At any rate, if you saw this question and realized that you had forgotten (or never been taught) the idea of direct proportionality, it would be a good idea to review it. The odds of it coming up on test day for you are relatively slim, but it’s such a simple concept that it can’t hurt to take a few moments to memorize it.
Page 907, Question 18
For this question, as for most SAT Math questions, we simply need to read carefully and look for ways to apply basic math ideas.
If there are 18 arcs of each length in the circle, and if the circle has a circumference of 45 units, then we can write this:
18(2 + b) = 45
That means that 36 + 18b = 45, which means that 18b = 9, so b = 1/2 unit.
If b is 1/2 unit and there are 45 units in the circumference of the circle, then 45 units corresponds to 360 degrees of arc. This means that each of the 45 units in the circle's circumference accounts for 45/360 degrees, or 8 degrees; since b is 1/2 of a unit, b must be 4 degrees, and the answer is (A).
Notice, as is often the case, that the answer choices represent mistakes that can be easily made. 16 degrees is the arc measurement of the 2-unit arcs, and 20 degrees is the arc measurement of a (2 + b) combination.
Page 908, Question 20
This question, like many questions near the end of an SAT Math section, is actually much easier than it might look. It’s also an especially clear example of a few of the SAT Math rules and patterns from this book, all wrapped into one test item.
One of the first things I’d want to notice is that the diagram isn’t drawn to scale. That means I should think about re-drawing it to scale. To figure out how to do that, I have to look at the text of the question, which mentions that each of the five line segments are congruent (which is just a fancy way to say they’re all the same length). That means that this diagram should show two equilateral triangles that share a side, like this:
The text of the question also includes another piece of information that was left out of the original diagram, which is the existence of line segment AC. So I’ll add that in:
Another fundamental thing that we should pay attention to is the set of answer choices. Notice that they all involve either √3 or √2 (choice (E) actually involves both).
We should ask ourselves what kinds of math concepts are related to the concepts we’ve encountered so far. We know the diagram involves two equilateral triangles, and we know equilateral triangles are 60o-60o-60o triangles. We also know that the answer choices involve √3 and √2; √3 is related to 30o-60o-90o triangles and √2 is related to 45o-45o-90o triangles.
At this point, we’d want to realize that line AC cuts the two 60o-60o-60o triangles in half, creating 4 new triangles, and each of these new triangles is a 30o-60o-90o triangle.
We know that the ratio of the legs of a 30o-60o-90o triangle is 1: √3, with √3 representing the long leg. (This is something we’d want to have memorized, but even if you forget it momentarily you can always look at the given information at the beginning of any SAT Math section).
The question asks for the ratio of AC to BD. AC includes two long legs of 30o-60o-90o triangles, while BD includes two short legs of 30o-60o-90o triangles. That means the ratio is 2√3:2, which is the same thing as √3:1, which is what (B) says, so (B) is correct.
Surveying all the answer choices, we’ll see that most of them start with √3, which is a good sign that starting with √3 is probably the correct option. But more of them end with something related to 2, not 1, so I would double-check myself to make sure that I hadn’t made a mistake in simplifying the ratio.
Page 919, Question 15
Most untrained test-takers will see this question and assume that they need to figure out the actual measurements of each angle in the answer choices, because that kind of solution might seem similar to something we would do in a trigonometry class. But, as trained test-takers, we’d want to realize that it’s possible to figure out which angle is smallest without figuring out their actual measurements—and, anyway, trig isn’t allowed on the SAT, so there has to be another way.
Since the two endpoints of each angle are the same (always X and Y), the farther away the vertex is, the smaller the angle will end up being. Since D is the farthest point from X and Y, XDY is the smallest angle, even if we never figure out its actual measurement. So (D) is the correct answer.
Page 919, Question 16
Most untrained test-takers will try to identify the actual values of x and y, because that’s what we would typically have to do in school if we saw a math question with multiple variables. But on the SAT, we’ll frequently be asked questions in which the variables don’t need to be identified, and this is one of those questions. (One of the major clues that we don’t need to identify the variables is that we weren’t asked to find either variable’s individual value; we were only asked to find the value of a complex expression that involves both variables.)