The SAT Prep Black Book(120)

Page 770, Question 8


Many test-takers miss this question, even though it involves nothing more than careful reading and basic arithmetic.

The question says that the new average is equal to the median number of siblings per student. That means we need to figure out what the median number of siblings is, and then figure out how to make the new average number of siblings equal to that median number.

So let’s start by figuring out the median number of siblings.

We see there are 3 students with 0 siblings, 6 students with 1 sibling each, 2 students with 2 siblings each, and 1 student with 3 siblings. So if we arrange the number of siblings for each student in a row from least to greatest, we’ll get this:

0,0,0,1,1,1,1,1,1,2,2,3

So the median number of siblings is 1, because the numbers in the middle of that series are both 1.

So when the new kid joins the class, the average number of siblings that everyone has is going to equal this median number of 1. When the new kid joins the class, there will be 13 total students in the class, which means the total number of siblings for all the kids in the class must also be 13 (because 13/13 is 1).

So now we add up the numbers of siblings of existing students, so we can see how far we are from a total of 13 students:

0 + 0 + 0 + 1 + 1 + 1 + 1 + 1 + 1 + 2 + 2 + 3 = 13

So actually, it turns out that the 12 students already in the class have a total of 13 siblings already, which means that the 13th student needs to have zero siblings in order for the average number of siblings in the class to be 1.

That means (A) is correct.

Now let’s review the math ideas that were necessary to answer this question. We had to know how to read a relatively simple table. We had to know what the words “median” and “average” meant. We had to arrange a lot of single-digit numbers from least to greatest. We had to figure out that a lot of single-digit numbers added up to 13. We had to know that 13/13 is 1. And we had to know that 13 + 0 is 13.

None of that would probably qualify as advanced mathematical knowledge at your school, but test-takers miss this question very, very often. We should ask ourselves why that is, so we can make sure we don’t make their mistakes.

In my experience, people who miss this question (or just give up on it in the first place) do so because they don’t understand what it’s asking when they read it. This is why I often say that critical reading skills are the most important thing on the entire SAT. And it’s because of the great number of questions like this—questions that present bizarre combinations of simple math ideas—that test-takers who try to prepare for the SAT by learning advanced math usually don’t see any improvement.





Page 773, Question 17


For this question, as for many abstract questions on the SAT Math section, there are two ways to go: we can come up with concrete values for the variables described in the question and run a test, or we can think about the question in the abstract. The concrete approach tends to take more time but will often feel more comfortable, while the abstract approach is faster but will be harder for many test-takers to manage.

In this explanation, I’ll start with a concrete example and then show the abstract one. You could use either on the test, of course, without bothering to think about the other.

Let’s assume that p is 3, r is 5, and s is 7, as those are three different prime numbers greater than 2, like the question asks us for.

In that case, n is 3 * 5 * 7, or 105.

Now let’s figure out what the factors of n are:

1, 3, 5, 7, 15, 21, 35, 105

There are 8 of them, so the answer to the question is 8.

Now, if we wanted to think about this in the abstract, we’d note that since n is the same as prs, and since p, r, and s don’t have any other factors besides 1 (since they’re prime), then the factors of prs will include every possible combination of p, r, and s, like this:

p, r, s, pr, ps, rs, prs

. . . plus, of course, 1.

That also makes for a total of 8 factors.

(Notice that the concrete example we did follows our abstract thinking perfectly if we substitute 3 for p, 5 for r, and 7 for s.)





Page 785, Question 2


People often miss this question, even though it’s fairly early in the section and doesn’t really involve any math. When they miss it, it’s usually as a result of not having read it carefully. Most people choose (A) because the figure in that choice has a line of symmetry, and they’re rushing through the question. They don’t notice that 3 other choices also have a line of symmetry, which might help them realize their mistake.

If we read carefully and realize we’re looking for a figure with 2 lines of symmetry, then we realize that (D) has one line of symmetry going vertically straight down its center and another line going horizontally straight through its center.