Simplifying Complex Fractions
There are some fractions whose numerator and/or denominator
themselves are expressions containing other fractions. We call
these fractions complex fractions.
We already know one meaning of "simplifying
fractions". That is, taking a simple fraction (i.e. a
fraction that contains no other fractions neither in its
numerator nor in its denominator) and manipulate it in order to
get an equivalent algebraic form which has less terms in its
numerator and denominator. However, when we talk about
simplifying a complex fraction what we want to do is to rearrange
the given complex fraction into an equivalent simple fraction
which is in simplest form. (Note that by the phrase simple
fraction, we mean a fraction which does not contain any
other fractions in its numerator or in its denominator.)
There is one case where simplifying a complex fraction is very
straightforward:
when the numerator and denominator of a complex fraction are
just single simple fractions themselves.
Therefore, 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:
As you can see, 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 to
check whether the simple fraction on the right can be simplified
any further.
Example 1:
Simplify:
solution:
This is a complex fraction in which both numerator and
denominator contain fractions. So we begin by simplifying each
part into fully factored single simple fractions. For the
numerator, we get
For the denominator, we get
So now, getting back to the original fraction, we have
Since neither numerator nor denominator of this last form can
be factored further, and since there are no common factors that
can be cancelled here between the numerator and the denominator,
this is as much as the original fraction can be simplified and so
this last form must be the required final answer.
Example 5:
Simplify:
solution:
This is a complex fraction similar in form to the one in
Example 1. The fact that the denominators in the fractions found
in the numerator and denominator of this complex fraction are
binomials does not change the overall strategy of simplification
here. We still start by reducing the expressions in the numerator
and denominator of the complex fraction to fully factored single
simple fractions. For the numerator this gives
For the denominator, we get
Substituting these two simplified forms for the numerator and
denominator, respectively, of the original complex fraction then
gives
as the final simplified form. We know we are done here
because:
 this is a single simple fraction (no other fractions are
present in either its numerator or its denominator)
 this simple fraction cannot be simplified further because
the numerator and denominator cannot be factored further
and there are no common factors present in the two parts
which can be cancelled.
