In this chapter we will learn method to add and subtract three or more rational numbers simultaneously with solved examples.

## How to add / subtract rational numbers ?

When three or more rational numbers are given, you have to **find common denominator with the help of LCM** and then do the required calculation.**Follow the below steps** for rational number calculation;

(a) **Find LCM of denominators**

(b) **Multiply each rational number to make denominator equal to LCM**

(c) Now do the **addition / subtraction of numbers**.

These steps are very easy to understand. Let us solve some questions to gain clarity of the above methods.**Example 01**

Calculate \mathtt{\frac{1}{6} -\frac{1}{3} \ +\frac{1}{2}} **Solution**

I will show you two method to solve the above problem.**Method 01**

Finding common denominator in one go.**(a) Find LCM of all denominators**

LCM ( 6, 3, 2 ) = 6**(b) Multiply each rational number to make denominator equals 6.****(i) Fraction 1/6**

Denominator is already 6. No need to do anything.**(ii) Fraction 1/3**

Multiply numerator & denominator by 2.

\mathtt{\Longrightarrow \ \frac{1}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1\times 2}{3\times 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2}{6}} **(iii) Fraction 1/2**

Multiply numerator & denominator by 3.

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

We have rational number with same denominators. Now simply do the calculation of numerator.**(c) Adding & subtracting the rational numbers**

\mathtt{\Longrightarrow \frac{1}{6} -\frac{2}{6} +\frac{3}{6}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1-2+3}{6}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-1\ +\ 3}{6}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2}{6}}

The rational number can be simplified further on division by 2.

\mathtt{\Longrightarrow \ \frac{2}{6}}\\\ \\ \mathtt{\Longrightarrow \frac{2\div 2}{6\div 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{3}}

Hence, **1/3 is the solution of given calculation**.

**Method 02****Solving the expression in parts**

Instead of calculating the rational numbers in one go, we will do calculation in pair of two.

Let’s solve the first two pairs under bracket

\mathtt{\Longrightarrow \left(\frac{1}{6} -\frac{1}{3}\right) \ +\frac{1}{2}}

\mathtt{\Longrightarrow \left(\frac{1}{6} -\frac{1}{3}\right) \ +\frac{1}{2}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{1-2}{6}\right) \ +\frac{1}{2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-1}{6} \ +\frac{1}{2}}

Now solve the left out rational numbers.

\mathtt{\Longrightarrow \ \frac{-1}{6} \ +\frac{1}{2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-1+3}{6}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2}{6}}\\\ \\ \mathtt{\Longrightarrow \frac{2\div 2}{6\div 2}}\\\ \\ \mathtt{\Longrightarrow \frac{1}{3}}

Hence,** 1/3 is the solution.**

**Example 02**

Solve \mathtt{\frac{-3}{7} +\frac{2}{5} \ -\frac{1}{4}}

**Solution****Method 01**

Finding common denominator in one go.**(a) Find LCM of denominators**

LCM (7, 5, 4) = 140**(b) Multiply each rational number to make denominator 140****(i) Number -3/7**

Multiply numerator & denominator by 20

\mathtt{\Longrightarrow \ \frac{-3}{7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-3\times 20}{7\times 20}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-60}{140}} **(ii) Number 2/5**

Multiply numerator & denominator by 28

\mathtt{\Longrightarrow \ \frac{2}{5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2\times 28}{5\times 28}}\\\ \\ \mathtt{\Longrightarrow \ \frac{56}{140}} **(iii) Rational number 1/4**

Multiply numerator and denominator by 35

\mathtt{\Longrightarrow \ \frac{1}{4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1\times 35}{4\times 35}}\\\ \\ \mathtt{\Longrightarrow \ \frac{35}{140}}

We have got rational number with same denominator. Now do the simple calculation of numerator.**(c) Adding / subtracting the rational numbers**.

\mathtt{\Longrightarrow \frac{-60}{140} +\frac{56}{140} -\frac{35}{140}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-60+56-35}{140}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-4\ -\ 35}{140}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-39}{140}}

Hence, **-39 / 140 is the solution.**

**Method 02****Solving in parts**

Let’s solve the first two rational numbers under bracket.

