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**

Follow the below steps;**(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**

Follow the below steps;**(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.**