I don’t know if you followed the kerfuffle between Jonathan Coulton and Glee a few weeks back. It’s old news now, but I watched it unfold at the time with great interest, and I’ve been thinking about it again the last few days. The incredibly short version: Jonathan Coulton is a fairly popular musician (on the Internet, anyway) who recorded a cover of Sir Mix-a-Lot’s “Baby Got Back” in 2005 (above). Glee did a note-for-note recreation of his cover without crediting him. Then Fox’s lawyers told him he should be thankful for the exposure he didn’t get because nobody credited him.

This is a case of morality and legality not completely overlapping, and that’s all very interesting if you’re into intellectual property law (which I know is very popular among high school students these days) but that’s not where I want to go with this. The reason I bring it up is that Mr. Coulton ended up announcing that rather than pursue recourse through the courts, he’d completely change direction and try to turn this into something positive for him, and for some great charities. And there’s an SAT lesson there: know when you’re beat, and do something about it.

Coulton’s indignation was justified, but he recognized early on that he’s not going to beat an army of Fox’s lawyers, so he shifted tactics. If what you’re doing isn’t working, try something else. This is what I’m talking about when I implore you to be nimble. It’s pretty good advice for life in general, and it’s particularly germane to the SAT, on which many of the most difficult questions are vulnerable to techniques that will allow you to sidestep the math solution, if you let them. Like this one, for example:

19. Yesterday, a group of y friends went to the mall and each purchased p pairs of gym socks. If y > x > 1 and p is a positive multiple of 3, how many fewer pairs of gym socks would they have purchased if x of the members of the group had purchased only a third as many socks as they actually did?
    (A)   
    (B)   
    (C)   
    (D)   
    (E)   

If you’re looking for a top score on SAT math, you should be able to solve this with algebra, and you should also be able to solve it by plugging in. Being nimble in this way is how you work around the fact that you’re likely to see at least one problem on test day that thwarts your first attempt to solve it. Being comfortable solving a question like this two ways is also the best way to avoid careless errors—check your work by solving the way that you didn’t solve it the first time. If you get the same answer both ways, you’re almost certainly right. Both solutions below.

Let’s start with plug in

Say 10 friends go to the mall (y = 10) and each buy 3 pairs of gym socks (p = 3). So what actually happened yesterday is that the group purchased 10 × 3 = 30 pairs of socks. Now say 2 of the friends (x = 2) purchased a third as many socks as they really did. So 2 friends bought only 1 pair of gym socks each. 8 friends buy 3 pairs: 8 × 3 = 24, and 2 friends buy 1 pair each: 2 × 1 = 2. Total pairs of socks purchased: 24 + 2 = 26, or 4 fewer pairs of socks than were actually purchased. Look to your answer choices, and see which one gives you 4 when you plug in y = 10, p = 3, and x = 2.

(A) Does:   
(B) doesn’t:   
(C) doesn’t:   
(D) doesn’t:   
(E) doesn’t:   

So (A) is clearly the answer.

Now let’s do the algebra

Note that, since you’ve already spent some time working through the problem logically with plug in, the algebra should be a bit more intuitive now than it might have seemed at first. First, create an expression for what was actually purchased. That’s easy: y people purchased p pairs of socks each. Total pairs purchased: yp. Now, figure out how many would be purchased in the alternate scenario where x of the friends purchased a third as many socks as they really did. y – x purchased p pairs of socks, and x purchased p/3 pairs of socks. Total pairs purchased in alternate scenario:

Now just do some subtraction to find out how many fewer pairs of socks would have been purchased:

Unsurprisingly, we arrive at the same answer either way. Again, if you’re shooting for an 800, you really should be able to breeze through this question (and ones like it) both ways.

Bonus solution

As you might have deduced from the fact that the correct answer doesn’t contain y at all, y totally doesn’t matter. All that really matters is how many fewer socks the x friends would have purchased. If they were to purchase 1/3 of what they did, they would purchase 2/3 less than they did. Since the x friends purchased xp socks in real life, they would purchase 2/3 of that is . [See also: “Is there a math way?“]