Finding the Least Common Denominator (LCD)

When denominators are different, you must use equivalent fractions as a tool to create new fractions with the same denominator. This will make them easy to compare. This new denominator is called the least common denominator (LCD). The least common denominator is the smallest number which is a common multiple of each of the original denominators.

Finding the LCD of a Set of Fractions

  1. Determine the prime factors for the denominator of each fraction.
  2. Note all prime factors that occur. For each prime factor that occurs, determine in which denominator it occurs the most. Write down the prime factor the number of times it occurs in that one denominator.
  3. Calculate the LCD of your fractions. To do this, multiply the factors written down in step 2.

To elaborate how these steps are done, let's work through these steps with two fractions. Let's find the LCD for 10/24 and 22/45:

  1. Determine the prime factors for the denominator of each fraction.
      Prime factors of 24 are: 2, 2, 2, and 3.
      Prime factors of 45 are: 3, 3 and 5.
       
  2. Note all prime factors that occur. For each prime factor that occurs, determine in which denominator it occurs the most. Write down the prime factor the number of times it occurs in that one denominator.
      The prime factor 2 occurrs most often, three times, in 24. We write the factor 2 three times. The prime factor 3 occurred most often, two times, in 45, so we write that two times, the 5 occurred only once in the denominator 45.
      The prime factors that occur are: 2, 2, 2, 3, 3, and 5.

     
  3. Calculate the LCD of your fractions. To do this, multiply the factors written down in step 2.
     
    2 x 2 x 2 x 3 x 3 x 5 = 360

      The LCD for these two fractions is 360.

Let's now look at another example of this.

Example

Find the LCD for the follow group of fractions: 9/12, 14/18, 66/100.
The answer to this is 900.

Let's work through the solution to this example.

  1. Determine the prime factors for the denominator of each fraction.
    We must write the prime factors of 12, 18, and 100.
    • Prime factors of 12 are: 2, 2, and 3
    • Prime factors of 18 are: 2, 3, and 3
    • Prime factors of 100 are: 2, 2, 5, and 5

     
  2. Note all prime factors that occur. For each prime factor that occurs, determine in which denominator it occurs the most. Write down the prime factor the number of times it occurs in that one denominator.
      In this case, the 2 is a prime factor for all three of the denominators. We must take 2 the number of times it occurs most in any one denominator. It occurs twice. This is also the case for both 3 and 5. So the prime factors that occur are 2, 2, 3, 3, 5, and 5.

     
  3. Calculate the LCD of your fractions. To do this, multiply the factors selected in step 2.
      The LCD for our fractions is 2 x 2 x 3 x 3 x 5 x 5 = 900.

Finding the LCD for a group of fractions is an important step to comparing fractions that have different denominators.