The SAT Prep Black Book(102)

(If that second approach didn’t make any sense, don’t worry about it. It was just another way to go, that’s all. The algebra works fine too, of course.)





Page 418, Question 16


This is a question that many test-takers don’t even understand when they first read it, because it sounds quite odd—usually we don’t describe a “four-digit integer” with a string of capital letters like WXYZ. But in these cases we should always trust that the College Board will explain what it means—it has to, or else the question can’t be asked. So let’s keep reading.

The question says that each capital letter represents a digit in the number, and then it tells us some things about the relationships of the values of those digits. Our job is to figure out what the values are.

Many people will try to approach this question like a system of equations. I suppose that might be possible with a lot of effort, but it won’t come easily to anyone.

Another popular (but largely unsuccessful) approach is to try picking numbers for each digit and play around with it until you hit on a working arrangement.

Let’s try something else, though. Let’s just look at what’s going on and try to think about what it says. And, above all, let’s remember not to panic just because this is a weird question. By now we know that the SAT likes to try to scare us by asking questions that look weird. No big deal.

One thing that I’d notice is that X is clearly the biggest number, because it’s the sum of all the other numbers. Okay, that might come in handy.

What’s a little less obvious, but still clear, is that Z is the smallest number. We know this because rule 2 tells us that Y is 1 less than W, but rule 3 tells us Z is 5 less than W.

So, since Z has to be the lowest digit out of these 4 digits, why not see what happens if we make it equal zero, which is the smallest digit possible? If we try setting Z equal to zero and it works, great! If it doesn’t work, we’ll probably figure out some more information about the question as we try to use 0 for Z. So let’s give it a shot and see what happens.

If Z is 0, then rule 3 tells us that W is 5.

If W is 5, then rule 2 tells us Y is 4.

Plugging all of that into rule 1, we would get that X equals 5 + 4 + 0, or 9. That makes sense, since we said before that X must be the biggest digit.

So the answer will be that WXYZ corresponds to 5940.

You might wonder how I knew to start with the idea of Z being 0. But I didn’t know beforehand that 0 would work; it was just an informed hunch. If I had started with Z as 2, I would have seen my mistake, adjusted, and tried again. And I didn’t even have to start with a value for Z. I might have started with W being 8 or something. In any case, the important thing isn’t to try to nail the question on the first guess; the important thing is to be willing to play around with the results until you get something to work.

And let me make something else extremely clear about this question: the lesson to be learned here is not how to approach future SAT Math questions about mysterious 4-digit numbers whose digits have particular relationships to one another. The chance of you ever seeing a question like this on the test again is basically zero. The much more important thing to try to pick up on is the general thought process—the way we read carefully, think about what the words mean, avoid using formal solutions whenever possible, and just kind of experiment with the question until we find an answer. That underlying skill set is critical on the SAT.





Page 418, Question 17


This question, like so many others, manages to be challenging even though it only involves a simple idea—in this case, the idea of equilateral triangles. As will often be the case on the SAT, there is no way to apply a formula or use a calculator. Instead, we just have to think about it.

One thing that we should note right away is that the diagram is not drawn to scale. We’ll want to focus on the part that doesn’t seem to be to scale and see if we can figure out what’s going on there.

Another important feature of this question is that it gives us information about the dimensions of the diagram in the text, but that information isn’t labeled on the diagram itself. This means that the information about those dimensions will be the key to solving the problem.

So let’s see what happens when we put all of that together. If CD is 10, and DE is also 10, and EF is also 10, then we know that each of the overlapping triangles has side lengths of 20. In other words, before they were overlapping, the two triangles were both 20 units on every side. And if DE is 10 units, then all the sides of the small triangle are also 10 units, because the two big triangles are equilateral. So a more accurate representation of the situation would be something like this: