Approaching Math IIC Questions
Approaching Math IIC Questions
Though there are four different types of questions on the Math IIC, there is a standard procedure that you should use to approach all of them.
  1. Read the question without looking at the answers. Determine what the question is asking and come to some conclusion about how to solve it. Do not look at the answers unless you decide that using the process of elimination is the best way to go (we describe how to use the process of elimination below).
  2. If you think you can solve the problem, go ahead. Once you’ve derived an answer, only then see if your answer matches one of the choices.
  3. Once you’ve decided on an answer, test it quickly to make sure it’s correct, and move on.
Working Backward: The Process of Elimination
If you run into difficulty while trying to solve a regular multiple-choice problem, you might want to try the process of elimination. On every question the answer is right in front of you, hidden among those 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
Given this question, you could build the equations:
Then, since y = (x – 5), you can make the equation:
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 was 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 you have 16 armchairs, then you would have 11 normal chairs and the room would contain 27 total chairs. We needed the total numbers of chairs to equal 31, so clearly C is not the right answer. But because the total number of chairs was too few, you can also eliminate A and B, the answer choices with smaller numbers of armchairs. If you then plug in 18, you have 13 normal chairs and 31 total chairs. There’s your answer: D. In this instance, plugging in the answers takes less time and, in general, just seems easier.
Notice that the last sentence began with the words “in this instance.” 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. A good rule of thumb for deciding whether to work backward is:
  • 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 quite difficult—more difficult than working out the problem would be. 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) pq
(C) p2 + q2
(D) p2 q2
(E) p2 + q
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? Once you begin this substitution it quickly becomes clear that
(A) p + q = 5 + 3 = 8
(B) pq = 5 – 3 = 2
(C) p2 + q2 = 25 + 9 = 34
(D) p2 q2 = 25 9 = 225
(E) p + q2 = 5 + 9 = 14
By picking two numbers that fit the definition of the variables provided by the question, it becomes clear that the answer has to be p2 q2 (D), since it multiplies to 225. By the way, you could have answered this question without doing the multiplication to 225 since two odd numbers, such as 9 and 25, when multiplied, will always result in an odd number.
Substituting numbers can help you transform problems from the abstract into 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. Also, when picking numbers to use as substitutes, pick wisely. Choose numbers that are easy to work with and that fit the definitions provided by the question.
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