Understanding the “Spectrum of Sufficiency” is a key advanced strategy on Data Sufficiency questions and yet very few people seem to be aware of the technique. Nevertheless, understanding the concept and how to apply it can really help you answer some of the most difficult DS questions correctly and can also help prevent you from falling into traps.
First, before explaining the concept, let’s begin by discussing an underlying assumption upon which this strategy is based: one of the key features of Data Sufficiency questions is that they are meant to trick test takers and therefore obvious answers on DS questions are almost always wrong (this is true even for most easy and medium level DS questions). Now, it is not always easy to tell what an “obvious” answer is on DS, but as you get more accustomed to DS questions you will start to get a sense for what is just too straightforward and obvious. Furthermore, if you are scoring pretty high on Quant, chances are that the questions you are seeing will be quite difficult, so in that case especially obvious answers will tend to be sucker answers that you don’t want to pick.
Now, enter the “Spectrum of Sufficiency.” Basically, the answer choices on DS questions can be arranged from left to right on a scale of “how much information is needed to answer the question.” In other words, choice D indicates that each statement alone was sufficient, so you really didn’t need that much information to achieve sufficiency. Next to that would be choices A and B, which indicate that one of the statements was sufficient but the other not. Next would be choice C, which of course indicates that you needed yet more information to arrive at sufficiency: both statements were needed together. Finally, you have choice E, which indicates that no matter what you did (using the statements both alone and combined) there was not enough information to answer the question. I personally arrange the answer choices in this order from left to right:
The key here is that if you arrive at what appears to be a somewhat obvious answer, not only can you infer that it is probably wrong, but you can also get some sense for what a probable answer would be based on the “Spectrum.” In other words, if you are considering choice C, but feel like it is a bit too obvious, the right answer is probably just to the right or just to the left of choice C, since its unlikely that you are going to jump completely to the other end of the spectrum (though that definitely does happen in some cases). In fact, I find that people often think that there is not enough information to answer the question when sometimes there is, so with my students I often find myself advising them to “stay to the left.” In other words, when they think the answer is E, it is actually C. Or when they think it is A or B, it is usually D.
Let me give an example to illustrate how powerful this can be. Consider the following question (from the GMATPrep exams):
This is a very difficult question, but if you are skilled at the art of Data Sufficiency then there is a good chance that you can get it right even if you can’t properly “solve” it. What do you think the obvious (read sucker) answer is on this question? The sucker answer is C, because obviously if you have the values of n and k, you can answer the question since then you would know exactly what number is being divided by 10 (some DS neophytes miss even that because they think that they would actually need to solve for the remainder and think that it would be impossible to do that with such a large number, when of course it doesn’t matter what that remainder actually is so long as we know that we would be able to solve for a specific remainder).
Now, if C is not the correct answer (which at this point we can’t say with 100% certainty it is not), what are the next most likely answers. Well, if we look to the right on the Spectrum of Sufficiency, that would put as at choice E, but that is impossible because for sure if we have the values of n and k we could answer the question. So E is out. Looking to the left, we have A or B. It is unlikely that the answer would be D at this point since both statements appear at first to be insufficient (so for them to both all of a sudden become sufficient is unlikely). Therefore, it would be best to consider choices A and B and ask yourself which of the two statements is more likely to be sufficient (given that we know that E is out and that C is possible but extremely unlikely). Well, if you look at the statements, statement 2 really cannot be sufficient because even if we know that it is 3^22 + k, without knowing the value of k we could never know the remainder. Just think about it: 3^22 is some number, but we could add 1 or 2 or 3 or whatever on to that and that will change what the remainder will be when divided by 10 (or any number really).
At this point, you have to look at statement 1 and consider that it is very likely sufficient given that E, B, and D are out (remember that D is out because statement 2 is not sufficient so that eliminates both B and D) and that choice C is just a little too easy and straightforward (it would be too straightforward even if this were an easy DS question, which it is not). So now you can look at choice A not blindly trying to surmise if it is sufficient or not, but with the foreknowledge that it probably IS sufficient – you just need to figure out how. Here is the thing: even if you don’t have the time or the wherewithal to figure it out, I would just guess A and move on. In other words, you could basically guess correctly on this question (and it is a very difficult one at that) without even really knowing how to solve it. That is pretty powerful stuff and it demonstrates how strategy and technique on Data Sufficiency itself are as important or more important than the content that underlies the questions themselves.
I won’t go into a full explanation here of why statement A is sufficient, but just briefly, since n must be a positive integer and since n is multiplied by four, this ensures that the powers will increase by increments of 4 (you could see this by plugging in integer values of n). And there is a pattern in the units digit of powers of 3 such that the units digit is always the same as you advance by jumps of 4 (from, say, 3^6 to 3^10 to 3^14). If you knew exactly how to solve this, that is obviously great, but the truth is that you can answer this question without knowing how to solve it and without even spending the time that it would take to solve it, just by understanding the Spectrum of Sufficiency and the way it relates to the trickery of the question type.
So when you are solving Data Sufficiency questions, be very wary of “obvious” answers and consider the alternatives by using the Spectrum of Sufficiency. Often you will find, as in the above case, that the some of the alternatives can be completely ruled out and this will often put you in a position between two answers: the more straightforward, obvious one and an answer that sits immediately to the left or right on the Spectrum. At that point, go back and consider that that other answer might very well be correct or if you need to guess, avoid the obvious choice and select the most reasonable alternative. Again, based on my experience people much more often think that they don’t have enough information when in fact they actually do, so if you are really stuck and not sure which way to go, take this advice: Stay to the Left!