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The Speedy Road to Insufficiency

Posted in Test Prep on October 10, 2011 by

Here’s another quick tip for those of you still struggling with the pitfalls of Data Sufficiency.

Let’s take a look at the following Official Guide DS problem:

If p and q are positive integers and pq = 24, what is the value of p?

(1)  q/6 is an integer

(2)  p/2 is an integer

It just so happens that this is a value-based question:  When we’re asked “what is the value of p?”, we’re being asked to provide a single value for p.  If a statement or combination of statements cannot give us a single value, then it must be insufficient.

So, before you go all crazy trying to set up equations, think about what I like to call “The Speedy Road to Insufficiency.”  What does that mean?  Well, in a nutshell, it means that when approaching a statement or combination of statements, you should treat it as “insufficient until proven sufficient.”  In other words, go into it trying to demonstrate insufficiency.  Why?  Well, in short, because it’s faster.  How so?  Well, in order to prove insufficiency, all we have to do is find two different possible values for p that satisfy the statement(s).  If we can quickly locate two such values, we don’t have to do any more work.  We know the statement must be insufficient.

Can we do this for Statement (1)?  Well, if q/6 is an integer and pq = 24, then q could be 6, 12, or 24, and the corresponding values of p would be 4, 2, and 1, respectively.  We just found three possible values of p.  Guess what?  We’re done with Statement (1).  Definitely insufficient.  And in truth, you could have stopped the moment you realized p could be 4 or 2.  Two possible values of p are enough to demonstrate insufficiency.

What about Statement (2)?  Well, if p/2 is an integer and pq = 24, then p could be 2, 4, 6, 8, 12, or 24.  Definitely not just one value of p, so we know Statement (2) is insufficient.

Now, I’ll just give this away and tell you that Statements (1) and (2) are still insufficient when combined.  But before reading the next paragraph, see if you can spot the speediest way to determine that.

Did you find it?  Well, notice that Statements (1) and (2) both say that p could be 2 or 4.  That’s it…don’t do any more work!  You’ve got two possible values of p, and thus you’ve shown insufficiency.

This strategy also works for “Yes/No” questions, which include phrases such as “Is x odd?” or “Is y > 2?”  These questions don’t ask for a specific value but instead ask you to answer “yes” or “no” to a specific question.  But the strategy is the same:  “Insufficient until proven sufficient”.  If you can show quickly that the answer could be either “yes” or “no”, you’ve shown insufficiency and can move on.  See if you can apply the strategy to this official problem:

If x ≠ -y, is (x-y) / (x+y) > 1 ?

(1) x > 0

(2) y < 0

Don’t just post your solution!  Also include the way you made things speedy.