Thinking Carefully Through Data Sufficiency

 I *was* going to write a post on zero, but I wanted to write this one first. Hey, it's my blog. :)

A student recently came to me to review a question that was a lot like this:

Does 4x + 7y = 19?

(1) 8x/ 7-y = 6

(2)  x= 3

Now, whether you find this question easy, hard, or somewhere in between, stay with me while I explain how and why the student got the question wrong, and how he could have gotten it right, because I think that his mistakes were ones students frequently make in DS.

I asked him to explain his thoughts on the problem. He said,

"Well, first I see that there is an x and a y, so I know I need to know more about x and y, so see, I wrote down, x = ? and y = ?. And then looking at (1), I simplified it and got 4x + 7y =21, so that was wrong. Then, looking at (2), if you know x, then you can do it, so it's sufficient. That's why I got B."

I asked the student, "You just said that with statement 2, you can 'do it.' What can you 'do'?" 

The student said, "I can solve plug in x."

"And that makes statement two sufficient?"

"Yes."

I asked: "Sufficient to do what, exactly?" 

The student paused, started to say something, then laughed, and said, "I get it. Wow, I really lost the thread, huh?"

We still had to talk about it some more, but I was really glad that he got what went wrong in the question -- that he had lost the thread. Do you get it? 

Let's go back to the beginning. The question itself was,

"Does 4x + 7y = 19?"

The student thought: "I need to know more about x and y."

The better approach would have been to think: "This is a yes or no style question. I need to know if it's true or false that 4x + 7y =19. If I'm given x and y each separately - great. But I might not need to know their individual values, just whether 4x + 7y = 19 or not.

Statement (1):  8x / 7-y = 6

The student had a good start to this one. He multiplied both sides by 7-y, to get 8x = 6(7-y).

Distributing, he got 8x = 42 - 6y.

He added the 6y over and got 8x + 6y = 42.

He divided both sides by 2, and got 4x + 3y = 21.

So far? Faultless. But his next step is where he went wrong, and it was all in how he explained the situation to himself. He looked at the simplified equation, thought, "that's wrong," and marked the statement insufficient.

You may have heard this before, but it bears repeating, and in bold letters: The statements are never, ever, ever wrong. They just might be insufficient to answer the question. It's imprecise, and therefore not a good idea, to say that the statements are "right" or "wrong" when you mean "sufficient" or "insufficient."

Here's how the student could have been accurate in his assessment of the statement. He could have done all that good algebra he did, and then he could have asked himself the original question again and attempted to answer it. His thoughts might have sounded like this:

"Does 4x + 3y =19? No, it doesn't. I just found out that it equals 21. That means I have enough information to answer the question with a resounding confident NO. I can answer this question, so it's sufficient."

Let's look at statement (2).

(2)  x= 3

The student thought: "I have x, so I can do it." 

As I mentioned above, this was imprecise thinking. What "it" could he "do"? He wasn't even really sure! It's so easy to confuse yourself when you are not clear in your thinking. He wouldn't have gotten lost if he had thought, "The question asked, 'Does 4x + 7y = 19?' If I know x, can I figure out if that equation is true or false? I really can't, because it depends on y. Statement 2 is insufficient." 

(So, in case it wasn't clear here, the answer was choice A, statement one only is sufficient.)

So there are two morals of the story today:

Moral #1: The statements are never wrong.

Wrong the statements never are! Right they must be. This means they aren't wrong.... hey, am I making the point clear and memorable? :)

Moral #2: Always ask yourself the original question and attempt to answer yourself in Data Sufficiency. This way you'll never "lose the thread" and confuse yourself into the wrong answer.

Two incidental notes:

*I asked the student if I could talk about this on my blog, and he said sure. I wouldn't just write this without asking first.

*Incidentally, that question is one I just made up for the purposes of this blog entry, but it's based off of dozens of questions I've seen just like it in Official Guide and real GMAT questions over the years.