# How to add rational numbers with different denominators ?

In this post we will learn the step by step method to add rational number with different denominator.

Here the process is similar to adding fraction with different denominators.

## Adding rational numbers with different denominator

In this case, we will try to multiply each rational number so that we get the numbers with same denominator.

(a) Find LCM of denominators.

(b) Multiply each rational number to get the denominator equal to LCM.

(c) Now simply add the numerator and get the solution.

Let us understand the above process with the help of examples.

Example 01
Add the rational number \mathtt{\frac{2}{5} \ +\ \ \frac{3}{9} \ \ }

Solution
You can see that both the rational numbers have different denominators.

(a) Take LCM of denominators.

LCM ( 5, 9 ) = 45

(b) Multiply each rational number to get the denominator 45.

(i) Fraction 2/5

Multiply numerator and denominator by 9

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

(ii) Fraction 3/9

Multiply numerator and denominator by 5

\mathtt{\Longrightarrow \frac{3}{9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3\times 5}{9\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{15}{45}}

Now we have got the fraction with same denominator, we have to simply add the numerators.

\mathtt{\Longrightarrow \ \frac{18}{45} \ +\ \frac{15}{45}}\\\ \\ \mathtt{\Longrightarrow \frac{18+15}{45} \ }\\\ \\ \mathtt{\Longrightarrow \frac{33}{45}}

Hence, 33/45 is the solution.

The number can be further simplified by dividing numerator & denominator by 3.

\mathtt{\Longrightarrow \frac{33\div 3}{45\div 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{11}{15}}

Hence, 11 / 15 is the solution of given addition.

Example 02
Add the rational numbers \mathtt{\frac{15}{11} \ +\ \ \frac{4}{3}}

Solution
Here both the rational numbers have different denominators.

(a) Find LCM of denominators.

LCM (11, 3) = 33

(b) Multiply each rational number to make denominator 33.

(i) Rational number 15 / 11

Multiply numerator & denominator by 3

\mathtt{\Longrightarrow \frac{15}{11}}\\\ \\ \mathtt{\Longrightarrow \ \frac{15\times 3}{11\times 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{45}{33}}

(ii) Rational number 4/3

Multiply numerator and denominator by 11.

\mathtt{\Longrightarrow \frac{4}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4\times 11}{3\times 11}}\\\ \\ \mathtt{\Longrightarrow \ \frac{44}{33}}

We have got rational number with same denominator. Now simply add the numerator.

\mathtt{\Longrightarrow \ \frac{45}{33} \ +\ \frac{44}{33}}\\\ \\ \mathtt{\Longrightarrow \frac{45+44}{33} \ }\\\ \\ \mathtt{\Longrightarrow \frac{89}{33}}

Hence, 89 / 33 is the solution of given addition.

Example 03
Add the rational numbers \mathtt{\frac{3}{2} \ +\ \ \frac{4}{5} \ +\frac{5}{6} \ \ }

Solution
Here we got rational numbers with different denominators.

(a) Take LCM of denominators.

LCM (2, 5, 6) = 30

(b) Multiply each rational number to make denominator 30.

(i) Rational number 3/2

Multiply numerator and denominator by 15.

\mathtt{\Longrightarrow \frac{3}{2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3\times 15}{2\times 15}}\\\ \\ \mathtt{\Longrightarrow \ \frac{45}{30}}

(ii) Rational number 4 / 5

Multiply numerator & denominator by 6

\mathtt{\Longrightarrow \frac{4}{5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4\times 6}{5\times 6}}\\\ \\ \mathtt{\Longrightarrow \ \frac{24}{30}}

(iii) Rational number 5/6

Multiply numerator and denominator by 5

\mathtt{\Longrightarrow \frac{5}{6}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5\times 5}{6\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{25}{30}}

Now we have got fraction with same denominator. Simply add the numerator and get the solution.

\mathtt{\Longrightarrow \ \frac{45}{30} \ +\ \frac{24}{30} +\frac{25}{30}}\\\ \\ \mathtt{\Longrightarrow \frac{45+24+25}{30} \ }\\\ \\ \mathtt{\Longrightarrow \frac{94}{30}}

Hence, 94/30 is the solution.

You can further reduce the fraction by dividing numerator and denominator by 2.

\mathtt{\Longrightarrow \frac{94\div 2}{30\div 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{47}{15}}

Hence, 47/15 is the solution.

Example 04
Add the rational number \mathtt{\frac{15}{4} \ +\ \ \frac{-13}{8}}

Solution
To solve the abode rational number, follow the below steps;

(a) Find LCM of denominators.

LCM (4, 8) = 8

(b) Multiply each rational number to make denominator 8.

(i) Rational number 15 / 4

Multiply numerator & denominator by 2.

\mathtt{\Longrightarrow \frac{15}{4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{15\times 2}{4\times 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{30}{8}}

(ii) Rational number -13 / 8

Since the denominator is already 8, we don’t need to do anything.

Now we have fraction with same denominator, we can simply add the numerator and find the solution.