# Common Denominator

## Definition

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 Common Denominator 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:

As we know,
the denominator is common here, i.e. 19.
${\therefore \dfrac{14}{19}+\dfrac{3}{19}=\dfrac{17}{19} }$

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

Solution:

As we know,
the denominator is common here, i.e. 21.
${\dfrac{11}{21}-\dfrac{8}{21}}$
${\therefore \dfrac{11}{21}-\dfrac{8}{21}=\dfrac{3}{21} }$
${=\dfrac{1}{7} }$

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

Solution:

Here the common denominator is 26. So we arrange the fraction according to the numerators in ascending order.
${\dfrac{15}{26} <\dfrac{17}{26} <\dfrac{21}{26}}$

## How to Find a Common Denominator for Unlike Fractions

We can add or subtract unlike fractions more easily by converting them to like fractions. It is done by multiplying the numerator and denominator of the fraction with the denominator of the other fraction.

For example, in the fractions ${\dfrac{2}{3}}$ and ${\dfrac{4}{9}}$, we multiply 9 with both 2 and 3, and 3 with 4 and 9 as shown below.

Multiply top and bottom of each fraction by the denominator of the other.

Let us learn this through some examples.

For example, to add ${\dfrac{4}{5}}$ and ${\dfrac{7}{10}}$, we will find the LCD (Least Common Denominator).

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

Now LCM of 5 and 10 is 10.

${\therefore \dfrac{4\times 2+7\times 1}{10}}$
${=\dfrac{8+7}{10}}$
${=\dfrac{15}{10}=\dfrac{3}{2}}$
${=1\dfrac{1}{2}}$

Now subtracting two unlike fractions: ${\dfrac{9}{13}}$, and ${\dfrac{7}{26}}$.

LCM of 13 and 26 is 26.

${\therefore \dfrac{9}{13}-\dfrac{7}{26}}$

${=\dfrac{9\times 2-7\times 1}{26}}$

${=\dfrac{18-7}{26}}$

${=\dfrac{11}{26}}$

Subtract: ${\dfrac{8}{15}-\dfrac{3}{20}}$

Solution:

As we know,
LCM of 15 and 20 is 60.
${\therefore \dfrac{8}{15}-\dfrac{3}{20}}$
${=\dfrac{8\times 4-3\times 3}{60}}$
${=\dfrac{32-9}{60}}$
${=\dfrac{23}{60}}$

## 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:

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