SAT Prep Black Book: The Most Effective SAT Strategies Ever Published (35 page)

That means the correct answer here is pi/4. Again, pi is the area of the entire circle, and the shaded region is 1/4 of the circle, so its area is pi/4. And (A) is correct.

There are a lot of ways to mess this question up, and most of them will involve either thinking that the radius is 2 or accidentally thinking that the shaded portion is 1/2 of the circle (this can happen if a person tries to work out the area mathematically and makes a mistake in the process, instead of just looking at the picture). Notice that both mistakes are reflected in the wrong answer choices. In fact, (E) is what you’d get if you made both mistakes together.

This is a question I would definitely double-check, or even triple-check, for several reasons. The first reason is that the answer choice patterns aren’t really pointing to (A) being right, even though it is. The patterns are really suggesting that (B) would be right (it’s in the middle of a geometric series with (A) and (C) and (E), and there are more 2’s in the answer choices than 4’s). Also, even apart from the answer choices, this is a question where it would be very easy to make a simple mistake and be off by a factor of 2 or 4, and the answer choices are clearly waiting for that.

This is a great example of the kind of question that causes the most trouble for the most test-takers. It’s something the vast, vast majority of test-takers have the skills and knowledge to answer correctly, but it’s also something where a simple mistake or two can easily be made, wrecking the question. If you want to improve your SAT Math score, this kind of question is where you should probably focus your energy first. Most people would significantly increase their scores if they just stopped giving away points on questions like this, rather than focusing on questions that seem harder.

This question asks about perpendicular lines. We need to know that when two lines are perpendicular to one another, their slopes are opposite reciprocals (this was an idea discussed in the toolbox earlier in this book). This idea is crucial to solving this question (at least when it comes to the way most people will solve it), but notice that the idea is a
property
of perpendicular lines, not a
formula
. Remember that challenging math questions on the SAT rely much more heavily on properties and definitions than on formulas.

To solve this question in the most straightforward way, it would probably be best to get the expression for the original line into slope-intercept format (the one that looks like
y
=
mx
+
b
). If we do that, we get this:

x
+ 3
y
= 12

3
y
= -
x
+ 12              (get the
x
on the right-hand side)

y
= -
x
/3 + 4              (isolate
y
)

So the slope of the given line is -
1/3.

A line perpendicular to this must have a slope that is the opposite reciprocal of -1/3, which is 3. So (C) must be correct
, because it gives a coefficient of 3 for
x
in the
y
=
mx
+
b
format, where
m
is the slope.

Note that we have the usual kinds of wrong answers we’d expect on an SAT Math question.  The slope of (B) is the opposite of the correct slope, and we can probably imagine how a person might make that mistake. The slope of (D) has the right sign (it’s positive) but uses 1/3 instead of 3, and, again, that’s an understandable mistake.

Notice, also, that there’s no other slope in the answer choices that has both its reciprocal and its opposite in the choices. This is a manifestation of the imitation pattern, and it’s a good sign that (C) is, indeed, correct.

A less straightforward way to answer this, but one that wouldn’t require knowing that perpendicular slopes are opposite reciprocals, would be to use your calculator. You could graph the original line on your calculator, and then graph each answer choice. This would let you see on your own which choice generated a perpendicular line. Of course, this would be a little time-consuming, and you’d have to make sure you didn’t key in the graphs wrong. But it could be a viable way to solve the question if you ha
d forgotten perpendicular lines have slopes that are negative reciprocals of each other.

Finally, note that the
y
-intercepts don’t matter here. The question asks about perpendicular lines, and that idea only requires us to consider the slopes, not the intercepts. This question provides yet another example of a situation in which we must consider the answer choices as part of the question, rather than hoping to predict the correct answer choice completely from scratch—in this case, there are an infinite number of valid
y
-intercepts that could be part of a line perpendicular to the given line.

This question bothers a lot of people—but, as we often see on the SAT Math section, it really just comes down to basic properties and definitions. In this case, we need to know what a triangle is. (In case you’ve forgotten, it’s a closed three-sided figure.)

If we go through each choice and imagine trying to draw it, we’d see that they can all be drawn except for (E)—if you try to draw a triangle with two sides of 5 units and a third side of 10 units, you end up with just a line segment of 10 units, because in order to reach the endpoints of the long ‘side,’ the two short ‘sides’ have to open up completely to make a straight line.

We could also think of this in terms of the so-called “triangle inequality,” which says that the length of any side of a triangle must be less than the sum of the other two sides. But I deliberately wanted to talk about it without referring to that idea because, as I keep saying, it’s very important to learn to attack SAT Math in an informal way.

There’s another important element here, too. Notice that the question uses the word “EXCEPT.” I can’t tell you how many times people have missed questions like this because they overlooked that word, and ended up just choosing the first answer choice that would work under normal circumstances. This is one more example of how important it is to read everything on the SAT very, very carefully. It’s also a good example of the importance of checking over the other answer choices before you move on—if you overlook the word “EXCEPT” and choose (A) because it’s the first thing you see that works to make a triangle, you can hopefully catch your mistake if you glance through the other choices and notice that (B), (C), and (D) also work.

