Least Common Denominator

Definition

The least or lowest common denominator (LCD) is the smallest of all the common denominators in the given fractions.

In the fractions, ${\dfrac{3}{7}}$, ${\dfrac{5}{7}}$, and ${\dfrac{6}{7}}$, the common denominator is 7.

Why do we need to find a Common Denominator?

We need to find the common denominator because we cannot add or subtract fractions with different denominators.

But 24 slices are too many. Can we have a manageable number of slices?

Yes, we can. That’s where we calculate the LCD.

How to the Find Least Common Denominator

To find the LCD, we calculate the lowest common multiple (LCM) of the denominators for the fractions with different denominators (unlike fractions).

Considering the above example of pizzas, we will see how we find the LCD.

Finding the LCD with the Greatest Common Factor

Now,  let us calculate the least common denominator of 6, 8, and 15.

We will find the LCM of the three numbers, 6, 8, and 15.

It is 120.

For example, when adding ${\dfrac{4}{9}}$ and ${\dfrac{5}{12}}$, we need to find the greatest common factor (GCF) of 9 and 12, which is 3.

We can do it in 2 ways:

1. Either we multiply the denominators and divide by the GCF, i.e., 9 × 12 = 108, 108  ÷ 3 = 36
2. Or, we divide one of the denominators by the GCF, and then multiply the result with the other denominator, i.e., 9 ÷ 3 = 3, 3 × 12 = 36.

Therefore, LCD = 36

Now we will rename the fractions. Here is how we do it:

• We divide the LCD by one denominator. i.e., 36 ÷ 9 = 4,
• Following step 1, 36 ÷ 12 = 3,
• Adding the renamed fractions:
• ${\dfrac{4}{9}\times \dfrac{4}{4}+\dfrac{5}{12}\times \dfrac{3}{3}}$

${=\dfrac{16}{36}+\dfrac{15}{36}}$

${=\dfrac{16+15}{36}}$

${=\dfrac{31}{36}}$

${\therefore \dfrac{4}{9}+\dfrac{5}{12}=\dfrac{31}{36}}$

When the denominators were made equal, it was easy for us to add the two different fractions.

Now let us subtract the above 2 fractions:

${\dfrac{4}{9}-\dfrac{5}{12}}$
${\dfrac{4}{9}\times \dfrac{4}{4}-\dfrac{5}{12}\times \dfrac{3}{3}}$
${=\dfrac{16}{36}-\dfrac{15}{36}}$

${=\dfrac{16-15}{36}}$
${=\dfrac{1}{36}}$
${\therefore \dfrac{4}{9}-\dfrac{5}{12}=\dfrac{1}{36}}$

Solved Examples