Least Common Denominator: Definition & Examples

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  • 0:03 LCD: Analogy & Definition
  • 1:09 How to Find the LCD
  • 3:16 Sample Problem #1
  • 3:46 Sample Problem #2
  • 4:16 Sample Problem #3
  • 5:43 Lesson Summary
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Lesson Transcript
Instructor: Joshua White

Josh has worked as a high school math teacher for seven years and has undergraduate degrees in Applied Mathematics (BS) & Economics/Physics (BA).

In this lesson, we'll explore the least or lowest common denominator, including its definition and examples of how to find it. Our examples will include both numerical and algebraic problems.

LCD: Analogy & Definition

Imagine that you and a group of friends are ordering some pizzas for dinner. However, each person wants a different topping. For example, if five people want pepperoni, you'll probably need to order more than one pepperoni pizza to make sure that each one has enough to eat. But if only one person wants sausage, one pizza with that topping will probably be sufficient.

The process you just went through, determining how much of an item from a group of items is needed, is similar to finding the least common denominator among a group of fractions in math. Let's see how.

The least common denominator (LCD), also known as the lowest common denominator, is the least common multiple for all the denominators in a group of fractions or rational expressions. In a group of numbers, the least common multiple is the smallest number that will divide evenly into all of the numbers in the group. Rational expressions are just fractions that have variables in their denominators. For example, the pizza order you placed was similar to an LCD in that the order was the smallest number of pizzas that allowed each person to have a sufficient amount of pizza with the toppings he or she desired.

The main reason to find the LCD in math is when you have a problem where you need to add or subtract fractions or rational expressions. Since the numerators of fractions can only be combined when the fractions have identical denominators, finding the LCD is something that needs to be done quite frequently.

How to Find the LCD

To find the LCD for two or more fractions, first determine the factors for each denominator by identifying the prime factors. For example, if a fraction has a denominator of 60, you might first factor it as 6 * 10. Here, 2 time 3 equals 6 and 2 time 5 equals 10, so 60 = 2 * 2 * 3 *5.

Repeat this same process for all the denominators in your problem until you have the prime factors for each one. Then, find the LCD by multiplying the maximum number of each prime factor together.

For example, let's say you have 60 and 42 as the two denominators. You already know that 60 = 2 * 2 * 3 *5. Factoring 42 gives you 42 = 2 * 3 * 7. Now, let's count up the factors. The number 2 appears at most twice in one of the factorizations, while 3, 5, and 7 appear just once in each of the factorizations. Therefore, the LCD will be 2 * 2 * 3 * 5 * 7 or 420.

There are a couple of important things to note here. The first is that you only need two 2s, not three, because two is the maximum number of times that a 2 appears in either of the factorizations. The second thing to note is that you could have just multiplied to get the common denominator: 60 * 42 = 2520. However, 2520 is not the least or smallest common denominator.

Although it's mathematically sound to use 2520 as the common denominator instead of 420, it will make the numbers in both of the numerators larger than they need to be and most likely require spending extra time simplifying the final answer for accuracy. With that in mind, let's take a look at some more examples.

Sample Problem #1

Find the LCD for 2/15 and 3/8.

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