How to subtract rational numbers with different denominators ?

In this chapter we will learn to subtract rational numbers with different denominators.

In the end of chapter, we have also solved some problems for better conceptual understanding.

Subtracting Rational numbers with different denominators

To subtract the numbers, we have to first make all the denominators same.

This can be done by using LCM concept.

Follow the below steps to subtract the rational numbers;

(a) Find LCM of denominators

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

(c) Now simply subtract the numerator and retain the same denominator.

I hope you understood the above three steps. Let us see some solved examples for better clarity.

Example 01
Subtract \mathtt{\frac{15}{7} -\frac{2}{3}}

Solution
Observe that both rational numbers have different denominators.

To subtract, follow the below steps;

(a) Find LCM of denominator

LCM (7, 3) = 21

(b) Multiply each rational number to make denominator 21.

(i) Fraction 15 / 7

Multiply numerator & denominator by 3.

\mathtt{\Longrightarrow \ \frac{15}{7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{15\times 3}{7\times 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{45}{21}}

(ii) Fraction 2/3

Multiply numerator and denominator by 7

\mathtt{\Longrightarrow \ \frac{2}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2\times 7}{3\times 7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{14}{21}}

We have got the fraction with same denominator. Now simply subtract the numerator and get the solution.

(c) Subtract the numerator

\mathtt{\Longrightarrow \frac{45}{21} -\frac{14}{21}}\\\ \\ \mathtt{\Longrightarrow \ \frac{45-14}{21}}\\\ \\ \mathtt{\Longrightarrow \ \frac{31}{21}}

Hence, 31/21 is the solution of given subtraction.

Example 02
Subtract \mathtt{\frac{9}{20} -\frac{1}{4}}

Solution

(a) Find LCM of denominators.

LCM (20, 4) = 20

(b) Multiply each rational number to make denominator 20.

(i) Fraction 9 / 20

The denominator is already 20. No need to do anything.

(ii) Fraction 1 / 4

Multiply numerator and denominator by 5.

\mathtt{\Longrightarrow \ \frac{1}{4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1\times 5}{4\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5}{20}}

We have got fractions with same denominator. Now simply subtract the numerator to get the solution.

(c) Subtract the numerator

\mathtt{\Longrightarrow \frac{9}{20} -\frac{5}{20}}\\\ \\ \mathtt{\Longrightarrow \ \frac{9-5}{20}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4}{20}}

Here we got 4 / 20 as solution.

The fraction can be further reduced by dividing numerator and denominator by 4.

\mathtt{\Longrightarrow \ \frac{4}{20}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4\div 4}{20\div 4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{5}}

Hence, 1/5 is the solution.

Example 03
Subtract \mathtt{\frac{1}{15} -\frac{2}{9}}

Solution

(a) Find LCM of denominators.

LCM (15, 9 ) = 45

(b) Multiply each rational number to make denominator equal to 45.

(i) Fraction 1/ 15

Multiply numerator and denominator by 3

\mathtt{\Longrightarrow \ \frac{1}{15}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1\times 3}{15\times 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3}{45}}

(ii) Fraction 2/9

Multiply numerator and denominator by 5.

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

We have got the fractions with same denominator. Now simply subtract the numerator to get the solution.

(c) Subtract the numerator

\mathtt{\Longrightarrow \frac{3}{45} -\frac{10}{45}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3-10}{45}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-7}{45}}

Hence, -7/45 is the solution of given expression.

Example 04
Subtract \mathtt{\frac{10}{23} -\frac{-6}{5}}

Solution
Do the following steps;

(a) Find LCM of denominators

LCM (23, 5) = 115

(b) Multiply the rational numbers to make denominator equals 115

(i) Fraction 10 / 23

Multiply numerator and denominator by 5.

\mathtt{\Longrightarrow \ \frac{10}{23}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10\times 5}{23\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{50}{115}}

(ii) Fraction -6 / 5

Multiply numerator and denominator by 23

\mathtt{\Longrightarrow \ \frac{-6}{5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-6\times 23}{5\times 23}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-138}{115}}

We have got both fractions with same denominator. Now simply subtract the numerator.

(c) Subtracting the numerators

\mathtt{\Longrightarrow \frac{50}{115} -\frac{-138}{115}}\\\ \\ \mathtt{\Longrightarrow \ \frac{50-( -138)}{115}}\\\ \\ \mathtt{\Longrightarrow \ \frac{50\ +\ 138}{115}}\\\ \\ \mathtt{\Longrightarrow \ \frac{188}{115}}

Hence 188/115 is the solution.

Example 05
Subtract \mathtt{\frac{-1}{6} -\frac{-1}{3}}

Solution
(a) Find LCM of denominators.

LCM (6, 3) = 6

(b) Multiply each rational number to make denominator equals to 6.

(i) Rational number -1/6

The denominator is already 6, so no need to do anything.

(ii) Rational number -1/3

Multiply numerator and denominator by 2.

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

(c) Now subtract the numerators.

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

Hence, 1/6 is the solution of given subtraction.