How to Make Abstract Data Sufficiency Questions More Concrete

On data sufficiency problems, it’s easy to feel overwhelmed by the abstract possibilities presented by the question. Since you don’t actually have to calculate an exact solution, frequently you are faced with problems that would be too tedious to solve without a calculator. However, just because you don’t have to actually solve them, doesn’t mean it isn’t comforting to do so when faced with abstract problems (just add a little concrete).

As a simple example, consider a question that tells you that Y is the product of the first four prime numbers. You don’t actually need to calculate that it’s 2 x 3 x 5 x 7 = 210, but it’s quick enough that you aren’t handicapped by executing the math either. Then, instead of thinking of the abstract number Y, you can always just replace it with 210. Sometimes, something as innocuous as this can help make abstract problems much more palpable.

Let’s look at an actual GMAT Data Sufficiency problem that highlights this issue:

A collection of 36 cards consists of 4 sets of 9 cards each. The 9 cards in each set are numbered 1 through 9. If one card has been removed from the collection, what is the number on that card?

(1) The units digit of the sum of the numbers on the remaining 35 cards is 6.
(2) The sum of the numbers on the remaining 35 cards is 176.

(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.

In this question, we are tasked with determining whether we can accurately predict the card that has been removed from an arbitrary set based on what’s left. (Statistically, it’s the ace of clubs!) Without doing any math, your inkling might be that it’s solvable, because removing one specific value from a larger specific value should leave yet another specific value. However, this is the type of problem where you’re likely to start second guessing yourself and you might oscillate from D to E to C. To avoid this type of indecision, let’s just calculate the actual values of the variables!

If there are 4 sets of 9 cards each, with each card being numbered from 1 to 9, then we can easily calculate the sum of each set. The brute force approach of adding 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 will work, but is slow and error-prone. A better solution is to identify that these are consecutive integers, which means the mean will be equal to the median. Since the median is clearly 5, the mean must be 5 as well. Combining with the formula that total = mean x number of elements and we have a sum of 5 x 9 = 45. Since each set is identical, the sum of each set is 45, and the total sum of the four sets is 45 x 4 = 180.

So the mystery abstract sum the question set up is actually 180. It cannot be any other number, and as such we can stop referring to it as X (or Y or the other), and start referring to it as 180. Let’s now evaluate the statements one at a time:

Statement 1 says that the units digit of the sum of the numbers on the remaining 35 cards is 6. This means that the value of the subtracted card must be 4, as all cards have a single-digit value. No other card value would leave a units digit of 6 (smaller than 14) for the sum of the remaining numbers, so 4 must be the subtracted number. Statement 1 is sufficient on its own.

Statement 2 says that the sum of the numbers on the remaining 35 cards is 176. This is very similar to statement 1, and it even gives more detail! If the sum was 180, then you’d have to subtract 4 to get to 176. This again confirms that the missing card is a 4, nothing else will do. Statement 2 is equally sufficient.

Unsurprisingly, since statements 1 and 2 are so similar, they produce either an answer of D or E. In this instance, each statement alone provided enough information to get the correct answer. In data sufficiency, it’s important to know that you don’t have to calculate these sums to answer questions, but you certainly can if you want to make sure you have the GMAT’s number.

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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.