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Number System : Arithmetic operations on Fractions

by Wonder Boy · Published October 1, 2020 · Updated September 12, 2021

Fraction : A fraction is a number in the form $\frac{x}{y}$ , where x & y are both integers and y $\ne$ 0. In the form $\frac{x}{y}$ , ‘x’ is called the numerator of the fraction and ‘y’ is called the denominator of the fraction.

We can also describe a fraction as a part of a whole. The denominator denotes how many equal parts the whole is divided into & the numerator denotes how many of the parts we are taking.

Similar to integers, we can do arithmetic operation on fractions:
Addition of Fractions : To add 2 fractions, convert the denominators of the two fractions to same number. To do this we first require to calculate the Lowest Common Multiple(LCM) of the denominators.
For Example:
$\frac{3}{4}$ + $\frac{5}{6}$ = ?
Let’s first calculate the LCM of denominators 4 & 6 which is equal to 12.
$\frac{3}{4}$ + $\frac{5}{6}$ = $\frac{9}{12}$ + $\frac{10}{12}$ = $\frac{19}{12}$ .

Subtraction of Fractions : Similar to addition operation, to subtract one fraction from other, convert the denominators of the two fractions to same number. To do this we first require to calculate the Lowest Common Multiple(LCM) of the denominators.
For Example:
$\frac{5}{6}$ – $\frac{3}{4}$ = ?
Let’s first calculate the LCM of denominators 4 & 6 which is equal to 12.
$\frac{5}{6}$ – $\frac{3}{4}$ = $\frac{10}{12}$ – $\frac{9}{12}$ = $\frac{1}{12}$ .

So, Calculation of LCM is very important for addition or subtraction operation on fraction. After calculation of LCM of denominators, the aforementioned arithmetic operation becomes very easy.

Addition or Subtraction operation on a mixed fraction: To add or subtract a mixed fraction, collect the integral part and the fractional part separately.
For Example:
$5\frac{7}{10}$ + $7\frac{3}{20}$
= 5 + 7 + $\frac{7}{10}$ + $\frac{3}{20}$
= 12 + $\frac{14}{20}$ + $\frac{3}{20}$
= 12 + $\frac{17}{20}$
= 12 $\frac{17}{20}$

For Example:
$7\frac{3}{20}$ – $5\frac{7}{10}$
= 7 – 5 + $\frac{3}{20}$ – $\frac{7}{10}$
= 2 + $\frac{3}{20}$ – $\frac{14}{20}$
= 1 + 1 – $\frac{11}{20}$
= 1 + $\frac{9}{20}$
= 1 $\frac{9}{20}$

Addition or Subtraction operation on improper fraction: To add or subtract improper fractions, firstly convert them into Mixed fractions and then continue operation as explained for Mixed fractions.
For Example:
$\frac{57}{10}$ + $\frac{143}{20}$
= $5\frac{7}{10}$ + $7\frac{3}{20}$
= 5 + 7 + $\frac{7}{10}$ + $\frac{3}{20}$
= 12 + $\frac{14}{20}$ + $\frac{3}{20}$
= 12 + $\frac{17}{20}$
= 12 $\frac{17}{20}$

Multiplication of Fractions:
Multiply the numerators of the two fractions to get the numerator of the resultant numerator.
Multiply the denominators of the two fractions to get the denominator of the resultant denominator.
For Example:
$\frac{5}{13} \times \frac{6}{11}$
= $\frac{30}{143}$

Reciprocal of a number (or fraction): One divided by that number.
For Example:
Reciprocal of 3 is $\frac{1}{3}$ .
Reciprocal of $\frac{5}{3}$ is $\frac{3}{5}$ .
Reciprocal of a number will help us in division of fractions.

Division by a Fraction: It is same as multiplication by reciprocal of the fraction.
For Example:
$\frac{3}{13} \div \frac{5}{11}$
= $\frac{3}{13} \times \frac{11}{5}$
= $\frac{33}{65}$