How to Find the Least Common Denominator
Why Find a Common Denominator?
When we find ourselves needing to add or subtract fractions, or compare fractions, the first thing we have to do is to find a common denominator so that we can compare pieces that are the same size. While any common denominator will work, we prefer to use the least common denominator as finding a larger one can cause us to do more work unnecessarily.
Let's look at how to find the least common denominator (or lowest common denominator).
What is a Common Denominator?
A common denominator of two fractions is simply the smallest value that the denominators (that is, the bottom numbers) of both fractions divide into evenly. For example, given the fractions 1/4 and 5/6, any number that is a multiple of both four and six, such as 12, 24, or 36, is a common denominator. The smallest such number is called, unsurprisingly, the smallest, lowest, or least common denominator,
In general, given two or more numbers, we can find the lowest common multiple of those numbers; when dealing with fractions, even though we talk about how to find the least common denominator, the process is the same. The simplest way is to just list the multiples of every number and choose the first value that appears in every list. For example, suppose we wanted to find the least common denominator of the fractions 2/3, 3/4, and 4/5. We can make the following lists:
3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60
4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60
5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60
The first number that appears in all three lists is 60, so 60 is the lowest common denominator. Note that in this case, multiplying the three numbers together would have given us our answer: 3 x 4 x 5 = 60. This is not generally true, however; while multiplying our numbers together will give a common multiple, it will only give the least common multiple if the numbers have no factors in common.
Finding LCD with a Factor Tree
While the above method works, it can take quite a long time to find a common multiple, particularly when using multiple values; fortunately, there is an easier way. If we find the prime factorization of the numbers, we can simply calculate the lowest common multiple. Suppose we want to find the least common multiple (LCM) of 4, 5, and 6. We start by finding a prime factorization of each number (which we can do using a factor tree).
4 = 2 x 2
5 = 1 x 5
6 = 2 x 3
Now that each number is expressed as the product of its prime factors, we simply take the maximum number of times any factor appears in one list. Looking at the above, the factor 2 appears at most twice, 3 appears at most once, and 5 also appears at most once. Thus, the LCM of 4, 5, and 6 is 2 x 2 x 3 x 5, or 60; if we started with the fractions 1/4, 1/5, and 1/6, we would now have the equivalent fractions 15/60, 12/60, and 10/60.