Simplifying Complex Fractions
By the phrase simple fraction, we mean a
fraction which does not contain any other fractions in its
numerator or in its denominator. We call complex
fractions to those fractions whose numerator and/or
denominator themselves contain other fractions. The goal of
simplifying complex fractions is to rearrange them into
equivalent simple fractions which are in simplest form.
Remark:
The word “simple” is being used in two slightly
different ways here. On one hand, a simple fraction
is a fraction that contains no other fractions as part of its
numerator or its denominator. Nevertheless, we may have a simple
fraction (i.e. it contains no other fractions in any of its
parts) that can still be simplified to another simple fraction.
What we mean by simplify in this case is to
manipulate the given simple fraction into an equivalent algebraic
form which has less terms, etc. in its numerator and denominator.
That is why it makes sense to refer to “a simple fraction in
simplest form”.
Sometimes the numerator and denominator of a complex fraction
are just single simple fractions themselves. Then, for the first
step in simplifying the complex fraction, we just use the
wellknown “invert and multiply” rule: multiply the
fraction in the numerator by the reciprocal of the fraction in
the denominator:
You see that the initial complex fraction on the left has been
turned into a single simple fraction on the right. This step is
justified only if the numerator and denominator of the original
complex fraction are both single simple fractions. When the
pattern in the box above is valid, all that is left to do in
simplifying the original complex fraction is to use methods
already illustrated many times in the last few documents in this
series to check whether the simple fraction on the right can be
simplified any further.
Example 1:
Simplify:
solution:
Since the numerator and denominator of the main fraction here
are each simple fractions themselves, we are justified in
applying the pattern in the box above. This gives us
This is now a simple fraction because neither its numerator
nor its denominator contain fractions. To check for possible
further simplification, we need to make sure that both the
numerator and denominator are completely factored, and then we
must cancel any common factors that we detect. Since the
numerator is already a product of prime numbers, it cannot be
factored further. However, for the denominator, we have
(y)(15y) = 15y^{ 2} = (3)(5)(y^{ 2})
so now
This last fraction cannot be simplified further, so it must be
the required final answer.
A strategy now suggests itself for more complicated complex
fractions. We start by simplifying the expressions in the
numerator and denominator separately until both are at worst
single simple fractions (and it is probably to our advantage to
simplify the numerator and denominator separately as much as
possible as well). Then we can apply the method illustrated
above.
Example 2:
Simplify:
solution:
This is a complex fraction because the denominator is an
expression containing a fraction. However, since the denominator
is not just a single simple fraction, the method of Example 1
cannot be applied here immediately. However, we can do the
following. First, the numerator is easily rewritten as a single
simple fraction:
For the denominator, we can write
We can get rid of the awkward minus signs in this last form by
multiplying top and bottom by 1, giving
This last result is what the denominator of the original
complex fraction looks like when written as a single simple
fraction that has been simplified. So, now we have
which has the form of the pattern in the box at the beginning
of this document. Proceeding as in Example 1, we get
as the final answer. In this example, the complex fraction
simplifies down to an expression which isn’t even a
fraction!.
