One final rule—perhaps the most important—is one that you must always remember when working with fractions that have an expression (more than one term) in the numerator or denominator. Three examples are:
In example (a), the numerator is expressed as a sum.
In example (b), the denominator is expressed as a sum.
In example (c), both the numerator and the denominator are expressed as sums.
When simplifying fractions that incorporate sums (or differences), remember this rule: You may split up the terms of the numerator, but you may never split the terms of the denominator.
Thus, the terms in example (a) may be split:
But the terms in example (b) may not be split:
Instead, simplify the denominator first:
The terms in example (c) may not be split either:
Instead, simplify both parts of the fraction:
Often, GRE problems will involve complex fractions with variables. On these problems, it is tempting to split the denominator. Do not fall for it!
It is tempting to perform the following simplification:
This is wrong because you cannot split terms in the denominator.
The reality is that
cannot be simplified further.
On the other hand, the expression
can be simplified by splitting the difference, because this difference appears in the numerator. Thus:
Simplify the following fractions:
