Data Sufficiency Language

by on September 18th, 2009

If an official looking man with a clipboard (clipboards automatically make the kookiest of the kooky look official) were to stop you on the street corner and ask for a number, what would you say? If you’re like most people, you would probably respond by saying 2 or 5 or, maybe, 12. Having given your response, you’d be happy to wander off to attend to your business.

But instead, let’s watch Clipboard Guy for a bit longer. For whatever reason, he’s been standing on that street corner asking passersby for a number for a long time now. He’s dutifully written each person’s response down. He’s gotten a lot of expected responses and is starting to get a little bored with his task. Suddenly, he asks his question and the respondent says, “1/2”. Clipboard Guy smiles and writes down the number.  Feeling rejuvenated, he stays on the street corner a while longer asking his question. Eventually, a respondent says “pi”. Clipboard Guy chuckles and writes the number down.

So, what does the story of Clipboard Guy have to do with GMAT Data Sufficiency questions? I’m glad you asked. Data Sufficiency questions often take advantage of standard ways that most people think about math.

Consider this data sufficiency question:

What is the value of x ?

(1) x > 20

(2) x < 22

The correct answer is, of course, (E). Thanks to the story about Clipboard Guy, you probably spotted the trap built into the question. However, you may have been briefly tempted by (C). After all, most people’s first thought when putting the statements together is that x = 21. Had you not just read about Clipboard Guy and realized from the story that ‘number’ and ‘integer’ are not synonymous, you may have actually gone with (C).

Notice that the test writers can adjust the difficulty of a question simply by inserting (or omitting) a single word. For the data sufficiency question above, if the question stem were changed to “What is the value of integer x?” the question becomes much easier. Most people read both versions of the question as though they ask the same thing.

Let’s ask Clipboard Guy to stand on that street corner again and ask a few other questions. Here are some of his discoveries:

• If he asks people to draw a triangle, he discovers that almost everybody draws either an equilateral or a right triangle.
• When he asks passersby to name an even number, he finds that most people say 2 or 4. He had to wait a very long time before somebody said 0 or a -2.
• Almost everybody erroneously answered 1 when he asked for the first prime number.
• Many people had obviously forgotten that remainders are always integers because so many people responded that the remainder of 10 divided by 4 is 0.5.
• More than a few respondents said that the first multiple of 10 was 20 rather than 10. A fair number of people also mixed up the terms factor and multiple because they answered 2. (Those same people had also forgotten that 1 is a factor of every integer greater than 1.)

The test writers are obviously privy to Clipboard Guys results and they are hard at work building questions based on his results.

Here’s an example of how the test writers might use Clipboard Guy’s results.

If a ≠ 0, does xa = a ?

(1) x is a single digit prime integer

(2) 0< a < 10

The question is equivalent to asking “Does x = 1?” For the first statement, x could be 2, 3, 5 or 7. In each case, given that a ≠ 0, the answer to the question “Does xa = a ?” is ‘no’ so the statement is sufficient. Statement (2) on the other hand is no help – x could be 1 for any number chosen for a or x could be some other number. So, the correct answer to this question is (A).

However, anybody who forget that 1 is not prime, would probably think that the first statement could lead to either a ‘yes’ or a ‘no’ answer to the question. That person would most likely pick (E) as their answer.

The moral to the story is that the GMAT test writers employ very precise language when they write questions. In order to avoid the traps, you need to know the precise definitions of the terms employed and you need to take the time to explore all the possibilities implied by those terms.

• Very nice one! Very helpful!!

• >More than a few respondents said that the first multiple >of 10 was 20 rather than 10
Is 0 not the first multiple of 10 ??

• In the Official Guide, the test writers are perhaps a little ambiguous when they state their definition of multiple (and factor). They state that "y is said to be divisible by x or to be a multiple x" if "y = xn for some integer n". That's a pretty standard math definition of the terms. They never state which set of numbers they are defining factors and multiples over, however, and that produces some ambiguity.

In answer to the question "Is 0 not the first multiple of 10?", the definition in the Official Guide would suggest ruling that case out. We'd get 10=(0)(n) and since no integer multiplied by 0 will produce a result of 10, the definition provided by the test writers would not count 0 as a multiple of 10. Note, however, that the definition would allow for negative numbers as multiples.

In my experience, writers of standardized tests generally act as though the word 'multiple' and 'factor' are defined over the set of positive integers. Note that most of the time they also say something like "the first positive multiple of 10" to avoid any chance of ambiguity.

Consider two data sufficiency statements. If the statement said "x is a multiple of 10", I would say that {...-20, -10, 10, 20...} satisfy the statement. However, if the statement said "x is a positive multiple of 10", then only {10, 20, 30,...} should satisfy the statement. I'd be surprised if I ever found a problem that depended on counting x=0 as satisfying either statement.

In my article, I actually meant to have Clipboard Guy ask for the "first positive multiple of 10" but it looks like I omitted the word positive. Without the word positive, there really wouldn't be a "first" multiple because we'd have to count negative multiples.

Thanks for catching the omission and giving me a chance to state things more precisely!

• Hi John, according to me there is again an error in the explanation. You very well stated the standard definition i.e "y is said to be divisible by x or to be a multiple x" if "y = xn for some integer n".
But you did not put the values(10 and 0) correctly in the equation y=xn . We are checking whether 0 is a multiple or not, so zero should be on the right hand side like-
0=(10)(n). And this is true for n=0 . So this proves that 0 is indeed a multiple. Please share your views.

• Just adding to my previous comment...what you have checked here is whether 0 is a factor of 10. But what we need to check is whether 0 is a multiple of 10!!