When Do You Have Enough Information on Data Sufficiency Questions?

by on April 5th, 2015

math_3Habitually, data sufficiency questions give students cause for concern on the GMAT quantitative section. This is primarily due to the fact that data sufficiency questions are rarely seen in high school and college, and are therefore relatively unknown to most prospective test takers. If you remember the first data sufficiency question you encountered while studying for the GMAT, it may have looked like it was written in another language.

In many ways, data sufficiency questions are like being in a foreign land. Even if you understand the rules, you are often not as comfortable as in your native environment that you’ve acclimated to over many years. It is normal to feel a little discombobulated, especially at first. However, once you’ve done a few  data sufficiency questions, you tend to get a feel for the question type. One issue still eludes a lot of test takers: When is it enough?

Data sufficiency is asking about when the data is sufficient. It is pretty easy to disprove something if you can find a counter-example right away, but if you struggle with finding definitive proof, how long should you try to work at it.

Suppose a question asks whether X^2 = Y^3, that is asking whether any perfect square is also a perfect cube, you could spend a lot of time meandering towards a solution. What if we try 2^3, which gives 8? Well 8 isn’t a perfect square of any number, so we keep going. 3^3 is 27, which isn’t a perfect square of any number either. How far should we go? The next number, 4^3, gives 64, which is a perfect square, so we found an example relatively quickly, but we could conceivably spend several minutes calculating various permutations. Imagine a question asking if X^2 = Z^5 and see how long it would take to find an example.

The good news is that the question is almost always solvable using logic, algebra and mathematical properties. The bad news is it’s not always obvious how to proceed with these definitive approaches, and the brute force strategy is often employed. We can try various options and see if any of them work, while at the same time looking for patterns that tend to repeat or signal the underlying logic of the situation. While this strategy certainly has its place, it can sometimes be very wearisome.

Let’s look at a data sufficiency question that highlights this issue:

W, X, Y and Z represent distinct integers such that WX * YZ = 1,995. What is the value of W?

   WX

* YZ

_____

1,995

  • X is a prime number
  • Z is not a prime number
  • Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.
  • Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.
  • Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.
  • Each statement alone is sufficient to answer the question.
  • Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

This question can be very tempting to start off with brute force. We can limit our choices by looking at the unit digits. If the unit digit of the product is 5, then there are only a few digits that are possible for X and Z. They all have to be odd, and, more than that, one of them must be exactly 5, as no other digits combine to give a 5. If one of them is 5, the other one is some odd number, 1, 3, 5, 7 or 9. Unfortunately, multiple options exist at both prime (3, 5 and 7) and non-prime (1, 9) for these digits, so it will be hard to narrow down the choices.

Let’s look at this problem another way, which is: these two numbers must multiply to 1,995. We know one number ends with a 5, so we arbitrarily set it to be 25 and see what that gives if we set the other number to be 91. That comes to 2,275, which is way above what we need. How about 25 * 81, that yields 2,025. That’s too big, but just barely. How about 25 * 79? That will give us 1,975, which is slightly too small. We can’t get 1,995 with 25, but that’s all we’ve demonstrated so far. We can eliminate some choices as number like 15 can never be multiplied by a 2-digit number and yield 1,995, but there are still numerous choices to test.

It’s pretty easy to see how the brute force approach when you have dozens of possibilities will be very tedious. There’s another element that’s even worse, which is let’s say you manage to find a combination that works (such as 21 * 95), how can you be sure that this is the only way to get this product? Short of trying every single possibility, you can’t be sure of your answer.

This problem thus requires a more structured approach, based on mathematical properties and not dumb luck. If two numbers multiply to a specific product, then we can limit the possibilities by using factors. We thus need to factor out 1,995 and we’ll have a much better idea of the limitations of the problem.

1,995 is clearly divisible by 5, but the other number might be hard to produce. The easiest trick here is to think of it as 2,000, and then drop one multiple of 5. Since 2,000 is 5 x 400, this is 5 x 399. Now, 399 is a lot easier than it looks, because it’s clearly divisible by 3 (since the digits add up to 21, which is a multiple of 3). Afterwards, we have 133, which is another tough one, but you might be able to see that it’s divisible by 7, and actually comes to 7 x 19. Finally, since 19 is prime, we have the prime factors of 1,995: 3 x 5 x 7 x 19.

How does this help? Well there may be 16 factors of 1,995, but the limitations of the problem tell us that we only have two two-digit numbers. Thus something like 15 * 133 breaks the rules of the problem. Our only options to avoid 3-digits are 19*3 and 5*7 or 19*5 and 3*7. This gives us either 57 * 35 or 95 * 21. At least at this point we’re 100% sure that these are the only two-digit permutations that combine to give 1,995.

Let’s get back to the problem. Statement 1 tells us that X (the unit digit of the first number) is prime, which knocks out 21 from the running. However the three other options all end with a prime unit digit, meaning that any of them are still possible. At this point it’s very important to note that the problem specified that W, X, Y and Z were all distinct integers. Since they must all be different, the option of 57 * 35 is not valid because the 5 is duplicated. As such, the only option is 95*21, and the prime number restriction confirms that it’s really 95 * 21 (and not 21 * 95). Variable W must be 9, and thus this statement ends up being sufficient.

Statement 2 essentially provides the same information, as Z is not a prime number and thus necessarily 1 given our choices. This confirms that the multiplication is 95 * 21 and W is still 9. Either statement alone is sufficient, so answer choice D is the correct option here. It’s important to note how close this question was to being answer choice B, as the non-prime limitation ensured we knew where the 1 was. But the fact that these digits had to be distinct changed the answer from B to D, reinforcing the adage that you should read the questions carefully.

This question can be solved without factors, but it is very hard to confidently answer it using only a brute-force approach. Solving through mathematics and number properties is not always the easiest route to success on data sufficiency. Sometimes you can write down a few options and see exactly how the problem will unfold, but if you use concrete concepts, you’ll know when it’s been enough.

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