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# Don’t Start with Answer Choice C when “Back-Solving” on Certain GMAT Quant Questions! (or Should You?)

Warning: What I’m about to teach you in this article may go against everything you’ve been taught about the GMAT math strategy I call “Working Backwards.”

Okay, with that disclaimer out of the way, let me also say this: What I’m about to explain may also save you precious seconds on the GMAT when applying this important strategy, which may in turn result in more right answers for you on the GMAT overall.

I’ll let you ultimately decided whether or not to apply this “wrinkle” to your usage of this GMAT strategy or not, but I think you’re going to like what you see.

### Back-Solving Strategy Overview

Since the beginning of time (or at least since the advent of the current iteration of the GMAT), GMAT instructors like myself have been teaching a strategy called “Working Backwards” (or, in some circles, “Back Solving”) for certain GMAT problem solving question types.

You can watch a free GMAT video session that explains the full strategy in detail here: http://offers.dominatethegmat.com/Comprehensive-GMAT-Online-Course—Free-Trial

The basic premise of the strategy is that instead of solving qualifying GMAT math problems in the traditional algebraic way your high school algebra teacher may have taught you, why not start with the answer choices, plug them back in to the original question, and see which one works? One of those five multiple choice answer choices is correct, after all! For many students, this method is easier, faster, and more assured than a traditional algebraic approach.

Now here’s where it gets interesting. As part of that strategy, we’ve long taught (myself included) that it’s best to start by testing answer choice C when working backwards. Why? Because the multiple choice answers on the GMAT, when they’re numbers and not variables, are always in either ascending or descending (usually descending) order and so answer choice C represents the mid-range value. Therefore, if you test answer choice C and it’s too big, you can then test one of the “smaller” answer choices; if you test “C” and determine that it’s too small, you can then test one of the other “larger” answer choices. Of course if “C” works, you’re golden! It makes your life a lot easier and reduces the number of possible answer choices you have to test by starting with answer choice C.

### The Strategy’s “Wrinkle”

I was recently talking with a GMAT tutor in Denver named Mike Meresman and he pointed out something interesting about this strategy that deserves consideration. He suggests, and has the mathematical “proof” to back it up, that it’s actually more efficient to start by testing answer choice B (or D) when working backwards, not C.

Blasphemy, you say?

Hold that thought and take a look at this logic. In probability theory, there’s a concept called expected value that refers, intuitively, to the value of a random variable one would “expect” to find if one could repeat the random variable process an infinite number of times and take the average of the values obtained. It’s calculated by finding the weighted average of all possible values. In other words, each possible value that the random variable can assume is multiplied by its assigned weight, and the resulting products are then added together to find the expected value.

What does that mean in plain English? And what bearing does it have on the GMAT? It basically means that we can calculate the number of expected attempts that will be required when applying the “working backwards” technique before arriving at the correct answer. Let’s calculate the expected value in each scenario:

If you start with answer choice C, 20% of the time it will be correct and you will be finished after only one attempt. The other 80% of the time, you will have to make a second attempt with another answer choice (either larger or smaller) to determine the correct answer. Therefore, 20% times one attempt (1 * 0.2) plus 80% time two attempts (2 * 0.8) = an expected value of 1.8 attempts.

If you start with answer choice B, 20% of the time it will be correct and 20% of the time it will be too high making A correct, so that’s 40% of the time you are done in one attempt. The other 60% of the time you will need a second attempt: Even though three possible answers remain, if you test answer choice D next, it will reveal the correct answer one way or the other; if it works, you’re done; if it’s too large, the answer is C; if it’s too small, the answer is E. Therefore, 40% times one attempt (1 * 0.4) plus 60% times two attempts (2 * 0.6) = an expected value of 1.6 attempts. Starting with D works the same as with B.

Therefore, as you can see, it’s more efficient to start working backwards with answer choice B (or D) instead of C because you can expect to have to test fewer answer choices (1.6 vs 1.8) before getting the right answer.

Caveat: It will probably still make the most sense for you to start with answer choice C if it’s an easier number and will therefore make the math easier, saving you time in the long run. For example, if answer choice C is a nice round number like 10 and answer choice B (or D) is a more difficult number like 7.62, go ahead and start with C. The extra brain power you conserve by avoiding unnecessarily difficult math will be worth it!

### Application Example: Working Backwards

Here is the example I used in the free GMAT video lesson you just watched (if you haven’t done so yet, get it here) to illustrate the “Working Backwards” strategy:

Jason uses two different mixtures of windshield washer fluid for his car. In summer the mixture is one part washer fluid to three parts water; in winter the mixture is two parts washer fluid to one part water. How many ounces of washer fluid should Jason add to 24 ounces of the summer mixture in order to produce the winter mixture?

(A) 3
(B) 12
(C) 24
(D) 30
(E) 36

In the video, I solved it the traditional way starting by testing answer choice C. This time, try solving the problem again but instead start with either answer choice B or D and see if you don’t get to the answer faster.