The SAT Prep Black Book(111)

(8 + k)x = 5kx (deal with the k terms on both sides first)

(8 + k) = 5k (divide both sides by x)

8 + k = 5k (simplify)

8 = 4k (combine like terms)

2 = k (isolate k)

and, therefore:

m = 8k and k = 2

m = 8(2) (substitute 2 for k)

m = 16 (simplify)

So (B) is correct.

That’s the algebraic approach. It’s kind of ugly, at least by SAT standards, but it works.

There’s another approach we can use that will probably be a little easier on our brains, though it might not be any faster. That would be to look at each answer choice and try to solve backwards; only one of the choices will make this possible.

For instance, if we want to test out choice (A), we would see what has to happen if m is 8. Since m is the product of 8 and k, according to FOIL, then we know that k would have to be 1 if we made m be 8. But if we FOIL out the expression on the left with k equal to 1, the rest of the expression doesn’t end up matching with the rest of the expression on the right.

Then we could try with (B), (C), and so on. We would see that the process can be made to work when m equals 16, but not for any other choice.

Another way to go, which is probably faster, but which requires a bit more awareness of algebra, is to notice the relationship between m and 8k, and the relationship between 8x, kx and 5k. Basically, we can see that k must be a value such that 8+k and 5k are equal, which means k must be 2, as we figured out before. And that would mean that m has to be 16.

Again, this is one of the ugliest questions in the book, but we can work out the answer if we stick to the fundamentals of algebra, and if we’re willing to play around with an unfamiliar presentation of those fundamentals.

Depending on the approach you take, this question might eat up a little (or a lot) more time than the average SAT Math question. While we should always try to find the most efficient solutions during practice sessions, remember that it’s normal for some questions to take longer than others on test day. Questions like this one are a big part of the reason why it’s so important to try to save time on questions that seem easier to you. That way, you have enough time left to play around with the harder questions, or to go back over your work and check your answers.





Page 530, Question 18


This question is a little complicated for an SAT Math question, but, as trained test-takers, we expect that we might see a little more complication in the last question or two of the grid-in section. As always, we’ll pay close attention to what the question is asking and see what we can figure out.

This question asks about the value of a, and the only place in the whole question where we can find any reference to a is in the expression y = ax2. So that means we’re going to need values for y and x that we can plug in to that expression, so we can solve for a.

Apart from (0,0), which won’t help us figure out a, there are only two points on the graph that have (x,y) coordinates we can probably figure out: points Q and R, which are also in square PQRS.

We’re told that PQRS has an area of 64. We know this is an important dimension to be aware of because it was included in the text of the question after being omitted from the diagram itself. If the area is 64, that means each of the square’s sides is 8 units long. And that, in turn, means that point R is located at (4,8), because OS is half of PS.

Now we have values for x and y that we can plug in for point R, which is on the graph of y = ax2:

y = ax2 (given equation for the graph)

8 = a(42) (substitute x and y values for the point (4,8))

8 = 16a (simplify expression on the right)

8/16 = a (divide both sides by 16 to isolate a)

1/2 = a (simplify expression on the left)

Notice that all of the ideas in this question are relatively simple ideas on their own; the trick was to trace the proper approach back through the wording of the question to figure out how to string together the relevant ideas.

Notice, also, that this question is actually very similar to question 18 on page 717 of the College Board’s Blue Book. (The two questions aren’t identical, but the only real difference between them is that ABCD isn’t a square in that other question, and the function in this question isn’t a parabola.) Do not be misled by this coincidence into assuming that the College Board frequently repeats a limited number of question types in each SAT Math section. On the contrary, in the entire Blue Book these two questions are among the only examples of question material being repeated so closely on the SAT Math section.





Page 547, Question 13


When you have a question like this where the answer choices are all relatively small numbers, that's a sign from the test that counting the options is the best way to go. On the other hand, if the number of options was much higher, that would be a sign that there's a solution where you don't actually have to count things.