Table of Contents

Last modified on August 3rd, 2023

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**.

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.

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:

- Either we multiply the denominators and divide by the GCF, i.e., 9 × 12 = 108, 108 ÷ 3 = 36
- 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:

${\dfrac{4}{9}-\dfrac{5}{12}}$

${\dfrac{4}{9}\times \dfrac{4}{4}-\dfrac{5}{12}\times \dfrac{3}{3}}$

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

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

${=\dfrac{1}{36}}$

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

**Find the least common denominator of 12, 15, and 18.**

Solution:

LCM of 12, 15, and 18 = 180.

∴ LCD of 12, 15, and 18 = 180.

**${\dfrac{3}{4}+\dfrac{13}{30}}$**

Solution:

As we know, 4 × 15 = 60, and also, 30 × 2 = 60

${\therefore \dfrac{3}{4}+\dfrac{13}{30}}$

${\dfrac{3}{4}\times \dfrac{15}{15}+\dfrac{13}{30}\times \dfrac{2}{2}}$

${=\dfrac{45}{60}+\dfrac{26}{60}}$

${=\dfrac{45+26}{60}}$

${=\dfrac{71}{60}}$

${=\dfrac{11}{60}}$

**${\dfrac{8}{9}-\dfrac{2}{7}}$**

Solution:

As we know, there is no common factor for 9 and 7, so we straight away multiply: 9 × 7 = 63 ${\therefore \dfrac{8}{9}-\dfrac{2}{7}}$

${\dfrac{8}{9}\times \dfrac{7}{7}-\dfrac{2}{9}\times \dfrac{9}{9}}$

${=\dfrac{56}{63}-\dfrac{18}{63}}$

${=\dfrac{56-18}{63}}$

${=\dfrac{38}{63}}$

Last modified on August 3rd, 2023