# Addition and Subtraction of Rational numbers

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.