The SAT Prep Black Book(123)

This is also a great example of an SAT Math question for which no formula exists. There’s no way you could use a calculator on this question; there’s not even any real way you could use basic arithmetic. This question is literally about braiding, but untrained test-takers in advanced calculus classes miss it frequently in practice.

Page 852, Question 19

Like the previous question, this one ends up involving no actual math. All we need to do is think carefully about the definition and properties of the term “median.”

If you double the value of each number or increase each number by 10, you must end up changing the median number (because you either double it or increase it by 10, as the case may be).

In other words, if the 11 numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, then doubling them all makes the median 12 and adding ten makes it 16. The median is the value in the middle--that's the definition of the term. Changing the number that's in the middle of the range of numbers changes the median.

(C) or (D) might also end up affecting the median, if we increase the smallest number by such an amount that it becomes greater than the original median, or if we decrease the largest number by such an amount that it becomes less than the original median.

Only (E) has no possibility of changing the median. Increasing the largest number still causes it to remain the largest number, so the number in the middle of the series can’t change.

Page 861, Question 16

Many students try to use some kind of formula here, but that’s a mistake, of course. For one thing, the SAT can't ask us to find the area of a trapezoid because it doesn't give that formula in the beginning of the math section or in the question itself. So there must be some other way to solve the problem that is easier and more direct.

If you draw out the rectangle and the trapezoid, we'll see that the trapezoid takes up 3/4 of the area of the rectangle, like so (I dashed in some other lines so you can see that ABED takes up 3/4 of ABCD more clearly):



So that means that we can do this:

ABED = 3/4(ABCD)

2/3 = 3/4(ABCD)

8/9 = ABCD

This means that the answer is (C).

Notice that one of the wrong answers is 3/4. This is a good sign for us, because it indicates that we were probably on the right track when we figured out that ABED was 3/4 of ABCD.





Page 888, Question 8


This question is often frustrating for untrained test-takers because it seems at first as though it will be impossible to figure out the product of the slopes if we don’t know the slopes themselves, and the question doesn’t tell us the slopes.

But if we remember the definition of the term “rectangle” and the properties of perpendicular lines in a coordinate plane, then we can realize that the slopes of lines that are perpendicular to one another are opposite reciprocals (like, for instance, 2 and -1/2), so they'll have to multiply together to give us -1.

There are two pairs of perpendicular sides, in a rectangle, so the final product of all of these slopes is 1, because -1 * -1 is 1. (For instance, if the slopes were 2, -1/2, 2, -1/2, they would multiply together to make 1.) So (D) is the right answer.

So, in the final analysis, this question really just involved knowing that rectangles have 2 pairs of perpendicular sides, knowing that perpendicular lines have slopes that are opposite reciprocals, and knowing that -1 * -1 is 1. Each of those facts is pretty basic on its own, but most people who look at this question never realize that those facts are involved in answering it. This is why it’s so important to develop the instincts for taking these questions apart by reading carefully, thinking about definitions and properties, and ignoring formulas and calculators for the most part.





Page 906, Question 12


This question basically hangs on the definition of the term “directly proportional.” We need to know that when n is directly proportional to q, n = kq, where k is some unknown proportional constant that we usually need to figure out.

So, in this case, if y is directly proportional to x2, then y = k(x2).

So let’s substitute and see what happens:

1/8 = k(1/2)2 (plug in the given values for x and y)

1/8 = k(1/4) (simplify on the right)

4(1/8) = k (isolate k)

1/2 = k (simplify)

Now that we know what k equals, we can plug it in with a y value of 9/2, to find our new x:

9/2 = 1/2(x)2 (plug in values for y and k)

9 = x2 (combine like terms)

3 = x (isolate x)

So the answer is (D). Notice that our other answer choices include 9, which is what we’d get if we accidentally solved for x2 instead of x, which is a mistake people often make on this question.

It’s not common for the SAT to test a math concept as obviously as it’s testing direct proportionality in this question—the only wrinkle in this question is the idea of being proportional to x2 instead of x. I think this particular question is so direct because direct proportionality is a concept that many algebra teachers no longer cover for some reason. (This is just a theory on my part, but I think it matches the evidence. Math concepts that are more commonly encountered on the SAT aren’t usually tested this directly.)