# Unit Fraction

A unit fraction is a fraction in which the numerator is always one, and the denominator is any positive integer. The term â€˜unitâ€™ means â€˜one,â€™ representing one part of the total number of parts.

${\dfrac{1}{6}}$, ${\dfrac{1}{2}}$, ${\dfrac{1}{7}}$, ${\dfrac{1}{9}}$, and ${\dfrac{1}{5}}$ are unit fractions.

When a pizza is divided into 4 equal slices, one of its slices ${\dfrac{1}{4}}$ is represented mathematically as a unit fraction.

## Difference between Unit and Non-Unit Fractions

Any fraction which is not a unit fraction is a non-unit fraction. Thus, a non-unit fraction has a numerator other than 1, while the denominator can be any positive integer.

${\dfrac{7}{8}}$, ${\dfrac{3}{5}}$, ${\dfrac{9}{11}}$ are non-unit fractions.

### With the Same Denominators

When adding or subtracting two or more unit fractions with the same denominator, we keep the denominators the same and add or subtract the numerators.

On adding ${\dfrac{1}{7}+\dfrac{1}{7}}$

${\dfrac{1}{7}+\dfrac{1}{7}}$

= ${\dfrac{1+1}{7}}$

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

When subtracting two fractions with the same denominator, we always get the difference 0.

${\dfrac{1}{9}-\dfrac{1}{9}}$

= ${\dfrac{1-1}{9}}$

= ${0}$

### With Different Denominators

If the unit fractions have different denominators, we first find their Least Common Multiple (LCM) and then make them equivalent with the LCM as their common denominator. Once the denominators are the same, we add their numerators to get the sum.

Now, let us add ${\dfrac{1}{8}+\dfrac{1}{4}}$

Here, the denominators are 8 and 4

Their LCM is 8

Since the denominators of ${\dfrac{1}{8}}$ and ${\dfrac{1}{4}}$ are different, we will first convert them to equivalent fractions and then add the numerators.

Converting ${\dfrac{1}{4}}$ to an equivalent fraction with 8 as the denominator, we get

${\dfrac{1\times 2}{4\times 2}}$ = ${\dfrac{2}{8}}$

Thus, the sum is

${\dfrac{1}{8}+\dfrac{1}{4}}$

= ${\dfrac{1}{8}+\dfrac{2}{8}}$

= ${\dfrac{1+2}{8}}$

= ${\dfrac{3}{8}}$

Now, let us subtract ${\dfrac{1}{8}}$ from ${\dfrac{1}{4}}$

Here, the denominators are 8 and 4

Their LCM is 8

The denominator of the fraction ${\dfrac{1}{8}}$ is 8 (which is the same as the LCM), while in the other fraction ${\dfrac{1}{4}}$ it is 4, which is different.

Thus, we will convert ${\dfrac{1}{4}}$ to a fraction with 8 as its denominator to make it equivalent.

${\dfrac{1\times 2}{4\times 2}}$ = ${\dfrac{2}{8}}$

Now, the difference is

${\dfrac{1}{4}-\dfrac{1}{8}}$

= ${\dfrac{2}{8}-\dfrac{1}{8}}$

= ${\dfrac{2-1}{8}}$

= ${\dfrac{1}{8}}$

## Multiplying

### With Unit Fractions

When multiplying two or more unit fractions, we simply multiply the numerators and the denominators separately.

Multiplying ${\dfrac{1}{2}\times \dfrac{1}{6}}$

= ${\dfrac{1\times 1}{2\times 6}}$

= ${\dfrac{1}{12}}$

### With Non-Unit Fractions

To multiply a unit fraction with a non-unit fraction, we multiply their numerators and the denominators separately and simplify the product when necessary.

On multiplying ${\dfrac{1}{2}\times \dfrac{5}{6}}$

= ${\dfrac{1\times 5}{2\times 6}}$

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

## Dividing

### By a Whole Number

To divide a unit fraction by a whole number, we multiply the fraction with the reciprocal of the whole number.

Let us divide ${\dfrac{1}{6}}$ by ${3}$

${\dfrac{1}{6}\div 3}$

= ${\dfrac{1}{6}\times \dfrac{1}{3}}$

= ${\dfrac{1\times 1}{6\times 3}}$

= ${\dfrac{1}{18}}$

## Solved Examples

Add the unit fractions: ${\dfrac{1}{18}+\dfrac{1}{9}+\dfrac{1}{18}}$

Solution:

Given ${\dfrac{1}{18}+\dfrac{1}{9}+\dfrac{1}{18}}$
The LCM of 18, 9, and 18 is 18
Thus, the sum is
${\dfrac{1}{18}+\dfrac{2}{18}+\dfrac{1}{18}}$
= ${\dfrac{1+2+1}{18}}$
= ${\dfrac{4}{18}}$
= ${\dfrac{2}{9}}$

Multiply the unit fractions: ${\dfrac{1}{6}}$ and ${\dfrac{1}{5}}$

Solution:

${\dfrac{1}{6}\times \dfrac{1}{5}}$
= ${\dfrac{1\times 1}{6\times 5}}$
= ${\dfrac{1}{30}}$