How Would You Solve This Data Sufficiency GMAT Question?

The question format least familiar to most prospective GMAT students is unquestionably Data Sufficiency. As a test exclusive (it has a no trade clause) question type, it is unlikely that you have come across such a question without having at least glanced at a GMAT prep book. However the format is completely logical. The question is asking when do you have sufficient data to answer a question, be it “always yes”, “always no” or “specific value x”. The enemy is uncertainty; any definitive answer will suffice to answer the question and move on to the next hurdle.

As anyone who’s actively studying for the GMAT knows, you must determine whether you have sufficient data with each statement separately, and then possibly combine them if you still have not determined sufficiency. This leads most assiduous students to spend most of their time determining the relationship between the statements and the question stem. If the question were true (which it always must be), would that guarantee one specific answer? Would such a definitive answer be guaranteed if I used the other statement instead? What if I used both statements?

Allow me to pose one more rhetorical question: what happens when the exam throws a spanner in the works? The exam is designed to zigzag to avoid always asking questions in the same way. Sometimes these winding paths lead to counter-intuitive questions, which can confound unprepared test takers. One such tactic is to provide too much information (#TMI) so that test takers get perplexed as to what they’re supposed to solve.

Let’s look at an example that isn’t particularly difficult, but can cause students to feel stress and spend undue time on a question they inherently know how to solve:

If the average (arithmetic mean) of the five numbers x, 7, 2, 16 and 11 is equal to the median of the five numbers, what is the value of x?

(1)  7 < x < 11

(2) x is the median of the five numbers

(A)   Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.

(B)   Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.

(C)   Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.

(D)   Each statement alone is sufficient to answer the question.

(E)    Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

Looking at the question, we are being asked to solve for x. One specific value is needed here, as a range of values would be useless. Ignoring the statements, a lot of information is provided in the question stem. The average of the five numbers is also the median of the same numbers, so it behooves us to put them in order to give loose boundaries on x. The question specifically doesn’t put them in order for us to not necessarily see the limits as easily. In order, the set would be {x, 2, 7, 11, 16}.

Once we have an ordered set, we can easily solve for x. The first hint is that the mean and the median are the same, which we know to be true for sets that are equally spaced. That isn’t very helpful here as the spacing is not even between the four elements we already have, much less when we introduce x, but it’s a natural place for our thinking to initially go. The next step might be to use the logic that x is also the mean of the set, which can be solved algebraically or logically within a couple of steps.

Using algebra, we know that the sum of the five terms is equal to the average times the number of terms. We can then set up the equation: (x+2+7+11+16)/5=x

Which can then be mathematically combined: (36+x)/5=x

Multiplying both sides by 5 to eliminate the denominator: (36+x)=5*x

Moving x to the same side: 36=4*x

Thus: 9=x

We can also get the answer using logic, especially since the GMAT usually gives integers in this situation, so you only have a couple of values of x to plug in to find that it must be 9.

At this point, after a four step algebraic problem or a couple of educated guesses, we have done everything necessary to correctly answer this problem. (Gasp!) We have, in fact, solved the value of x without using either statement! I know the answer must be 9 from the information given uniquely in the question stem (is that answer choice F?) After solving the question, let’s look at the two statements and see which of the five answer choices we should select.

Statement 1 tells us that x is between 7 and 11. This was given in the question stem because the x was the median. In other words, statement 1 doesn’t give any new information, so it seems that it’s somewhat superfluous (TMI?). However, the question format specifically asks: “If statement 1 were true, could we solve for x”? And the answer is that, yes, absolutely we can solve that x is 9 if statement 1 were true. The fact that we can solve it without statement 1 doesn’t invalidate that we can solve it with statement 1. Specifically, statement 1 alone is sufficient to answer the question, which narrows the possible correct answers to A and D.

Statement 2 tells us that x is the median of the five numbers, which is the same information as statement 1. Statement 2 thus implies statement 1, and whatever the answer to statement 1, the same will hold for statement 2. The answer on such questions can thus only be D or E, since both statements give redundant information. Since statement 1 was true, statement 2 must also be true. Thus, each statement alone is sufficient, which is a verbatim transcript of answer choice D.

In actuality, you can solve this question without using either statement, but that option is not valid in Data Sufficiency. It’s not so much do I need the statement, but rather if the statement were true, would that guarantee the uniqueness of the answer. Since either statement alone guarantees one definitive answer, the answer must be D. On test day, you don’t want to waste undue time or second guess yourself if the question pattern isn’t exactly what you expect. Understand the rules of the game and approach each question logically. Those two tenets should be sufficient to get the right answer, even if you feel that the question has given you TMI.

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Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you weekly advice for success on your exam.  After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.

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