Permissible vs. Sufficient: Picking Numbers on Yes/No Data Sufficiency Questions
One of the hallmark points of confusion on the GMAT is the dreaded Yes/No Data Sufficiency question. In a Value question, such as “What is the value of ?” the question of sufficiency is a familiar one: if you can solve for , you have sufficiency. But in a Yes/No question, especially when variables are involved, finding a solid answer can be a much cloudier process.
The best way to clear this fog is with a concrete example. Let’s look at this Data Sufficiency question, along with its first statement:
Is Statement (1) sufficient to answer the question? Unless you have a comprehensive understanding of the underlying Number Properties at work here, your first reaction to this statement is likely to try out different numerical values for , because working with real numbers instead of variables will be a much more comfortable place for most of us. We are free to try out any value for , but our first consideration in checking this statement should be that the number we pick is permissible, according to the statement. If it is not, we can’t even consider the number as an example.
Is zero a permissible number to use here? Well, if , then is also 0, and this statement tells us that has to be greater than 1. You have to take the statements as true, so zero is NOT a number we can use here (not permissible).
How about ? That puts us into permissible territory, because , and . But even that is only half the battle. Now that we know is a permissible example, we have to see what answer it yields to our original question, “Is positive?” Since 2 is a positive number, the answer here is “Yes.”
Now we have one example in the bank, and we know that, given the information in this statement, the answer to the question can be “Yes.” But is that enough to declare sufficiency? Unfortunately, it is not. If this statement is sufficient to answer the question, it will give us an unequivocal Yes or No answer; we know now that the answer could be Yes, but could it also be No?
Well, if the answer could be No, then that would mean could be negative or zero. We’ve already seen that can’t be zero (because it’s not permissible, remember?), so what about being negative? Let’s take the flip side of our other example and try . It would certainly answer the question stem with a No, but is it permissible?
Remember, the statement mandates that . Working a little calculation, equals , and since the product of two negative numbers is a positive number, when . So our second example is permissible after all, and it answers the question “Is positive?” with a resounding “No.” Since we have answered the question with a potential “Yes” (when ) and a potential “No” (when ), this statement is actually insufficient in the end; we require further information to determine whether or not is positive.
As we see, it is absolutely necessary to remember what must be assumed as true (the statements) and what may or may not be true (the question stem) when Picking Numbers in these types of problems. While this specific example is not as challenging as some, and you may have logically thought through it with number properties rules from the outset, this thought process is vital to learn for these types of questions, and will be most helpful with the most challenging questions, where you cannot gather the potential scenarios quickly at a glance without doing some scratch work. Questions like this are exactly why we’ve kept such a close watch on the methods used in Data Sufficiency questions throughout the new GMAT revision. When the Yes/No monster rears its ugly, convoluted head, never forget when picking numbers: First permissible, then Sufficient!