply fractions to make denominator 30

Fraction 1/2
Multiply numerator and denominator with 15

\mathtt{\Longrightarrow \frac{1\ \times \ 15}{2\ \times \ 15} \ =\ \frac{15}{30}}

Fraction 1/3
Multiply numerator and denominator by 10

\mathtt{\Longrightarrow \frac{1\ \times \ 10}{3\ \times \ 10} \ =\ \frac{10}{30}}

Fraction 1/5
Multiply numerator and denominator by 6

\mathtt{\Longrightarrow \frac{1\ \times \ 6}{5\ \times \ 6} \ =\ \frac{6}{30}}

Now all the denominator have same fraction.
Simply add the numerator and retain the denominator.

\mathtt{\Longrightarrow \ \frac{15}{30} \ +\ \frac{10}{30}} \ +\ \frac{6}{30}\\\ \\ \mathtt{\Longrightarrow \ \frac{15\ +\ 10+6}{30}}\\\ \\ \mathtt{\Longrightarrow \ \frac{31}{30}}

Hence, 31/30 is the solution