Strategies for SAT II Mathematics

If you run into difficulty while trying to solve a multiple-choice problem, you might want to try the process of elimination. For every question, the answer is right in front of you, hidden among five answer choices. So if you can’t solve the problem directly, you might be able to plug each answer into the question to see which one works.

Not only can this process help you when you can’t figure out a question, there are times when it can actually be faster than setting up an equation, especially if you work strategically. Take the following example:

• A classroom contains 31 chairs, some of which have arms and some of which do not. If the room contains 5 more armchairs than chairs without arms, how many armchairs does it contain?
• A. 10
• B. 13
• C. 16
• D. 18
• E. 21

Total chairs (31)=armchairs(x) + normal chairs (y)
normal chairs (y) = armchairs(x) -5
Then, since y = x – 5 you can make the equation:
31 x + (x-5)
31 = 2x-5
36= 2x
x =18

There are 18 armchairs in the classroom.

This approach of building and working out the equations will produce the right answer, but it takes a long time! What if you strategically plugged in the answers instead? Since the numbers ascend in value, let’s choose the one in the middle: C 16. This is a smart strategic move because if we plug in 16 and discover that it is too small a number to satisfy the equation, we can eliminate A and B along with C. Alternatively, if 16 is too big, we can eliminate D and E along with C.

So our strategy is in place. Now let’s work it out. If we have 16 armchairs, then we would have 11 normal chairs and the room would contain 27 total chairs. We needed the total number of chairs to equal 31, so clearly C is not the right answer. But because the total number of chairs is too few, we can also eliminate A and B, the answer choices with smaller numbers of armchairs. If we then plug in D, 18, we have 13 normal chairs and 31 total chairs. There’s our answer. In this instance, plugging in the answers takes less time, and just seems easier in general.

Now, working backward and plugging in is not always the best method. For some questions it won’t be possible to work backward at all. For the test, you will need to build up a sense of when working backward can most help you. Here’s a good rule of thumb:

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 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 × q²since 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.