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Common Denominator ? Quickly and Easily!

May 6th 2006

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By: Victor Guskov

When adding fractions we need the lowest common denominator (LCD). We can get it by decomposing each bottom into primes and multiplying the all the distinct primes and the common primes to highest power together. This method is general, but slow and troublesome for students. In nine cases of ten it is possible to obtain the LCD much more easily and quickly.

At first we must think about the greatest common factor (GCF). It is the largest number that divides evenly both given denominators. If the GCF equal to 1, we need to multiply the bottoms to get the LCD. Suppose we needed to add 3/8 and 2/9 together. The GCF of 8 and 9 is 1. So the LCD of 3/8 and 2/9 is 8 x 9 = 72.

If the GCF is greater then 1, start the next procedure. Take the largest of the denominators. If the other bottom divides it evenly, it is the LCD. Else multiply the largest of denominators by 2. If the other bottom divides evenly the result, you get the LCD. Else multiply the largest of denominators by 3, try the result and so on. Let us consider two examples.

1) Add 3/8 to 5/12.
8 divides not 12 evenly. Multiply 12 by 2. The product is 24 (you can also add 12 to 12). 8 divides evenly 24: 24/8 = 3. So 24 is the LCD of 3/8 and 5/12.
Take notice of that we have the both factors for our fractions already. Here they are: 2 for 5/12 and 3 for 3/8.
Finally, 3/8 + 5/12 = 9/24 + 10/24 = 19/24.

2) Add 5/12 to 3/16.
12 divides not 16 evenly. Multiply 16 by 2: 16 x 2 = 32 (or 16 +16 = 32). 12 divides not 32 evenly. Multiply 16 by 3: 16 x 3 = 48 (or 32 +16 = 48). 12 divides evenly 48: 48/12 = 4. So 48 is the LCD of 5/12 and 3/16. The factor for 5/12 is 4, and the factor for 3/16 is 3.
Finally, 5/12 + 3/16 = 20/48 + 9/48 = 29/48.

As I have said earlier this method works well in 9 cases of 10. Do not apply it, if you are needed to add 7/112 and 11/240 together. But I think that school math can successfully manage without fractions with so large denominators.