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.
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).
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.
you’ve decided on an answer, test it quickly to make sure it’s correct, and
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
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:
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?
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
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 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?
||p + q
||p – q
||p2 + q2
||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
||p + q =
5 + 3 = 8
||p – q =
5 – 3 = 2
||p2 + q2 =
25 + 9 = 34
||p2 q2 =
25 9 = 225
||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
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.