Simplifying Complex Fractions

Complex fractions are fractions whose numerator and denominator themselves contain fractions. The goal of simplifying a complex fraction is to rewrite it as an equivalent fraction in which the numerator and denominator do not contain any other fractions. When we’re considering numerical fractions, this means that the numerator and the denominator of the final form are each just single numbers.

The most effective procedure consists of several steps involving the operations we have already

1. simplify the expression in the numerator of the complex fraction to a single fraction in simplest form

2. simplify the expression in the denominator of the complex fraction to a single fraction in simplest form

3. the original complex fraction now looks like one fraction divided by another fraction. Carry out the division and simplify the result.

Example: Simplify .

solution: This is a complex fraction because the numerator is a sum of two fractions and the denominator is a difference of two fractions. You might be tempted to immediately do some cancellations (since, for example, the numerator and denominator both contain , etc.), but none of the common “things” in any of the fractions present are factors of either the numerator or the denominator, and so cancellations to attempt to simplify things here would be an error. Instead, we follow the systematic approach described above.

(1) simplify the expression in the numerator of the complex fraction to a single fraction in simplest form

(2) simplify the expression in the denominator of the complex fraction to a single fraction in

(3) now:

This last sequence includes whatever simplification appears possible. Thus, the final answer is

Example: Simplify

solution: In steps

So

as the final answer.

The method described here also covers the case where either a whole number is divided by a fraction, or a fraction is divided by a whole number.

Examples:

You can see that the general rule for this kind of problem is:

When a fraction is divided by a whole number, you just multiply that whole number into the denominator of the fraction.