So if you’d like a little mental exercise, see if you can figure out a way to eliminate values of 2.5 or smaller in this question without doing the actual math. I’m pretty sure that will leave you with the fastest possible solution to the question.
(If, like me, you can’t see a way to do that, there’s no real problem—we should still have plenty of time left over from answering most of the other questions as quickly and efficiently as possible.)
Page 586, Question 20
This question offers a classic example of the kinds of information we can glean from a question by thinking of the answer choices from the very beginning.
Most test-takers will try to answer this question by coming up with an algebraic expression on their own and then looking in the answer choices to find a match. That approach can work if you do it perfectly, but it’s very challenging for some people.
What I would recommend instead is to think about the similarities and differences in the answer choices and how they might be relevant to the concepts in the question.
If we look at the elements in the answer choices, we see that each choice is a fraction, with either n or 100n on the top, and either n + 75 or 2n + 75 on the bottom (choice (C) also has 100 on the bottom).
So we basically only need to figure out the answers to these questions:
1. Should 100 be involved in the correct expression? If so, in the numerator or in the denominator?
2. Should the denominator contain the expression n + 75 or 2n + 75?
Let’s think about that. As for the issue with 100, we’d want to realize that the question is asking for a percentage but the answer choices are all fractions. Ideally, thinking about the percentages-versus-fractions issue along with the idea of 100 should remind us that we have to multiply fractions by 100 to turn them into percentages (because fractions describe a portion of a single unit, and percentages describe a portion of 100 units). So we should have 100 in the correct expression, and it should be in the numerator because we’re multiplying by it.
Now, should the denominator have n or 2n? For a lot of people, the temptation is to say that 2n doesn’t make any sense, because it doesn’t appear in the text of the question. But there are a couple of clues in the answer choices that should make us re-examine that assumption. For one thing, if 2n were just some pointless, random mistake, we wouldn’t expect to see it repeated across multiple answer choices; instead, we’d expect that other choices would have other random values like 3n or 4n. On top of that, 2n + 75 actually appears more often in the denominator than just n + 75, which would suggest, according to the answer choice patterns we talked about for the SAT Math section, that 2n is actually the correct version. (Just to be clear, there’s no guarantee that 2n is correct just because it shows up more often, but in general the elements that show up more often will tend to be part of the correct expression.)
So we really want to revisit this idea of 2n. Is there any way it could make sense—either as the correct answer, or as an understandable mistake?
Actually, it makes sense as the correct answer when we realize the question is asking us to compare the number of male students to the entire number of students in the whole college. So the number in the denominator needs to reflect both the male and female students together. We know that the number of female students is given by n + 75, and the number of male students is given by n. So the sum of the female and male students will be n + 75 + n, or 2n + 75.
That means (E) must be correct.
(Notice, by the way, that 100 appears in 3 of the 5 answer choices, which strongly suggests it’s part of the right answer. In those 3 appearances, it shows up twice in the numerator, which suggests—but, again, does NOT guarantee—that it ought to be in the numerator in the correct answer.)
Page 595, Question 8
I remember being asked about this question for the first time in a live class I was teaching literally the first weekend after the second edition of the Blue Book came out. I had never seen the question before and for some reason I completely panicked in front of my students (which I almost never do). I had to admit sheepishly that I wasn’t seeing whatever I needed to be seeing, and ask the students to allow me to give them the solution over lunch break when I would have time to think more clearly. (This, by the way, is the only time in my life I ever needed to do that. But sometimes these things happen. As it would turn out, all of my troubles were caused by straying from my normal game plan because I let myself get stressed out. More on that in a moment.)
The difficulty in this question, for most people, arises from the fact that it looks like the small triangle at the top of the figure has a nearly horizontal base. We really want that base to be horizontal, because then it will be parallel to the base of the large triangle, and then c would just be 180 - a - b, like choice (C) says.