Finding the Least Common Denominator (LCD): Fifth Grade Mathematics

Finding the Least Common Denominator (LCD) allows you to rename fractions with unlike denominators into equivalent fractions of equal piece size so they can be added or subtracted.

What is the Least Common Denominator?

Fractions cannot be added or subtracted directly if their pieces are different sizes (for example, thirds and fourths). The Least Common Denominator (LCD) is the Least Common Multiple (LCM) of the denominators of the fractions.

+--------------------------------------------------------------------------+
|                  FINDING THE LCD FOR 1/4 AND 5/6                         |
|                                                                          |
|   Step 1: List multiples of each denominator:                            |
|           Multiples of 4: 4, 8, 12, 16, 20, 24, ...                      |
|           Multiples of 6: 6, 12, 18, 24, 30, ...                         |
|   Step 2: Identify the smallest shared multiple:                         |
|           LCD = 12                                                       |
|   Step 3: Rename each fraction into equivalent twelfths:                 |
|           1/4 = (1 * 3) / (4 * 3) = 3/12                                 |
|           5/6 = (5 * 2) / (6 * 2) = 10/12                                |
+--------------------------------------------------------------------------+

Three Scenarios for Finding Common Denominators

When inspecting two denominators, one of three relationships will always apply:

+--------------------------------------------------------------------------+
|                  THREE DENOMINATOR RELATIONSHIPS                         |
+---------------------+----------------------------------------------------+
| Relationship        | Action to Take                                     |
+---------------------+----------------------------------------------------+
| One denominator is  | The larger denominator is the LCD! Only rename     |
| a multiple of other | the fraction with the smaller denominator.         |
| (e.g., 3 and 6)     | (LCD is 6; rename 1/3 to 2/6).                     |
| Denominators share  | Find the LCM using multiples or prime factors.     |
| common factors      | (e.g., 6 and 8 share factor 2; LCD is 24, not 48). |
| Denominators share  | Multiply the two denominators together.            |
| NO common factors   | (e.g., 3 and 5 share no factors; LCD is 3 * 5 = 15)|
+---------------------+----------------------------------------------------+

Avoiding Unnecessarily Large Denominators

While multiplying two denominators together (e.g., 8 * 12 = 96) always produces a common denominator, using the Least Common Denominator (LCD = 24) keeps numbers smaller and reduces the amount of simplifying needed at the end.

Exercises and Solutions

Exercise 1: Identifying the LCD and Renaming. Find the Least Common Denominator (LCD) for the fractions 3/8 and 5/12. Rename both fractions as equivalent fractions using that LCD.

Solution 1: Find the Least Common Multiple of denominators 8 and 12. Multiples of 8: 8, 16, 24, 32... Multiples of 12: 12, 24, 36... The smallest common multiple is 24, so the LCD is 24. Rename 3/8: multiply numerator and denominator by 3: (3 * 3) / (8 * 3) = 9/24. Rename 5/12: multiply numerator and denominator by 2: (5 * 2) / (12 * 2) = 10/24. The renamed equivalent fractions are 9/24 and 10/24.

Exercise 2: One Denominator Multiple of the Other. Find the LCD for 2/5 and 7/20, and explain why only one fraction needs to be renamed.

Solution 2: Examine denominators 5 and 20. Because 20 is a multiple of 5 (5 * 4 = 20), the Least Common Denominator is 20. The fraction 7/20 already has 20 as its denominator, so it does not need to be changed. Only 2/5 must be renamed: multiply numerator and denominator by 4: (2 * 4) / (5 * 4) = 8/20. The two fractions with a common denominator are 8/20 and 7/20.