There are a couple of ways of
finding the LCD. Take for example 
. We could list the multiples of 12 and 14—the
first number that appears on each list will be the LCD:
12, 24, 36, 48,
60, 72, 84 and 14, 28, 42, 56, 70, 84.
Because 84 is the first number on each list, 84 is the LCD for 1/12 and 9/14. This method works fine as long as your lists are not too long. But what if your denominators are 6 and 291? The LCD for these denominators (which is 582) occurs 97th on the list of multiples of 6.
We can use the prime factors of
the denominators to find the LCD more efficiently. The LCD will consist of
every prime factor in each denominator (at its most frequent occurrence). To
find the LCD for 1/12 and 9/14 factor
12 and 14 into their prime factorizations: 12 = 2 ⋅
2 ⋅ 3 and 14 =2 ⋅ 7. There
are two 2s and one 3 in the prime factorization of 12, so the LCD will have two
2s and one 3. There is one 2 in the prime factorization of 14, but this 2 is covered
by the 2s from 12. There is one 7 in the prime factorization of 14, so the LCD
will also have a 7 as a factor. Once you have computed the LCD, divide the LCD
by each denominator. Multiply each fraction by this number over itself.
LCD = 2 ⋅
2 ⋅ 3 ⋅ 7 = 84
Examples
6 = 2 ⋅
3 and
15 = 3 ⋅ 5
The LCD is 2 ⋅ 3 ⋅ 5 = 30; 30 ÷ 6 = 5 and 30 ÷ 15 = 2. Multiply 5/6
by 5/5 and 4/15 by 2/2.
24 = 2 ⋅
2 ⋅ 2 ⋅ 3
and 36 = 2 ⋅
2 ⋅ 3 ⋅ 3
The LCD = 2 ⋅ 2 ⋅ 2 ⋅
3 ⋅ 3 = 72; 72 ÷ 24 =
3 and 72 ÷ 36 = 2. Multiply 17/24
by 3/3 and 5/36 by 2/2.
Practice
Solutions
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