The SAT Prep Black Book(104)
A Selection of Challenging Questions
At this point we’ve taken a look at a variety of Math questions, but you’re probably interested in seeing more. I understand where you’re coming from with that, and I’m about to show you some more Math solutions, but . . . before I do, I want to make sure I reiterate what you should be learning from each solution.
I am NOT doing these solutions in an effort to show you something formal that you should memorize and then expect to repeat on future SAT Math questions, as we would normally do for a math class in school. Remember that the SAT Math section will show you very strange combinations of very basic ideas, and specific questions and questions ‘types’ are not repeated for the most part. The odds are that when you take the SAT for real, you’ll see a whole test full of math questions that don’t look like any other SAT Math questions you’ve ever seen before, at least not on the surface. And this will be the case no matter how many practice SAT Math questions you work on beforehand. Every test combines the same basic stuff in new ways.
You may have noticed I’m repeating this idea a lot. There’s a reason for that:
It’s very important, and most people ignore it.
So if you want to get better at SAT Math, the goal is to learn the underlying process, and to practice using it against real SAT Math questions written by the College Board. That’s why I’ve included these solutions: to give you an idea of the right general approach so you can continue to refine your instincts, not to give you cookie-cutter instructions for specific question types, because specific question types basically don’t exist on the SAT Math section, practically speaking.
With that important reminder out of the way, let’s take a look at some of the SAT Math questions that students have typically asked about.
As with other question explanations in this book, you’ll need a copy of the second edition of the College Board’s Official SAT Study Guide (otherwise known as the Blue Book) to follow along. Let’s get started.
Page 399, Question 11
As with many SAT Math questions, there are basically two ways to do this: we can pick a concrete number that satisfies the requirements for k and then look to see what happens to k + 2, or we can think in the abstract about properties relating to the concept of remainders. In general, abstract solutions will be faster to reach but harder for many students to execute, while concrete solutions will give most students added confidence but end up taking more time. For this particular question, though, the amount of time spent on each solution is likely to be roughly the same.
If we go the concrete route of picking a number to be k, we have to make sure that it satisfies the setup. In this case, a number like 13 would work, because 13 divided by 7 gives a remainder of 6. So then we'd look to see what happens when we divide 15 by 7, since 15 is 2 more than 13; the result is a remainder of 1.
The other way to do this is to think in the abstract: if k has a remainder of 6 when it's divided by 7, then k is 6 more than some multiple of 7, and k + 2 will be 8 more than that multiple of 7. We know that 8 is more than 7, so one more 7 will "fit in" when k + 2 is divided by 7, and we’ll be left with a remainder of 8 - 7, or 1.
Either way, (B) is the right answer.
Of course, you wouldn’t need to do both solutions on test day. I’m just doing both of them to show that there are a variety of ways to attack this question successfully, as will be the case for most SAT Math questions.
Page 400, Question 17
This is one of those questions that lets us work on identifying a whole variety of the patterns and rules we talk about in this book.
First, we’d want to notice that the diagram is drawn to scale. That means that we might be able to figure out the answer by eyeballing the diagram—or, at the very least, we want to make sure that our final answer makes sense in the context of the scale of the diagram.
Also notice that the diagram and the answer choices have a lot of expressions with √2 in them. We know that √2 relates to 45o-45o-90o triangles, which seems relevant to the question since ABC, ADB, and BDC are all 45o-45o-90o triangles.
There are a variety of ways we could try to figure out the area of the shaded region. The most straightforward way is probably to work out the distance of EF, along with the distance from F to the base of the figure, and then multiply those two things together, since they would represent the length and the height of the rectangle. We might do that by realizing that BE and BF are each 5√2 units long, and that they are each the legs of a 45o-45o-90o triangle with EF as its hypotenuse. Since the ratio of the sides of a 45o-45o-90o triangle is 1:1:√2, the distance of EF is 5√2(√2), which is 10.