College Board released a bunch of sample questions this week for the new PSAT and SAT, which will make their debuts in October 2015 and March 2016, respectively. Over the next few days, I’ll be making posts working through each question, a few at a time, and commenting on them when I feel like I have something insightful to say.

In this post, I’ll deal with questions 12 through 15 in the “calculator permitted” section. See questions 1 through 5 here, 6 through 11 here, 16 through 20 here, 21 through 26 here, and 27 through 30 here.

Now we’re finally getting into some fun stuff! We know the company will bring in 12n dollars for n items sold, and we know it costs the company 7n + 350 to make n items. To figure out how many items need to be sold to make a profit, solve the following inequality:

12n > 7n + 350
5n > 350
n > 70

Again, the focus here is not on the algebra itself—it’s on whether you can figure out how to set the algebra up. That, I suppose, is what the College Board means by “Heart of Algebra.”

I’m flummoxed here because they’re calling this a medium difficulty question, and it seems far easier than most of the other easy questions we’ve seen together so far, as long as you remember that the number of males doesn’t necessarily equal the number of females. (Again, this is a reading question as much as it is a math question.) If the average age of the males is 15, and the average age of the females is 19. If the number of males equalled the number of females, then the average age would be exactly 17, but we don’t know that the preserve has the same number of males and females.

In spite of myself, I kinda like this question. It’s all about being cautious and modest with data. The only statement that data about 2000 16-year-olds in the US can support is a statement specifically about 16-year-olds in the US. If you want to draw conclusions about people in the world, then you have to sample the whole world. So cross off B and D.

Then, you have to remember something that will be drilled into your head in any social science class (and hopefully has already been drilled into your head a bit): correlation does not imply causation! Just because exercise and sleep are positively correlated (this question uses the word associated, which is a more general term, but association does not imply causation lacks that alliterative flair) we cannot conclude that one causes the other.

Before we start discussing this, note that the CB, at the time of my writing this post, has F’d up a bit. Answer choice D, which should be the right answer, is misprinted. It looks like that over there on the right, but it should look like this: . That’s a big boo-boo to have a typo in the right answer—hopefully they’ll fix it soon.

Anyhoo, you can plug in here if you like. You know the population is supposed to double every 12 years, and you know the starting population is 50. So if you say n = 12, the right answer should resolve to 100. Let’s check:

A:
B:
C:
D:

Only D works!

If you want to understand why D works, you need to first understand exponential growth in general. The basic setup of exponential growth (compound interest in a savings account is one classic application of this) is that you have a starting value (a), a growth rate over a certain period (r), and a number of periods (p). Where you’ll be after p periods is . Why?

Well, look at just the first 3 periods, one period at a time. You start with a. After 1 period, a grows by a factor of r, so you have ar. After a second period, ar grows by a factor of r again, so you have . After a third period,  grows by a factor of r again, and you get . Et cetera. I guess if this is going to start appearing on the new SAT, I should make a whole post about it. 🙂

Anyway, in this case, the growth period is 12 years, but we’re given n in years. That’s why we need our exponent to be . That way, in 24 years, when there have only been 2 growth periods, the exponent  will equal 2, as it should.