# Common Fractions

A common fraction, also known as a simple or vulgar fraction, is a fraction where both the numerator and denominator are integers separated by a line (fraction bar). The numerator indicates how many parts are taken from the whole, while the denominator shows how many equal parts the whole is divided into. It thus represents a part of the whole.

Common fractions can be proper, improper, mixed, or complex fractions.

${\dfrac{3}{5}}$, ${\dfrac{8}{7}}$, ${\dfrac{13}{25}}$, and ${\dfrac{2}{9}}$ are a few examples of common fractions.

## Simplifying

To simplify common fractions, we divide both the numerator and the denominator by their greatest common factor (GCF) to reduce them to their lowest terms.

For example,

Let us simplify the fraction ${\dfrac{48}{60}}$

Finding the GCF

The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

The GCF of 48 and 60 is 12

Dividing the Fraction by the GCF

On dividing the numerator and denominator by 12,

${\dfrac{48\div 12}{60\div 12}}$

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

Thus, the simplified fraction is ${\dfrac{4}{5}}$

## Comparing

To compare common fractions, we first find their common denominator, and then we compare the numerators to get the result.

Let us compare ${\dfrac{7}{5}}$ and ${\dfrac{3}{5}}$

Finding the Common Denominators

Here, the denominators are the same, which is 5

Comparing the Numerators

7 > 5

This gives ${\dfrac{7}{5}}$ > ${\dfrac{3}{5}}$

Compare the following fractions: ${\dfrac{3}{10}}$ and ${\dfrac{2}{7}}$

Solution:

Given, the fractions are ${\dfrac{3}{10}}$ and ${\dfrac{2}{7}}$
The denominators are 10 and 7, which are different.
The LCM of 10 and 7 is 70
Now, converting both fractions so that they have a common denominator of 70,
${\dfrac{3}{10}}$ = ${\dfrac{3\times 7}{10\times 7}}$ = ${\dfrac{21}{70}}$
${\dfrac{2}{7}}$ = ${\dfrac{2\times 10}{7\times 10}}$ = ${\dfrac{20}{70}}$
Comparing the numerators, we get
21 > 20
Thus, ${\dfrac{2}{7}}$ > ${\dfrac{3}{10}}$

To add or subtract common fractions, we first find their common denominator. Then, we add or subtract the numerators to get the result.

Let us add ${\dfrac{7}{5}}$ and ${\dfrac{3}{5}}$

Finding the Common Denominators

Here, the denominators are the same, which is 5

${\dfrac{7}{5}+\dfrac{3}{5}}$

= ${\dfrac{7+3}{5}}$

= ${\dfrac{10}{5}}$

= ${2}$, which is in the simplest form and thus is the sum.

${\dfrac{3}{5}}$ and ${\dfrac{8}{7}}$

Solution:

Given, ${\dfrac{3}{5}+\dfrac{8}{7}}$
The denominators are 5 and 7, which are different.
The LCM of 5 and 7 is 35
Now, converting both fractions so that they have a common denominator of 35,
${\dfrac{3}{5}}$ = ${\dfrac{3\times 7}{5\times 7}}$ = ${\dfrac{21}{35}}$
${\dfrac{8}{7}}$ = ${\dfrac{8\times 5}{7\times 5}}$ = ${\dfrac{40}{35}}$
${\dfrac{3}{5}+\dfrac{8}{7}}$
= ${\dfrac{21}{35}+\dfrac{40}{35}}$
= ${\dfrac{21+40}{35}}$
= ${\dfrac{61}{35}}$, which is in the simplest form and thus is the sum.

Simplify: ${\dfrac{11}{15}-\dfrac{3}{5}}$

Solution:

Given, ${\dfrac{11}{15}-\dfrac{3}{5}}$
The denominators are 15 and 5, which are different.
The LCM of 15 and 5 is 15
Now, converting both fractions so that they have a common denominator of 15,
${\dfrac{3}{5}}$ = ${\dfrac{3\times 3}{5\times 3}}$ = ${\dfrac{9}{15}}$
${\dfrac{11}{15}-\dfrac{3}{5}}$
= ${\dfrac{11}{15}-\dfrac{9}{15}}$
= ${\dfrac{11-9}{15}}$
= ${\dfrac{2}{15}}$, which is in the simplest form.
Thus, the difference is ${\dfrac{2}{15}}$

## Multiplying or Dividing

To multiply common fractions, we multiply the numerators and denominators together.

For example,

If ${\dfrac{2}{15}}$ is multiplied by ${\dfrac{5}{8}}$

${\dfrac{2}{15}\times \dfrac{5}{8}}$

Multiplying the Numerators and Denominators

= ${\dfrac{2\times 5}{15\times 8}}$

= ${\dfrac{10}{120}}$

Simplifying

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

To divide common fractions, we multiply the first fraction (dividend) by the reciprocal of the second fraction (divisor).

For example,

If ${\dfrac{2}{15}}$ is divided by ${\dfrac{5}{8}}$

Finding the Reciprocal of the Second Fraction

The reciprocal of ${\dfrac{5}{8}}$ is ${\dfrac{8}{5}}$

Multiplying the First Fraction by the Reciprocal

On multiplying ${\dfrac{2}{15}}$ by ${\dfrac{8}{5}}$,

${\dfrac{2}{15}\times \dfrac{8}{5}}$

= ${\dfrac{2\times 8}{15\times 5}}$

Simplifying

= ${\dfrac{16}{75}}$, which is in the simplest form.

Thus, the quotient is ${\dfrac{16}{75}}$

DIvide: ${\dfrac{5}{9}\div \dfrac{14}{21}}$

Solution:

Given, ${\dfrac{5}{9}\div \dfrac{14}{21}}$
The reciprocal of ${\dfrac{14}{21}}$ is ${\dfrac{21}{14}}$
On multiplying ${\dfrac{5}{9}}$ by ${\dfrac{21}{14}}$,
${\dfrac{5}{9}\times \dfrac{21}{14}}$
= ${\dfrac{5\times 21}{9\times 14}}$
= ${\dfrac{105}{126}}$
Simplifying,
${\dfrac{105\times 21}{126\times 21}}$
= ${\dfrac{5}{6}}$
Thus, the quotient is ${\dfrac{5}{6}}$

## Common Fractions to Decimals

To convert a common fraction into its corresponding decimal value, we divide the numerator by the denominator. Here is a chart with some common fractions and their equivalent decimals.