Lowest common denominator

Key Notes :

What is the Lowest Common Denominator (LCD)?

  • Definition: The Lowest Common Denominator (LCD) is the smallest number that is a multiple of each of the denominators in a set of fractions. It is the smallest common multiple (LCM) of the denominators.
  • The LCD allows fractions to have the same denominator so they can be added, subtracted, or compared easily.

Steps to Find the LCD:

  1. List the Multiples: Write down the multiples of each denominator.
    • Example: For fractions with denominators 4 and 6, list the multiples:
      • Multiples of 4: 4, 8, 12, 16, 20, …
      • Multiples of 6: 6, 12, 18, 24, …
  2. Find the Least Common Multiple (LCM): Identify the smallest common multiple in both lists.
    • The smallest common multiple of 4 and 6 is 12.
  3. Rewrite Fractions with the LCD: Once the LCD is found, rewrite each fraction with the LCD as the denominator by multiplying both the numerator and denominator by the necessary factor.
    • Example: To rewrite 1/4 and 1/6with a denominator of 12:
      • 1/4 = 3/12​ (multiply both numerator and denominator by 3)
      • 1/6 = 2/12​ (multiply both numerator and denominator by 2)

Learn with an example

➡️ What is the lowest common denominator of 1/6 and 8/9 ?

The denominators are 6 and 9. Find the lowest common multiple of 6 and 9.

Write the prime factorisation for each number.

  • 6 = 2 × 3
  • 9 = 3 × 3

Repeat each prime factor the most number of times it appears in any of the prime factorisations above.

The most the factor 2 appears is once.

The most the factor 3 appears is twice.

So, multiply:

2 × 3 × 3 = 18
The lowest common multiple of 6 and 9 is 18.

The lowest common denominator of 1/6 and 8/9 is 18.

➡️ What is the lowest common denominator of 2/5 and 7/10?

The denominators are 5 and 10. Find the lowest common multiple of 5 and 10.

Write the prime factorisation for each number. 5 is a prime number. You do not need to factorise 5.

5
10 = 2 × 5

Repeat each prime factor the most number of times it appears in any of the prime factorisations above.

The most the factor 2 appears is once.

The most the factor 5 appears is once.

So, multiply:

2 × 5 = 10
The lowest common multiple of 5 and 10 is 10.

The lowest common denominator of 2/5 and 7/10 is 10.

What is the lowest common denominator of 1/4 and 1/8?

The denominators are 4 and 8. Find the lowest common multiple of 4 and 8.

Write the prime factorisation for each number.

4 = 2 × 2
8 = 2 × 2 × 2

Repeat each prime factor the most number of times it appears in any of the prime factorisations above.

The most the factor 2 appears is three times.

So, multiply:

2 × 2 × 2 = 8

The lowest common multiple of 4 and 8 is 8.

The lowest common denominator of 1/4 and1/8 is 8.

Let’s practice!🖊️