Simplifying Complex Fractions
Before introducing the definition of complex fraction, let's
- A simple fraction is a fraction
containing no other fractions as part of its numerator or
- A simple fraction can be simplified. What we intend to do
when simplifying a simple fraction is to obtain an
equivalent fraction that looks simpler (i.e. it has less
terms in its numerator and denominator).
Now that we have this clear, we can learn that
- complex fractions are fractions whose
numerator and/or denominator themselves are expressions
containing other fractions.
- A complex fraction can also be simplfied. What we mean by
simplifying a complex fraction is to rearrange it into an
equivalent simple fraction which is in simplest form.
You should already know how to simplify a simple fraction.
But, how do we simplify a complex fraction?
When the numerator and denominator of a complex fraction are
just single simple fractions themselves, simplifying it turns out
to be very easy. We just use the wellknown invert and
multiply rule: multiply the fraction in the numerator by
the reciprocal of the fraction in the denominator:
We have turned the initial complex fraction on the left into a
single simple fraction on the right. After you have done this,
all that is left is to check whether the simple fraction on the
right can be simplified any further.
Be careful, because this step is justified only if the
numerator and denominator of the original complex fraction are
both single simple fractions.
The strategy illustrated in the previous examples is easily
applied to this complex fraction. First, the numerator can be
simplified to give
The denominator can be simplified to give
Substituting these simplified expressions back into the
original complex fraction then gives
after cancelling common factors in the resulting numerator and
denominator. Since no further factorization is possible in either
the numerator or the denominator of this fraction, and since no
common factors still exist that can be cancelled between the
numerator and denominator of this last expression, it must be the
required most simplified form of the original complex fraction.
You can use this example as a practice problem. Try carrying
out the required simplification first before you look at the
outline of the solution that follows. This example has
essentially the same structure as that of Example 1 above. So,
first simplify the numerator
Then, simplify the denominator:
The numerator and denominator of this last expression are both
fully factored and there are no common factors in them which can
be cancelled. So this last fraction must be the required
simplified form of the original complex fraction.