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 Ã— 2 Ã— 5 Ã— 11
and
88 = 2 Ã— 2 Ã— 2 Ã— 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 Ã— 3 Ã— 3 Ã—13
and
315 = 3 Ã— 3 Ã— 5 Ã— 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.