\mathtt{\Longrightarrow \left(\frac{-3}{7} +\frac{2}{5}\right) \ -\frac{1}{4}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{-15+14}{35}\right) \ -\frac{1}{4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-1}{35} \ -\frac{1}{4}}

Now complete the calculation.

\mathtt{\Longrightarrow \ \frac{-1}{35} \ -\frac{1}{4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-4-35}{140}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-39}{140}}

Hence, –**39 / 140 is the solution of given calculation.**

**Example 03**

Calculate \mathtt{\frac{9}{8} +\frac{7}{3} \ -\frac{1}{5} \ +\frac{1}{2}}

**Solution****Method 01**Finding common denominator in one go.

**(a) Find LCM of denominators**

LCM (8, 3, 5, 2) = 120

**(b) Multiply each rational number to make denominator 120**.

**(i) Fraction 9/8**

Multiply numerator and denominator by 15.

\mathtt{\Longrightarrow \ \frac{9}{8}}\\\ \\ \mathtt{\Longrightarrow \ \frac{9\times 15}{8\times 15}}\\\ \\ \mathtt{\Longrightarrow \ \frac{135}{120}}

**(ii) Fraction 7/3**

Multiply numerator and denominator by 40

\mathtt{\Longrightarrow \ \frac{7}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7\times 40}{3\times 40}}\\\ \\ \mathtt{\Longrightarrow \ \frac{280}{120}}

**(iii) Fraction 1/5**

Multiply numerator & denominator by 24.

\mathtt{\Longrightarrow \ \frac{1}{5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1\times 24}{5\times 24}}\\\ \\ \mathtt{\Longrightarrow \ \frac{24}{120}}

**(iv) Rational number 1/2**

Multiply numerator and denominator by 60.

\mathtt{\Longrightarrow \ \frac{1}{2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1\times 60}{2\times 60}}\\\ \\ \mathtt{\Longrightarrow \ \frac{60}{120}}

We have rational number with same denominator. Now do the calculation with numerator.

**(c) Do the calculation**.

\mathtt{\Longrightarrow \ \frac{135}{120} +\frac{280}{120} \ -\frac{24}{120} \ +\frac{60}{120}}\\\ \\ \mathtt{\Longrightarrow \ \frac{135+280-24+60}{120}}\\\ \\ \mathtt{\Longrightarrow \ \frac{451}{120}}

Hence, **451/120 is the solution.**

**Method 02****Solve in parts**

Solving the rational numbers under bracket.

\mathtt{\Longrightarrow \left(\frac{9}{8} +\frac{7}{3}\right) +\left( \ -\frac{1}{5} \ +\frac{1}{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{27+56}{24}\right) +\left(\frac{-2+5}{10}\right)}\\\ \\ \mathtt{\Longrightarrow \ \frac{83}{24} +\frac{3}{10}}\\\ \\ \mathtt{\Longrightarrow \ \frac{415+36}{120}}\\\ \\ \mathtt{\Longrightarrow \ \frac{451}{120}}

Hence, **451/120 is the solution.**

**Example 04**

Solve \mathtt{\frac{-6}{5} +\frac{-2}{3} +\ \frac{3}{5} \ +\frac{2}{7}} **Solution**

We will solve this using Method 02.**Form pair of two using brackets**.

Here -6/5 and 3/5 have same denominator. So we take them under one bracket as the calculation become easier.

\mathtt{\Longrightarrow \left(\frac{-6}{5} \ +\ \frac{3}{5}\right) +\left(\frac{-2}{3} \ +\frac{2}{7}\right)}

**Now solve the numbers inside brackets.**

\mathtt{\Longrightarrow \left(\frac{-6}{5} \ +\ \frac{3}{5}\right) +\left(\frac{-2}{3} \ +\frac{2}{7}\right)}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{-3}{5} \ \right) +\ \left(\frac{-14+6}{21}\right)}\\\ \\ \mathtt{\Longrightarrow \frac{-3}{5} \ +\ \left(\frac{-8}{21}\right)}\\\ \\ \mathtt{\Longrightarrow \frac{-3}{5} \ -\frac{8}{21} \ \ } **Now solve the remaining numbers**

\mathtt{\Longrightarrow \frac{-3}{5} \ -\frac{8}{21} \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{-63-40}{105}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-103}{105}}

Hence, **-103/105 is the solution.**