Arithmetic Operations on Fractions

Example 1: Add $\frac{5}{8}$ and $\frac{6}{8}$
As the denominator is 8 and add numerator of both fractions only
$=\frac{5}{8}+\frac{6}{8}$
$=\frac{\mathrm{5\; +\; 6}}{8}=\frac{11}{8}$

Example 2: Add $\frac{1}{4}$, $\frac{2}{4}$ and $\frac{6}{4}$
Here, as denominator is 4 for all fractions, so add numerators of all fractions only.
$=\frac{1}{4}+\frac{2}{4}+\frac{6}{4}$
$=\frac{\mathrm{1\; +\; 2\; +\; 6}}{4}=\frac{9}{4}$

Example 1: Add $\frac{3}{4}$ and $\frac{5}{6}$
Step 1: Take LCM of denominators of fractions. $\frac{3}{4}$ and $\frac{5}{6}$.
Step 2: Multiply both fractions by numbers to make their denominators same equal to LCM 12.
$\frac{3}{4}\times \frac{3}{3}=\frac{9}{12}$
$\frac{5}{6}\times \frac{2}{2}=\frac{10}{12}$
Step 3: Add the like fractions $\frac{9}{12}$ and $\frac{10}{12}$.
$=\frac{9}{12}+\frac{10}{12}$
$=\frac{\mathrm{9\; +\; 10}}{12}=\frac{19}{12}$
$=\frac{\mathrm{9\; +\; 10}}{12}=\frac{19}{12}$
∴ $\frac{3}{4}+\frac{5}{6}=\frac{19}{12}$

Example 2: Add $\frac{2}{3}$, $\frac{4}{6}$ and $\frac{1}{4}$
Step 1: Take LCM of denominators of fractions $\frac{2}{3}$, $\frac{4}{6}$ and $\frac{1}{4}$
Step 2: Multiply all fractions by numbers to make their denominators same equal to LCM 12.
$\frac{2}{3}\times \frac{4}{4}=\frac{8}{12}$
$\frac{4}{6}\times \frac{2}{2}=\frac{8}{12}$
$\frac{1}{4}\times \frac{3}{3}=\frac{3}{12}$
Step 3: Add the like fractions $\frac{8}{12}$, $\frac{8}{12}$ and $\frac{3}{12}$
$=\frac{8}{12}+\frac{8}{12}+\frac{3}{12}$
$=\frac{\mathrm{8\; +\; 8\; +\; 3}}{12}=\frac{19}{12}$
∴ $\frac{2}{3}+\frac{4}{6}+\frac{1}{4}=\frac{19}{12}$

Example 1: Subtract $\frac{8}{9}$ from $\frac{10}{9}$
As these are like fractions and denominator is 9 for both fractions.
So, subtract numerator of both fractions only.
$=\frac{10}{9}–\frac{8}{9}$
$=\frac{\mathrm{10\; –\; 8}}{9}=\frac{2}{9}$

Example 2: Subtract $\frac{11}{12}$ from $\frac{15}{12}$
Here, the two fractions are like fractions and their denominator is 12.
So, subtract numerators of both fractions only.
$=\frac{15}{12}–\frac{11}{12}$
$=\frac{\mathrm{15\; –\; 11}}{12}=\frac{4}{12}=\frac{1}{3}$

Subtract $\frac{6}{4}$ from $\frac{8}{3}$
Step 1: Take LCM of denominators of fractions.

LCM of 4 and 3 is 12
Step 2: Convert unlike fractions into like fractions by multiplying a number which makes denominator equal to LCM 12.
$\frac{8}{3}\times \frac{4}{4}=\frac{32}{12}$
$\frac{6}{4}\times \frac{3}{3}=\frac{18}{12}$
Step 3: Subtract $\frac{18}{12}$ from $\frac{32}{12}$
$=\frac{32}{12}–\frac{18}{12}$
$=\frac{\mathrm{32\; –\; 18}}{12}=\frac{14}{12}=\frac{7}{6}$
∴ $\frac{8}{3}–\frac{6}{4}=\frac{14}{12}$

Example 1: $\frac{7}{5}\times 2$
Here, $\frac{7}{5}$ is a fraction and 2 is a whole number.
2 is written as $\frac{2}{1}$
$=\frac{7}{5}\times \frac{2}{1}$
$=\frac{7\times 2}{5\times 1}=\frac{14}{5}$

Example 2: $\frac{6}{5}\times 4$
$=\frac{6}{5}\times \frac{4}{1}$
$=\frac{6\times 4}{4\times 1}=\frac{24}{5}$

Example 1: Multiply fraction $\frac{7}{5}$ by $\frac{3}{4}$
Here, multiply numerators 7 and 3. Also, multiply their denominators 5 and 4.
$\frac{7}{5}\times \frac{3}{4}$
$=\frac{7\times 3}{5\times 4}=\frac{21}{20}$

Example 2: Multiply fractions $\frac{2}{5}$, $\frac{4}{6}$ and $\frac{3}{2}$
$\frac{2}{5}\times \frac{4}{6}\times \frac{3}{2}$
multiply numerators 2, 4 and 3 and also multiply denominators 5, 6 and 2.
$=\frac{\mathrm{2\; \times \; 4\; \times \; 3}}{\mathrm{5\; \times \; 6\; \times \; 2}}=\frac{24}{60}$
Now $\frac{24}{60}$ should be reduced to its the lowest term by dividing with the common factor 12.
$\frac{24}{60}=\frac{2}{5}$

Example 1: Divide $\frac{2}{3}$ by $\frac{5}{7}$
$=\frac{2}{3}÷\frac{5}{7}$
$=\frac{2}{3}\times \frac{7}{5}$
$=\frac{2\times 7}{3\times 5}=\frac{14}{15}$

Example 2: Divide fraction $\frac{4}{5}$ by whole number 6
$=\frac{4}{5}÷\frac{6}{1}$
$=\frac{4}{5}\times \frac{1}{6}$
$=\frac{\mathrm{4\; \times \; 1}}{\mathrm{5\; \times \; 6}}=\frac{4}{30}$
$\frac{4}{30}$. should be reduced into its the lowest term by dividing with common factor 2
$=\frac{2}{15}$

Example 3: Divide 5 by $\frac{6}{7}$
$=5÷\frac{6}{7}$
$=\frac{\mathrm{5\; \times \; 7}}{\mathrm{1\; \times \; 6}}$
$=\frac{35}{6}$