There are a ton of things we can learn from this question, especially. Please,
please
read this whole description and pay attention!

First, this question is rated 5 out of 5 for difficulty by the College Board. In other words, this is a question that a ton of people miss. But, as we’ll see in a moment, it only involves basic arithmetic. In fact, it involves a specific application of an idea from arithmetic that has possibly never, ever come up in a normal math class before. What it does NOT involve is any kind of formula, or any way that a calculator is likely to be helpful. Remember, this is SAT Math, and it’s likely to involve tricks and misdirection more than formulas.

I’d also like to point out, before we even get started, that this is a question in which several of our reliable answer-choice patterns would have allowed us to predict the correct answer without even seeing the question, just from the answer choices alone. I am NOT saying that you should ever try to answer a question just from the answer choices—I’m just saying that the answer choices are hinting very strongly at (C) being the right answer, if we know how to read them. The fact that so many people missed this question shows us that most people aren’t paying any attention to the answer choices, which is part of the reason why most people have a really hard time on the SAT Math section.

This question, then, is a positively classic example of the way most test-takers throw points away for no reason
on the SAT Math section. People don’t miss this question because they don’t know basic arithmetic. They miss it because they don’t pay attention to details and they don’t check over the other answer choices and think about SAT patterns.

In other words, they miss this question because of a lack of SAT-specific skills. Don’t be like them.

Okay, enough preamble. Now let’s actually answer the question.

The question asks what percent of the votes were cast for Candidate 1 given that Candidate 1 received 28,000 more votes. Most people will try to solve this by figuring out that 28,000 is 1% of 2.8 million. So far, so good. Then they’ll assume that this must mean Candidate 1 got 51% of the vote. They’ll choose (D), and they’ll move on to the next question without giving this a second tho
ught.

And they’ll be wrong.

Here’s the mistake: if Candidate 1 pulls 51% of the vote, than Candidate 2 must pull 49%, and
51% is 2 percentage points more than 49%, not 1 percentage point more
. So in order for Candidate 1 to have a 1 percentage-point margin over Candidate 2, the correct split isn’t 51 – 49.

It’s 50.5 to 49.5.

That’s why (C) is correct.

Notice that there are 3 answer choices that end in a 5, and only 2 that end in a 1. This goes along with the imitation pattern, and it strongly suggests that the correct answer should end in a 5. Also, notice that 3 out of the 5 choices involve decimals, which also suggests that 51% isn’t correct. We could even say that 50.5% is the middle entry in a series of sorts, where the other numbers are 50.05 and 55—it’s not a traditional series in the mathematical sense, but there’s a clear progression with reference to the decimal places, and (C), the right answer, is in the middle of that progression.

This is yet another great example, then, of the tremendous importance of thinking about the answer choices as part of the question. They’re not just there to take up space—the College Board uses them deliberately, in ways that we can exploit.

Sometimes people make this question a lot harder than it needs to be, by coming up with a decimal approximation for
√18 or by converting it to 3√2. But it’s much easier just to start by squaring both sides, so we get 2
p
= 18. That means
p
= 9.

Remember that SAT Math gets a lot easier if we look for ways to keep it simple.

This question, like many SAT Math questions, is really just a matter of knowing definitions and doing basic arithmetic.

When we round the given number to the nearest whole number, we get 2. When we round it to the nearest tenth, we get 1.8. Since 2 – 1.8 is 0.2, the answer is just 0.2.

Notice that this question is pretty impossible to answer if we don’t understand the concept of rounding, or if we don’t know which decimal place represents tenths.
Also notice that the math is incredibly basic, but people will still miss this question because of its strange presentation.

As is often the case, there are many ways to answer this question. We could do it with pure algebra, letting
x
be the number of towels and writing 6 = 2
x
/5. But that would be a little more formal than I like to be on the SAT Math section.

So let’s just think through it instead. If 6 towels represents 2/5 of all the towels, than 3 towels must represent 1/5 (because 3 is half of 6 and 1/5 is half of 2/5). So if 3 towels are 1/5 of the towels, then there must be 15 towels, because 3*5 is 15.
So the answer is 15.

Just to be clear, there’s nothing inherently wrong with doing it algebraically, as long as you feel comfortable with that and you set it up correctly and don’t make any mistakes. I’m just more comfortable with mental math, and I find that it tends to work a lot better on the SAT, so I encourage it in my students as much as possible.

This question is very difficult for a lot of test-takers. But if we just remember to read carefully and pay attention to details, we can figure this out.

The question describes 5 points on a line, like this: A B C D E.

We know that AD is 4.5, and BE is 3.5. Why not try to plot that out?

When we do try to plot it, we realize that we don’t have enough information yet.  If AD is 4.5 and BE is 3.5, there’s an infinite number of ways we could draw that. For instance, it might look like this: