In this chapter we will learn how to add two or more rational numbers with examples.

To understand the chapter, you should have basic knowledge of rational numbers. Click the red link to know more about the topic.

## How to add rational numbers ?

Here we will discuss three different cases;

(a) Adding rational numbers with same denominator.

(b) Adding rational numbers with different denominator.

(c) Adding rational number by converting into decimals

We we discuss both the cases in detail with solved examples.

### Adding rational number with same denominator

Let \mathtt{\frac{a}{b} \ \&\ \frac{\ c}{b}} be the rational number with same denominator.

To do the addition, simply add the numerators & keep the denominator same.

\mathtt{\Longrightarrow \ \frac{a}{b} \ +\ \frac{c}{b}}\\\ \\ \mathtt{\Longrightarrow \frac{a+c}{b}}

This method is simple and straight forward. Let us see some solved examples for further understanding.

Example 01
Add \mathtt{\frac{2}{3} \ +\ \ \frac{5}{3}}

Solution
Here both the rational numbers have same denominator.

Hence, simply add the numerator and keep the denominator same to come at final solution.

\mathtt{\Longrightarrow \ \frac{2}{3} \ +\ \frac{5}{3}}\\\ \\ \mathtt{\Longrightarrow \frac{2+5}{3} \ }\\\ \\ \mathtt{\Longrightarrow \frac{7}{3}}

Hence, 7/3 is the final solution.

Example 02
Add \mathtt{\frac{15}{7} \ +\ \ \frac{4}{7}}

Solution
Note that both the rational numbers have common denominator.

Simply add the numerator to get the solution.

\mathtt{\Longrightarrow \ \frac{15}{7} \ +\ \frac{4}{7}}\\\ \\ \mathtt{\Longrightarrow \frac{15+4}{7} \ }\\\ \\ \mathtt{\Longrightarrow \frac{19}{7}}

Hence, 19 / 7 is the solution.

### Adding rational numbers with different denominator

When we have rational numbers with different denominators, we simply have to convert the rational numbers with common denominators with the help of LCM.

(a) Take LCM of denominators.

(b) Multiply each rational numbers to make denominator equal to LCM.

(c) Now we have rational number with same denominator. Simply add the numerator to get the solution.

I hope you understood the above process. Let us see some examples of adding the rational numbers.

Example 01
Add \mathtt{\frac{2}{3} \ +\ \ \frac{4}{5} \ }

Solution
Note that the rational numbers have different denominators.

(a) Take LCM of denominators.

LCM ( 3, 5 ) = 15

(b) Multiply each rational number to make denominator 15.

(i) Rational number 2 / 3

Multiply numerator and denominator by 5

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

(ii) Rational number 4/5

Multiply numerator and denominator by 3.

\mathtt{\Longrightarrow \frac{4}{5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4\times 3}{5\times 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{12}{15}}

We have got rational numbers with same denominators.

Now simply add the numerators and get the solution.

\mathtt{\Longrightarrow \ \frac{10}{15} \ +\ \frac{12}{15}}\\\ \\ \mathtt{\Longrightarrow \frac{10+12}{15} \ }\\\ \\ \mathtt{\Longrightarrow \frac{22}{15}}

Hence, 22/15 is the solution.

Example 02
Add \mathtt{\frac{6}{7} \ +\ \ \frac{2}{9} \ }

Solution
Here both the rational numbers have different denominator.

(a) Find LCM of denominators

LCM (7, 9) = 63

(b) Multiply each rational number to make denominator 63.

(i) Rational number 6 / 7

Multiply numerator and denominator by 9.

\mathtt{\Longrightarrow \frac{6}{7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{6\times 9}{7\times 9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{54}{63}}

(ii) Rational number 2 / 9

Multiply numerator and denominator by 7.

\mathtt{\Longrightarrow \frac{2}{9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2\times 7}{9\times 7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{14}{63}}

Now we have rational numbers with same denominators.

\mathtt{\Longrightarrow \ \frac{54}{63} \ +\ \frac{14}{63}}\\\ \\ \mathtt{\Longrightarrow \frac{54+14}{63} \ }\\\ \\ \mathtt{\Longrightarrow \frac{68}{63}}

Hence, 68/63 is the solution.

### Adding rational numbers using decimals

In this method we will convert each rational number into decimal and then add the numbers.

This method is suitable for students who can do the division fast.

Let us understand the method with the help of example.

Example 01
Add \mathtt{\frac{4}{9} \ +\ \ \frac{3}{5} \ }

Solution
Convert both the rational numbers into decimals.

4/9 ⟹ 0.44

3/5 ⟹ 0.6

⟹ 0.44 + 0.6

⟹ 1.04

Hence, 1.04 is the solution of given addition.

Example 02
Add \mathtt{\frac{1}{2} \ +\ \ \frac{5}{11} \ \ }

Solution
Convert both the fraction into decimals.

1/2 ⟹ 0.5

5/11 ⟹ 0.4545