Least common denominator

The least common denominator is also called as Lowest common denominator. It is abbreviated as LCD. In a fraction, the bottom number is said to be a denominator. Fractions are the number written in the form of p/q, where p is the numerator and q is the denominator.

It can be defined as the smallest number, which is a common denominator of a given set of fractions. During the addition or subtraction of fractions, we generally required the same or common denominators. But if they are different, we have to find the LCD of fractions before adding or subtracting them.

The least common denominator is the LCM of two numbers. It is generally used when we add, subtract, or compare the fractions. Least common denominator is required because we cannot add fractions with different denominators.

Let's try to add the following two fractions:

2/9 + 3/5

It is difficult to add them because the denominators are not same. So, we have to find a common number to simplify it. To calculate the LCD of the fractions, we have to find the LCM (least common multiple) of their denominators. There are two methods to find the LCD. Now, we will see both of the ways to calculate the LCD of fractions.

Method 1

It is a very lengthy method, so it is not usually used. Follow the steps given below to find the LCM using multiples of numbers.

• List all the multiple of each number until the first common multiple is found.
• Pick the smallest multiple that is common in all the given numbers.

Let's understand this method by using an example. Suppose we have to find the LCD of fractions that are 1/3 and 1/6.

As we can see, the denominators of the fractions are 3 and 6. Now, we have to write some multiples of both denominators. The multiples of 3 and 6 are -

Multiples of 3: 3, 6, 9, 12, 15, 18, ….

Multiples of 6: 6, 12, 18, 24, 30, ……

We can see the common multiples of both numbers are 6, 12, 18, ….. The least multiple among these multiples is 6. So, the least common denominator, or we can say the lowest common denominator of fractions 1/3 and 1/6, is 6.

Method 2

It is another method of finding the least common denominator of two fractions. Here, we are showing some steps of calculating the least common denominator -

• First, we have to count the number of times a prime factor appears in each of the factorizations.
• Then we have to take the largest count for each prime number.
• Then write down that prime number as many times as we have counted it in the second step.
• The LCD is the product of all prime numbers written down.

We will also understand it with an example. Suppose the two fractions are 1/8 and 7/12.

The denominators of both fractions are 8 and 12. We have to calculate their prime factorization. The prime factorization of both denominators is -

8 = 2 x 2 x 2

12 = 2 x 2 x 3

Prime number 2 is occurring three times in 8, and prime number 3 is occurring one time in 12.

Their product is: 2 x 2 x 2 x 3 = 24.

So, the LCD of fractions 1/8 and 7/12 is 24.

Let's see some questions of finding the least common denominator.

Ques - 1 Find the LCD of 3/4 and 9/10.

Ans - Here, 4 and 10 are the denominators of the fractions given in the question. The multiples of the denominators are:

4 = 2 x 2

10 = 2 x 5

So, the least common denominator of 3/4 and 9/10 is

2 x 2 x 5 = 20

Ques - 2 Find the LCD of 5/6 and 10/11.

Ans - Here, 6 and 11 are the denominators of the fractions given in the question. The multiples of the denominators are:

5 = 5

11 = 11

So, the least common denominator of 5/6 and 10/11 is

5 x 11 = 55

Ques - 3 What is the least common denominator of 3/14 and 2/63?

Ans - Here, 14 and 63 are the denominators of the fractions given in the question. The multiples of the denominators are:

14 = 2 x 7

63 = 3 x 3 x 7

So, the least common denominator of 3/14 and 2/63 is

2 x 3 x 3 x 7 = 126

Ques - 4 What is the lowest common denominator of 8/25 and 23/100?

Ans - Here, 25 and 100 are the denominators of the fractions given in the question. The multiples of the denominators are:

25 = 5 x 5

100 = 2 x 2 x 5 x 5

So, the least common denominator of 8/25 and 23/100 is

2 x 2 x 5 x 5 = 100.

Ques - 5 What is the result of 2/15 + 3/10?

Ans - Here, 15 and 10 are the denominators of the fractions given in the question. The multiples of the denominators are:

15 = 3 x 5

10 = 2 x 5

So, the least common denominator of 2/15, 3/10 is

2 x 3 x 5 = 30.

Then, we will add both fractions using the common denominator 30. For this, first, we change the given fractions into equivalent fractions using the denominator 30.

2/15 = 2 x 2/15 x 2 = 4/30

3/10 = 3 x 3/10 x 3 = 9/30

So, 2/15 + 3/10 = 4/30 + 9/30 = 13/30.

Ques - 6 What is the lowest common denominator of 1½, 3, 3/8, 1, and 5/6?

Ans - In the given question, there is an integer and an improper fraction. 1½ can also be written as 3/2, and 3 can be written as 3/1. Now, the fractions are 3/2, 3/1, 3/8, and 5/6.

The denominators of the given fractions are 2, 8 and 6. The multiples of the denominators are:

2 = 2

8 = 2 x 2 x 2

6 = 2 x 3

LCD = 2 x 2 x 2 x 3 = 24

So, the least common denominator of 3/2, 3, 3/8, and 5/6 is 24.