Reducing Numerical Fractions to Simplest Form
To reduce a numerical fraction to simplest form
means to rewrite it as an equivalent fraction which has the
smallest possible denominator. People refer to this operation as
reducing a fraction to lowest terms.
Recall that starting with a fraction, we can get an equivalent
fraction by either:
(i) multiplying the numerator and denominator by the same
nonzero value
or
(ii) dividing the numerator and denominator by the same
nonzero value.
Obviously if the goal of simplification is to find an
equivalent fraction with a smaller denominator, we will have to
use the second principle: dividing the numerator and denominator
by the same nonzero value.
The strategy for finding the values to divide into the
numerator and denominator is quite systematic:
Step 1: Factor both the numerator and
denominator into a product of prime factors using the method
described in the previous note in this series.
Step 2: If the numerator and denominator both
have a prime factor which is the same, then divide the current
numerator and denominator by that value. The result will be the
numerator and denominator of an equivalent fraction, but with a
smaller denominator.
Repeat step 2 as often as possible. When the numerator and
denominator have no further prime factors in common, they form
the desired equivalent fraction which has the smallest
denominator. We have then obtained the simplest form of the
original fraction.
Example:
Reduce the fraction to simplest form.
solution:
For this example, we’ll go through the process
stepbystep in some detail. Then we’ll illustrate the
familiar “shortcuts” in a couple of examples.
The prime factorizations of the numerator and denominator here
are easily obtained:
42 = 2 Ã— 21 = 2 Ã— 3 Ã— 7
70 = 2 Ã— 35 = 2 Ã— 5 Ã— 7
So, the numerator and denominator of the fraction we are given
both contain the prime factor 2.
Thus
The new fraction, 21 / 35 , is simpler than the old fraction,
42 / 70 , because its denominator, 35, is smaller than the
original denominator of 70. Still, 21 / 35 is equivalent to 42 /
70 because it was obtained by dividing both the numerator and
denominator of 42 / 70 by the same value, 2.
But 21 / 35 is still not in simplest form because the
factorization of its numerator and denominator (shown in brackets
above) indicates that they both still contain a common prime
factor of 7. So
Thus, we have
These two fractions, the original 42 / 70and the final 3 / 5 ,
are equivalent because we got 3 / 5 by dividing the numerator and
denominator of 42 / 70 by the same values (2 and 7 in turn, or
effectively, 14, if you think of doing it in one step).
Furthermore, the numerator and denominator of 3 / 5clearly do not
share any further common prime factors, and so this
simplification process cannot be carried further. Therefore,
3 / 5 is the simplest form of 42 / 70 .
Example:
Reduce the fraction to simplest form.
solution:
Yes! This is the same problem as the first one. What we want
to do here is show the “shortcut” form of the strategy
for simplifying fractions. We begin as before by rewriting both
the numerator and the denominator as a product of prime factors:
Now, if you study the steps of the previous example, you will
see that dividing the numerator and denominator by 2 results in
those two factors disappearing from each. We indicate this by
drawing “slashes” through them:
Dividing the numerator and denominator in the resulting
factored equation by 7 again just results in those two factors of
7 disappearing from each. So again,
Now there are no common factors left in the numerator and
denominator, so the process ends, and we conclude that 3 / 5 is
the simplest form of 42 / 70 .
In practice, this whole process is typically done in a single
step, crossing out pairs of factors without rewriting
intermediate forms:
