Simplifying Complex Fractions

Complex fractions are fractions whose numerator and/or denominator themselves are expressions containing other fractions. The goal of simplifying complex fractions is to rearrange them into equivalent simple fractions which are in simplest form. By the phrase simple fraction, we mean a fraction which does not contain any other fractions in its numerator or in its denominator. (Note that the word “simple” is being used in two slightly different ways here. A simple fraction is a fraction containing no other fractions as part of its numerator or its denominator. However, even when a fraction is a simple fraction – it contains no other fractions in any of its parts – it may still be possible to simplify that simple fraction. The word simplify is this case means to manipulate the given simple fraction into an equivalent algebraic form which has less terms, etc. in its numerator and denominator. Thus a “simple fraction” may well still need to be “simplified” in this sense – it is this that most of our work on algebraic fractions so far has involved. That is why it makes sense to refer to “a simple fraction in simplest form.” When the words “simple” and “simplify” are used in this way, you can have a simple fraction which still needs to be simplified.)

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:

This example is similar to the previous one. We have a complex fraction here because this expression overall is a fraction, and its numerator is an expression which also contains a fraction.

You might think that we could cancel the 4 and the 2 in the numerator against the 8 in the denominator, as in

However, this is not a valid operation because the 4 and the 2 are not factors of the entire numerator (at least as we’ve identified them here) and so are not eligible for cancellation. Before we can consider any sort of cancellations between the numerator and denominator of the original expression here, we need to make sure that each are in the form of single simple fractions whose parts are fully factored.

So, begin by simplifying the expression in the numerator to a single simple fraction in simplest form:

Now we can rewrite the original fraction as one fraction divided by a second fraction, and then simplify the result:

This is the final result, since no further simplification is possible. (Some people might be tempted to cancel the x ² terms here, but you know better, of course. x ² is not a factor in the numerator, and so cannot be cancelled against the factor x ² in the denominator. For the same reason, no cancellation of the 2 in the numerator into the 4 in the denominator is valid.)