Work backward when the question describes an equation of some sort and the answer choices are all simple numbers.
If the answer choices contain variables, working backward will often be more difficult than actually working out the problem. If the answer choices are complicated, with hard fractions or radicals, plugging in might prove so complex that it's a waste of time.
Substituting Numbers
Substituting numbers is a lot like working backward, except the numbers you plug into the equation aren’t in the answer choices. Instead, you have to strategically decide on numbers to substitute into the question to take the place of variables.
For example, take the question:
|
|
|
If p and q are odd integers, then which of the following must be odd? |
|
(A) |
p + q |
|
(B) |
p – q |
|
(C) |
p2 + q2 |
|
(D) |
p2  q2 |
|
(E) |
p + q2 | |
It might be hard to conceptualize how the two variables in this problem interact. But what if you chose two odd numbers, let's say 5 and 3, to represent the two variables? You get:
|
(A) |
p + q = 5 + 3 = 8 |
|
(B) |
p – q = 5 – 3 = 2 |
|
(C) |
p2 + q2 = 25 + 9 = 34 |
|
(D) |
p2 q2 = 25 9 = 225 |
|
(E) |
p + q2 = 5 + 9 = 14 | |
The answer has to be
D, p2 
q2 since it multiplies to 225. (Of course, you could have answered this question without any work at all, as two odd numbers, when multiplied,
always result in an odd number.)
Substituting numbers can help you transform problems from the abstract to the concrete. However, you have to remember to keep the substitution consistent. If you're using a 5 to represent p, don't suddenly start using 3. Choose numbers that are easy to work with and that fit the definitions provided by the question
