In this post we will learn about** adding fractions with different denominators**.

For the addition, you should have basic understanding of the fraction and LCM concept.

**Adding fractions with different denominator**

In order to add the fractions, **its important to have a common denominator**.

In this method we will try to **convert the fractions with the help of LCM so that they have same denominator**.

Follow the below steps for the whole process;

(a) **Find LCM of the denominator**.

(b) **Multiply the fractions** so that the denominator is same as LCM value.

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

**Why its important to have same denominator in fraction addition?**

It is very difficult to add fractions manually with different denominator.

Let us understand this with help of graphical representation.

Consider the **fraction 1/4 and 1/5**.

The fraction** 1/4 mean that the object is divided into 4 equal part and 1 part of it is shaded**.

While fraction **1/5 mean that the object is divided into 5 equal part and 1 part of it is shaded**.

Observe the above image.

You will note that **adding the fractions is quite difficult as the object is divided into different parts**.

In order to add fractions, it is important to have object which is divided into equal parts. (Hence, same denominator is necessary)

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

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

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

Since the **object is divided into equal parts the addition gets easy and straight forward**.

Observe the above image.

Note that the addition is done simply by adding numerator and retaining the denominator.

**Examples of fraction addition with different denominator**

**Example 01**

Add \mathtt{\frac{1}{3} \ \&\ \frac{1}{5}}

**Solution**

For fraction addition, follow the below steps:

**(a) Find LCM of the denominators**

LCM (3, 5) = 15

**(b) Multiply the fraction to make denominator equal to 15**

**For Fraction 1/3**

Multiply numerator and denominator by 5

\mathtt{\Longrightarrow \frac{1\ \times \ 5}{3\ \times \ 5} \ =\ \frac{5}{15}}

**For Fraction 1/5**

Multiply numerator and denominator by 3

\mathtt{\Longrightarrow \frac{1\ \times \ 3}{5\ \times \ 3} \ =\ \frac{3}{15}}

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

\mathtt{\Longrightarrow \ \frac{5}{15} \ +\ \frac{3}{15} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{5\ +\ 3\ }{15}}\\\ \\ \mathtt{\Longrightarrow \ \frac{8}{15}}

**Hence, 8/15 is the solution.**

**Example 02**

Add the fractions \mathtt{\frac{2}{6} \ \&\ \frac{1}{8} \ }

**Solution**

Follow the below steps:

**(a) Find LCM of denominators**

LCM (6, 8) = 24

**(b) Multiply the fraction to make denominator 24**

**Fraction 2/6**

Multiply numerator and denominator by 4

\mathtt{\Longrightarrow \frac{2\ \times \ 4}{6\ \times \ 4} \ =\ \frac{8}{24}}

**Fraction 1/8**

Multiply numerator and denominator by 3

\mathtt{\Longrightarrow \frac{1\ \times \ 3}{8\ \times \ 3} \ =\ \frac{3}{24}}

**(c) Now we have fractions with same denominator.**

Just add the numerator and leave the denominator as it is.

\mathtt{\Longrightarrow \ \frac{8}{24} \ +\ \frac{3}{24} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{8\ +\ 3\ }{24}}\\\ \\ \mathtt{\Longrightarrow \ \frac{11}{24}}

**Hence, 11/24 is the solution.**

**Example 03**

Add the fractions \mathtt{\frac{6}{13} \ \&\ \frac{7}{15}}

**Solution**

Follow the below steps:**(a) Find LCM of the denominators**

LCM (13, 15) = 195

**(b) Multiply the fractions to make denominator 195**

**Fraction 6/13**

Multiply numerator and denominator by 15

\mathtt{\Longrightarrow \frac{6\ \times \ 15}{13\ \times \ 15} \ =\ \frac{90}{195}}

**Fraction 7/15**

Multiply numerator and denominator by 13

\mathtt{\Longrightarrow \frac{7\ \times \ 13}{15\ \times \ 13} \ =\ \frac{91}{195}}

**(c) Now the fractions have same denominator.**

Add the numerators and retain the denominator.

\mathtt{\Longrightarrow \ \frac{90}{195} \ +\ \frac{91}{195} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{90\ +\ 91\ }{195}}\\\ \\ \mathtt{\Longrightarrow \ \frac{181}{195}}

**Hence, 181/195 is the solution.**

**Example 04**

Add the fractions \mathtt{\frac{2}{9} \ \&\ \frac{13}{45} \ }

**Solution**

Do the following steps:

**(a) Take LCM of denominator**

LCM (9, 45) = 45

**(b) Multiply the fractions to make denominator 45**.

**Fraction 2/9**

Multiply numerator and denominator by 5

\mathtt{\Longrightarrow \frac{2\ \times \ 5}{9\ \times \ 5} \ =\ \frac{10}{45}}

**Fraction 13/45**

The denominator is already 45.

No need to do anything.

**(c) Now we have fractions with same denominator.**

Add the numerator and retain the denominator.

\mathtt{\Longrightarrow \ \frac{10}{45} \ +\ \frac{13}{45} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{10\ +\ 13\ }{45}}\\\ \\ \mathtt{\Longrightarrow \ \frac{23}{45}}

**Hence, 23/45 is the solution.**

**Example 05**

Add the fractions; \mathtt{\frac{1}{5} \ ,\ \frac{1}{10} \ \ \&\ \ \frac{1}{15}}

Solution

Do the following steps:

(a) Find LCM of denominators.

LCM (5, 10, 15) = 30

**(b) Multiply fractions to make denominator 30**

**Fraction 1/5**

Multiply numerator and denominator by 6

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

**Fraction 1/10**

Multiply numerator and denominator by 3

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

**Fraction 1/15**

Multiply numerator and denominator by 2

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

**(c) Now we have fractions with same denominator.**

Add the fractions and retain the denominator.

\mathtt{\Longrightarrow \ \frac{6}{30} \ +\ \frac{3}{30} \ +\ \frac{2}{30} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{6\ +\ 3\ +\ 2\ }{30}}\\\ \\ \mathtt{\Longrightarrow \ \frac{11}{30}}

**Hence, 11/30 is the solution of the addition.**