# Simplifying Fractions

There are two major difficulties when simplifying fractions. One is finding a number that is a common factor when it's not "obvious", and the other is simplifying completely. To deal with either style problem it is helpful to have a plan and approach each problem systematically.

The easiest way is to try dividing both the numerator (top number) and the denominator (bottom number) by each prime number. The rules of divisibility will simplify this process:

Start with 2:   EVEN numbers (ones that end with 2, 4, 6, 8, or 0) can be divided by two without a remainder (ie. they are divisible by 2).

Then go to 3:   Find the SUM OF THE DIGITS (Add the digits together). If the sum can be divided by three then the number is divisible by 3.   [NOTE:   Since you can tell by looking if a number is divisible by two or by five, you may want to use the "eyeball approach" before checking three....]

Next try 5:   Numbers that end with 5 or 0 are divisible by by five.

Go on to 7, 11, 13, 17 and so on.   Unfortunately there is no easy way to determine whether the number will be divisible by these -- you just have to try dividing by each. But, you can stop trying when the answer is less than the divisor.

The most challenging fractions to simplify are generally the ones that don't look like they can be simplified. For example:

26
65

Twenty-six can be divided by two (because it's even), but 65 can't. Sixty-five can be divided by five, but 26 can't... the fraction looks like it can't be simplified --BUT WAIT-- factor either number and then try dividing by the other, less obvious, factor. And always double-check to see if you can simplify ONE MORE TIME. Simplify completely:
 28   70 check the answer 2/5 (GCF=14) 28 = 2 x (2 x 7) 70 = 5 x (2 x 7) 51   85 check the answer 3/5(GCF=17) 51 = 3 x (17) 85 = 5 x (17) 54   81 check the answer 2/3GCF=27 54 = 2 x (3 x 3 x 3) 81 = 3 x (3 x 3 x 3) 16   60 check the answer 4/15GCF=4 16 = (2 x 2) x (2 x 2) 60 = (3 x 5) x (2 x 2) 39   91 check the answer 3/7GCF=13 39 = 3 x 13 91 = 7 x 13 34   85 check the answer 2/5GCF=17 34 = 2 x 17 85 = 5 x 17