MATHS
QUERY

As in previous chapter, discussed about what is fraction and what are its types. Now in this chapter, here is topic operations on fractions. All basic operations can be done on fractions i.e addition, subtraction, multiplication and division of fractions. In this chapter, addition, subtraction, multiplication and division of fractions will be discussed.

In addition of fractions of like fractions, only numerators are added and keep the value of denominator as it is.

Example

**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.

$=\frac{1}{4}+\frac{2}{4}+\frac{6}{4}$

$=\frac{\mathrm{1\; +\; 2\; +\; 6}}{4}=\frac{9}{4}$

To add unlike fractions, convert unlike fractions into like fractions by taking LCM of their denominators.

Example

**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 1: 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
$\frac{3}{12}$

$=\frac{8}{12}+\frac{8}{12}+\frac{3}{12}$

$=\frac{\mathrm{8\; +\; 8\; +\; 3}}{12}=\frac{19}{12}$

∴
$\frac{2}{3}+\frac{4}{6}+\frac{1}{4}=\frac{19}{12}$

In subtraction of like fractions, only numerators are subtracted and keep the value of denominator as it is.

Example

**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.

$=\frac{10}{9}\u2013\frac{8}{9}$

$=\frac{\mathrm{10\; \u2013\; 8}}{9}=\frac{2}{9}$

**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.

$=\frac{15}{12}\u2013\frac{11}{12}$

$=\frac{\mathrm{15\; \u2013\; 11}}{12}=\frac{4}{12}=\frac{1}{3}$

To subtract the unlike fractions, convert unlike fractions into like fractions by taking LCM of their denominators.

Example

**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
$\frac{32}{12}$

$=\frac{32}{12}\u2013\frac{18}{12}$

$=\frac{\mathrm{32\; \u2013\; 18}}{12}=\frac{14}{12}=\frac{7}{6}$

∴
$\frac{8}{3}\u2013\frac{6}{4}=\frac{14}{12}$

To multiply a fraction with whole number, multiply only numerator by the given whole number and keep the denominator same. Then reduce it to its the lowest term.

Example

**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:
$\frac{6}{5}\times 4$
**

$=\frac{6}{5}\times \frac{4}{1}$

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

To multiply a fraction by another fraction, multiply their corresponding numerators and denominators. Then reduce the obtained fraction into its the lowest form.

Example

**
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.

$\frac{24}{60}=\frac{2}{5}$

To divide a fraction with another fraction, first the division is changed into multiplication
by changing the division sign into multiplication and taking reciprocal of the second fraction.

Let’s learn it by following examples.

Example

**Example 1: Divide
$\frac{2}{3}$
by
$\frac{5}{7}$
**

$=\frac{2}{3}\xf7\frac{5}{7}$

$=\frac{2}{3}\times \frac{7}{5}$

$=\frac{2\times 7}{3\times 5}=\frac{14}{15}$

**
Example 2: Divide fraction
$\frac{4}{5}$
by whole number 6
**

$=\frac{4}{5}\xf7\frac{6}{1}$

$=\frac{4}{5}\times \frac{1}{6}$

$=\frac{\mathrm{4\; \times \; 1}}{\mathrm{5\; \times \; 6}}=\frac{4}{30}$

$\frac{4}{30}$.
should be reduced into its the lowest term by dividing with common factor 2

$=\frac{2}{15}$

**
Example 3: Divide 5 by
$\frac{6}{7}$
**

$=5\xf7\frac{6}{7}$

$=\frac{\mathrm{5\; \times \; 7}}{\mathrm{1\; \times \; 6}}$

$=\frac{35}{6}$