# Common Denominator

When the denominators of two or more fractions are same, they are the common denominators. For example, in the fractions ${\dfrac{2}{9}}$, ${\dfrac{5}{9}}$, and ${\dfrac{8}{9}}$, 9 is the common denominator.

Why Is It Important

We can only add, subtract, or compare two or more fractions when the denominators are same. Fractions with common denominators are called like fractions. In contrast, fractions with different denominators are called unlike fractions.

For example, we can easily add the two fractions, ${\dfrac{4}{11}}$ and ${\dfrac{3}{11}}$, as their denominators are the same.

${\dfrac{14}{19}+\dfrac{3}{19}}$

Solution:

Given,
${\dfrac{14}{19}+\dfrac{3}{19}}$
The denominator is common here, that is, 19
Now, adding the numerators, we get
${\dfrac{14}{19}+\dfrac{3}{19}=\dfrac{14+3}{19}=\dfrac{17}{19}}$
Thus, the sum is ${\dfrac{17}{19}}$

Subtract ${\dfrac{8}{21}}$ from ${\dfrac{11}{21}}$

Solution:

Here, ${\dfrac{11}{21}-\dfrac{8}{21}}$
The denominator is common here, that is, 21
Now, subtracting the numerators, we get
${\dfrac{11}{21}-\dfrac{8}{21}=\dfrac{11-8}{21}=\dfrac{3}{21}}$
${=\dfrac{1}{7} }$
Thus, the difference is ${=\dfrac{1}{7} }$

Compare and order ${\dfrac{17}{26}}$, ${\dfrac{15}{26}}$, and ${\dfrac{21}{26}}$ in ascending order.

Solution:

Here, the common denominator is 26
Arranging the numerators in ascending order,
15 < 17 < 21
Thus, the given fractions are ordered as
${\dfrac{15}{26} <\dfrac{17}{26} <\dfrac{21}{26}}$

## How to Find the Common Denominator for Unlike Fractions

We can add or subtract the unlike fractions more easily by converting them to like fractions.

### LCM Method

This method involves finding the least common denominator (LCD) for the two fractions. To do this, we list the multiples of each denominator and identify the lowest common multiple (LCM) among them.

Let us add ${\dfrac{4}{5}}$ and ${\dfrac{7}{10}}$

Now, finding the LCD (Least Common Denominator) and Adding

${\dfrac{4}{5}+\dfrac{7}{10}}$

Finding the LCM of the Denominators

The denominators are 5 and 10

The LCM of 5 and 10 is 10

Finding Fractions With Common Denominator

${\dfrac{4}{5}}$ = ${\dfrac{4\times 2}{5\times 2}}$ = ${\dfrac{8}{10}}$

${\dfrac{7}{10}}$ = ${\dfrac{7\times 1}{10\times 1}}$ = ${\dfrac{7}{10}}$

Now, ${\dfrac{4}{5}+\dfrac{7}{10}}$

= ${\dfrac{8}{10}+\dfrac{7}{10}}$

= ${\dfrac{8+7}{10}}$

= ${\dfrac{15}{10}}$

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

Thus, the sum is ${\dfrac{3}{2}}$

Subtract by LCM Method: ${\dfrac{9}{13}-\dfrac{7}{26}}$

Solution:

Given,
${\dfrac{9}{13}-\dfrac{7}{26}}$
The denominators are 13 and 26
The LCM of 13 and 26 is 26
Now, the fractions with the common denominator are:
${\dfrac{9}{13}}$ = ${\dfrac{9\times 2}{13\times 2}}$ = ${\dfrac{18}{26}}$
${\dfrac{7}{26}}$ = ${\dfrac{7\times 1}{26\times 1}}$ = ${\dfrac{7}{26}}$
Here, ${\dfrac{9}{13}-\dfrac{7}{26}}$
= ${\dfrac{18}{26}-\dfrac{7}{26}}$
= ${\dfrac{18-7}{10}}$
= ${\dfrac{11}{10}}$
= ${1\dfrac{1}{10}}$
Thus, the difference is ${1\dfrac{1}{10}}$

Using LCM Method, Find the Difference: ${\dfrac{8}{15}-\dfrac{3}{20}}$

Solution:

Given,
${\dfrac{8}{15}-\dfrac{3}{20}}$
The denominators are 15 and 20
The LCM of 15 and 20 is 60
Now, the fractions with the common denominator are:
${\dfrac{8}{15}}$ = ${\dfrac{8\times 4}{15\times 4}}$ = ${\dfrac{32}{60}}$
${\dfrac{3}{20}}$ = ${\dfrac{3\times 3}{20\times 3}}$ = ${\dfrac{9}{60}}$
Here, ${\dfrac{8}{15}-\dfrac{3}{20}}$
= ${\dfrac{32}{60}-\dfrac{9}{60}}$
= ${\dfrac{32-9}{60}}$
= ${\dfrac{23}{60}}$
Thus, the difference is ${1\dfrac{23}{60}}$

### When the HCF of Denominators is 1

If the HCF of the denominators is 1, then the LCD of the given fractions is obtained by multiplying the denominators.

For example, in the fractions ${\dfrac{3}{4}}$ and ${\dfrac{2}{5}}$, the HCF of 4 and 5 is 1.

Finding the LCM

The LCM of 4 and 5 is 20

Finding the LCD (Least Common Denominator)

${\dfrac{3}{4}}$ = ${\dfrac{3\times 5}{4\times 5}}$ = ${\dfrac{15}{20}}$

${\dfrac{2}{5}}$ = ${\dfrac{2\times 4}{5\times 4}}$ = ${\dfrac{8}{20}}$

### Cross Multiplication Method

This method involves multiplying the numerator and denominator of each fraction by the denominator of the other fraction.

For example, in the fractions ${\dfrac{2}{3}}$ and ${\dfrac{7}{9}}$, we multiply 2 and 3 of the fraction ${\dfrac{2}{3}}$ by 9, and 7 and 9 of the fraction ${\dfrac{7}{9}}$ by 3

Let us subtract ${\dfrac{2}{3}}$ from ${\dfrac{7}{9}}$

${\dfrac{7}{9}-\dfrac{2}{3}}$

Finding Fractions with Common Denominator

${\dfrac{7}{9}}$ = ${\dfrac{7\times 3}{9\times 3}}$ = ${\dfrac{21}{27}}$

${\dfrac{2}{3}}$ = ${\dfrac{2\times 9}{3\times 9}}$ = ${\dfrac{18}{27}}$

Subtracting

Now, ${\dfrac{7}{9}-\dfrac{2}{3}}$

= ${\dfrac{7\times 3}{9\times 3}-\dfrac{2\times 9}{3\times 9}}$

= ${\dfrac{21}{27}-\dfrac{18}{27}}$

= ${\dfrac{21-18}{27}}$

= ${\dfrac{3}{27}}$

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

Thus, the difference is ${\dfrac{1}{9}}$

Here is another example of finding the common denominator of the fractions ${\dfrac{2}{3}}$ and ${\dfrac{1}{5}}$, as shown below.

## Least Common Denominator

The least common denominator is the smallest of all the denominators for two or more fractions with different denominators.

Find the least common denominator of 6, 8, and 15.

Solution:

The LCM of 6, 8, and 15 is 120
So, the least common denominator is 120.

What is the least common denominator of ${\dfrac{3}{4}}$, ${\dfrac{4}{5}}$, and ${\dfrac{2}{3}}$?

Solution:

The lowest common multiple of the denominators 4, 5, and 3 is 60.
So, the least common denominator is 60.