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!.