Operations With Fractions

Similar to operations with whole numbers, the four basic operations – addition, subtraction, multiplication, and division, are also performed on fractions to simplify problems involving them.

Adding and Subtracting

To add or subtract fractions, they must have the same denominator. If not, we need to find their common denominator. 

With Like Denominators

When adding or subtracting the fractions with the same denominator, we simply add or subtract the numerators.

Let us add ${\dfrac{1}{4}}$ and ${\dfrac{5}{4}}$

Identifying the Denominators

Here, the denominators are 4 and 4

Adding the Numerators 

Now, ${\dfrac{1}{4}+\dfrac{5}{4}}$

On adding the numerators,

= ${\dfrac{1+5}{4}}$

= ${\dfrac{6}{4}}$

Simplifying

${\dfrac{3}{2}}$

Similarly, now let us subtract: ${\dfrac{7}{3}-\dfrac{2}{3}}$

Given, ${\dfrac{7}{3}-\dfrac{2}{3}}$

Identifying the Denominators

Here, the denominators are 3 and 3

Now, ${\dfrac{7}{3}-\dfrac{2}{3}}$

Subtracting the Numerators

= ${\dfrac{7-2}{3}}$

= ${\dfrac{5}{3}}$, which is in its simplest form.

Thus, the difference is ${\dfrac{5}{3}}$

With Unlike Denominators

When adding or subtracting the fractions with different denominators, we find the common denominator first and then add or subtract the numerators.

Let us subtract ${\dfrac{5}{9}}$ from ${\dfrac{4}{3}}$

${\dfrac{4}{3}-\dfrac{5}{9}}$

Identifying the Denominators

Here, the denominators are 9 and 3, which are different.

Finding the LCM of the Denominators

The LCM of 9 and 3 is 9

Making the Denominators the Same 

Converting ${\dfrac{4}{3}}$ to its equivalent fraction with 9 as the denominator,

${\dfrac{4}{3}}$ = ${\dfrac{4\times 3}{3\times 3}}$ = ${\dfrac{12}{9}}$

Now, we have ${\dfrac{12}{9}-\dfrac{5}{9}}$

Subtracting the Numerators

On subtracting the numerators,

= ${\dfrac{12-5}{9}}$

= ${\dfrac{7}{9}}$

Now, let us add: ${\dfrac{4}{5}+\dfrac{2}{7}}$

Given, ${\dfrac{4}{5}+\dfrac{2}{7}}$

Identifying the Denominators

Here, the denominators are 5 and 7, which are different.

Finding the LCM of the Denominators

The LCM of 5 and 7 is 35

Making the Denominators the Same

Now, converting ${\dfrac{4}{5}}$ and ${\dfrac{2}{7}}$ to their equivalent fractions with 35 as the denominator,

${\dfrac{4}{5}}$ = ${\dfrac{4\times 7}{5\times 7}}$ = ${\dfrac{28}{35}}$

${\dfrac{2}{7}}$ = ${\dfrac{2\times 5}{7\times 5}}$ = ${\dfrac{10}{35}}$

Now, we have ${\dfrac{4}{5}+\dfrac{2}{7}}$

Adding the Numerators

= ${\dfrac{28}{35}+\dfrac{10}{35}}$

= ${\dfrac{28+10}{35}}$

= ${\dfrac{38}{35}}$, which is in its simplest form.

Thus, the sum is ${\dfrac{38}{35}}$

Multiplying and Dividing

To multiply fractions, we multiply the numerators and the denominators separately to get the product.

Let us multiply ${\dfrac{2}{3}\times \dfrac{5}{6}}$

Multiplying the Numerators and Denominators Separately

= ${\dfrac{2\times 5}{3\times 6}}$

= ${\dfrac{10}{18}}$

Simplifying 

= ${\dfrac{5}{9}}$

Thus, the product is ${\dfrac{5}{9}}$