Reducing Numerical Fractions to Simplest Form

There is one simplification process that is usually called cancelling common factors. You write the numerator and denominator as products of prime factors. Then, if they have a prime factor in common, simply “cancel” it out in each. This cancelling is the equivalent of dividing the numerator and denominator by that same value, so the fraction that is left after cancelling a prime factor common to both numerator and denominator will be equivalent to the original fraction, but of course, its denominator will be smaller. Then, just repeat this cancelling process for every common factor in the numerator and denominator.

Example:

Simplify

solution:

Very briefly, resolving the numerator and denominator into products of prime factors gives

220 = 2 × 2 × 5 × 11

and

88 = 2 × 2 × 2 × 11

So,

Thus, the simplest form of 220 / 88 is 5 / 2. (Notice that this simplification method works even for “improper fractions” – fractions in which the numerator is larger than the denominator.)

Example:

Reduce to lowest terms.

solution:

234 = 2 × 3 × 3 ×13

and

315 = 3 × 3 × 5 × 7

So,

Since 26 and 35 share no factors in common, this is the simplest form to which we can reduce the original fraction.

If you recognize a common factor in the numerator and denominator of a fraction, it is permissible to cancel that factor as a way to get the fraction into a simpler form immediately. However, to guarantee that your final result is the simplest form you must perform this systematic procedure. It will be essential to use the systematic approach when we deal with the simplification of algebraic fractions.