Page 717, Question 17
A lot of people get nervous when they see this question, because they’ve never encountered a shape like the one in the diagram. But as trained test-takers, we have to remember that we’re always going to see new stuff on the SAT, and our job is to figure out how to apply basic math knowledge in these new situations.
In this case, as in many other questions, it’s just a matter of reading carefully. The only line segments that will be on edges are the ones to the vertex directly to the left of V, directly to the right of V, and directly underneath V. That means that the lines to the other 8 vertices aren't on the edges of the figure.
Like many of the SAT Math questions that most test-takers struggle with, this question involves no formula and no chance to use a calculator. It’s just a matter of reading carefully and then subtracting 3 from 11 and arriving at 8 as the answer.
Page 717, Question 18
This question challenges many test-takers, usually because the drawing looks so complicated. But if we think carefully and work through the question, we’ll find that it only relies on basic math at its core, just like all SAT Math questions.
The question asks us to find p. Since the only mention of p in the whole question is in the equation y = px3, that’s pretty much where we’ll have to start.
The only way we can get p out of y = px3 is to know the values for y and x. So that means we need to be able to identify the x and y values for some point on the graph of the function.
Apart from (0,0), which would be useless for us here, the only clear points that the function passes through are the ones labeled A and C.
Since C has x and y values that are positive, let’s focus there. We know that C is at (1/2,c). So now we need to figure out the dimensions of triangle ABCD.
Since we know the area of ABCD is 4, and the diagram shows its width is 1 unit based on the x-coordinates, then we know that ABCD has a height of 4.
This means that point C is at (1/2 , 2). Now we just plug those x and y values into y = px3:
2 = p(1/2)3 (plug in 1/2 for x and 2 for y)
2 = p(1/8) (simplify on the right)
16 = p (isolate p)
This question was a bit more involved than many SAT Math questions. We know to expect that some time near the end of the grid-ins. But the solution still only involved a bizarre combination of basic facts—something else we’ve come to expect from SAT Math questions in general.
The real lesson to take away from this question is the idea of tracing the solution back through the concepts in the question. Here, they asked about p, so we looked in the question for a statement that was relevant to p, then we thought about things that were relevant to that statement, and so on. This general approach will come in handy on other SAT Math questions, even though this particular question will never appear on a real test in the future.
Notice, also, that this question is actually very similar to question 18 on page 530 of the College Board’s Blue Book. I’ll repeat here what I said earlier in the explanation for that question: 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 one of the only examples of question material being repeated so closely on the SAT Math section.
Page 732, Question 14
Many students get nervous over this question because they don’t like things that deal with ranges and inequalities, but it’s important to remember that we don’t have to do these questions in ways that we could defend to a math teacher. We just need to find the answer any way we can.
I would start by multiplying the smallest and greatest values of x and y together, and then seeing how those things relate back to the answer choices. The smallest and greatest values of x are 0 and 8, while the smallest and greatest values of y are -1 and 3. So:
0 * -1 = 0
0 * 3 = 0
8 * -1 = -8
8 * 3 = 24
So our range of possible values for xy at least needs to include -8 and 24.
That means (E) is the only choice that works.
If we look at the answer choices, it’s a good sign that many of the choices end in 24. It’s not a great sign that only one choice ends in -8, but we could easily pause to double-check the validity of -8 by realizing that 8 * -1 is, indeed, -8.
In fact, from a certain way of looking at it, once we’re positive -8 is a valid option we can actually stop doing any further work, because only one choice includes it.
(Now, if you’re a math teacher or a strong math student you might be shaking your head at my approach here, which is quite informal. In fact, you might shake your head at a lot of what we do in the Black Book because of its informality. But my goal isn’t to show my students how to attack SAT Math questions formally, because formal solutions to SAT Math questions are typically slower and more difficult than the simple, direct approaches I want my students to follow. I’m not teaching math in this book; I’m teaching how to answer SAT Math questions. The distinction is important.)