GMAT Data Sufficiency Arbitrage!

by Ryan Jacobs Oct 25, 2018

Fair warning: unless you’re the kind of game-theory geek who watches a football game with your buddies and contributes comments like “the expected value from the Jaguars kicking a field goal was way higher than the expected value from their decision to try to convert fourth down—what were they thinking?!”, you might not enjoy this article very much. Also, if you haven’t studied a lot of GMAT Data Sufficiency, you won’t get much out of what I’m going to say. But if you’re still here, I have a fun (well, fun-ish) way to save you a little time on GMAT Data Sufficiency.

Let’s say you know that:

(a) A certain GMAT Data Sufficiency statement is sufficient, and
(b) You know the answer to the GMAT Data Sufficiency question at the top of the problem, according to that statement

Then you should also know that there must be at least one example of the OTHER statement that provides the same answer to the question at the top.* So don’t waste time finding it; you know it exists.

That’s it! Fun, right?

Here’s one simple example:

Is x > 0?

(1) x > 5
(2) x > -5

Statement (1) is sufficient; that should be pretty clear (if x is bigger than 5 then it’s also bigger than 0). We also know that according to that statement, the answer to the GMAT Data Sufficiency question “Is x > 0?” is yes.

Now I have to evaluate statement (2) on its own. Normally, I’d do this by finding two scenarios where x > -5 that result in different answers to the question—I might use x = -2, in which case the answer is no, and x = 40, in which case the answer is yes, proving that statement (2) is not sufficient.

But my point today is that finding x = 40 is unnecessary. As soon as I find x = -2, in which case the answer is no, I’m done. I already know that there exists at least one example of this statement where the answer is yes, based on the work I did with the other statement. So I don’t need to find it—I already know that statement (2) is not sufficient, making the correct answer (A).

Admittedly, we didn’t save a whole bunch of time just now, but here’s another example that I saw recently while reviewing one of my students’ practice exams where we could save some time by applying this idea:

If list S contains nine distinct integers, at least one of which is negative, is the median of list S positive?

(1) The product of the nine integers in list S is equal to the median of list S.
(2) The sum of the nine integers in list S is equal to the median of list S.

Statement (1) is sufficient, because the only way I can have integer values in a list this big where the product equals the median is if I have a whole bunch of 1’s or if I have a 0 in there somewhere. Since the integers are distinct, I can’t have a whole bunch of 1’s (I can only have a single one), so the product and median must both be zero. That means that according to statement (1), the answer to the question “Is the median of list S positive?” is definitely no (since zero is not positive).

Now consider statement (2) on your own. This is where you save yourself time.

Here’s how: I already know that (a) statement (1) is sufficient and (b) according to statement (1), the answer to the question is no. Well, then I also know that there must exist an example of the other statement, statement (2), for which the answer is also no. So don’t bother finding it, because it’s going to take you a significant amount of time. Instead, just find an example of statement (2) where the answer is yes, and then you’ll know that both answers are possible (and that statement (2) is therefore insufficient).

One such example, by the way, is -9, -8, -7, -6, 1, 6, 7, 8, 9. The sum and the median are both 1, and the median is indeed positive. Statement (2) is therefore insufficient, and the correct answer is (A). Normally I would have had to spend time figuring out a list where the sum is the median and the median is negative. Now I don’t. Success!

*This is really what people mean when they say “GMAT Data Sufficiency statements can’t contradict each other.” 📝

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Ryan Jacobs is a Manhattan Prep instructor based in San Francisco, California. He has an MBA from UC San Diego, a 780 on the GMAT, and years of GMAT teaching experience. His other interests include music, photography, and hockey. Check out Ryan’s upcoming GMAT prep offerings here.