SAT Prep Black Book: The Most Effective SAT Strategies Ever Published (39 page)

If the probability of choosing red is 3 times that of choosing blue, that means that there are 3 times as many red beads as there are blue beads. Since there are 12 red beads, then, there must be 4 blue beads. Further, the number of glass beads altogether is 4 times the number of wooden ones, so if there are 16 total glass beads (12 red and 4 blue), then there are 4 wooden ones. Adding that all up, we get that there are 20 beads.

That’s the more mathematical way to approach this.

But an easier way to think about this might be to realize that there are more red beads than anything else, and there are only 12 of those, so 45 is already way too big of a number, and anything bigger than 45 is obviously also way too big. That means the only answer choice that can possibly work is (A).

Remember to pay attention to details and answer choices!

This is
another SAT Math question that a lot of people struggle with, even though it only involves one of the simplest ideas in all of geometry: the idea that there is an infinite number of points in a circle (or in any geometric figure).

I think the easiest way to approach
this is to say that every single point on the circumference of the circle could be a point that served as the corner of a rectangle like the rectangles in the original diagram. So there is an infinite number of rectangles with perimeter 12 that can be inscribed in the circle. Since infinity is bigger than 4, we know that the answer is (E).

For this question, once more, I would just read the question carefully and think about what it's describing.

20% of Tom's money was his spend on the hotel. He spent $240 overall, so he spent
$240 * 0.2 = $48 on the hotel.

If he only paid for 1/4 of the hotel, then the hotel cost $48
* 4, or $192.

The fact that the last three answer choices all differ from one another by $48 should alert us to the fact that we need to be really careful here, because there are ways to misread or miscalculate and end up on
either wrong answer. The far most common mistake is to misread the thing about sharing with 3
other
people, and treat it like it just says the room was split among 3 people.

This is one more situation in which paying attention to the relationships among the answer choices can alert us to mistakes that the College Board wants us t
o make, and can reassure us that (D) is the correct answer.

Like many other SAT Math questions, this is one that manages to be fairly challenging even though it only involves basic arithmetic. It’s also a question that will require very careful reading, and a question for which there is no ready-made formula. In other words, it’s a typical SAT Math question.

So let’s just think about what the question is describing, and how we might figure out what it’s asking us.

One approach would be to try to make a square board that would have a number of border tiles somewhat near each answer choice, and see if only one answer choice can be made to work like that. This will be a little tedious, and probably extremely time-consuming, but it will work if we do it right:

(A) doesn't work--if there are 10 on the boundary, it couldn't be 3 x 3 or 4 x 4.

(B
) doesn't work either. 7 x 7 would give you 24.

(C
) doesn't work because 9 x 9 would give you 32.

(D
) doesn't work because 11 x 11 would give you 40.

(E
) works because if the board is 14 x 14 there are 52 tiles on the border.

Another approach, possibly slightly faster, is to try drawing out a few small game boards to see if we can figure out some kind of pattern that would help us eliminate all the wrong answer choices.

If, for instance,
n
= 2, then the square would look like this:

In this case, the number of thi
ngs on the boundary would be 4.
If
n
is 3, then the board looks like this:

In that case, the
k
number is 8.

And so on. N
ow we need to try and understand what's going on here. Basically,
k
must always end up being a multiple of 4. So we need an answer choice that's a multiple of 4. Only (E) is.

There are other valid approaches here as well, but I think those two will be the ones that most people find.

This question presents me with another opportunity to remind you of what’s important to take away from these discussions. The goal of going over this solution is not to teach you a formal way to approach questions that ask about the border tiles on square game boards, because there will never be another SAT Math question that asks about the border tiles on square game boards.

Instead, the goal of going over this question is to deepen your understanding of the principles of SAT Math in general. There’s no way to predict exactly what kinds of things you’ll see on test day, but if we com
e to understand the importance of reading carefully, thinking about the answer choices as part of the question, and so on, then you’ll be able to take apart whatever weird combinations of basic facts the SAT presents you with. So it’s not about memorizing rigid steps for certain types of questions. It’s about developing a general feel, and confidence that you can work out whatever they throw at you by relying on the test’s design principles.

Just about everyone panics for a second when they run into this question, because it seems to be asking us to figure out the area of a shape we’ve never seen before. This is one of those moments when knowing the unwritten rules of the test really comes in handy.
Remember that the College Board can only ask you to find the areas of rectangles, triangles, and circles, and it gives you the formulas for those areas at the beginning of each SAT Math section.

So i
f a question looks like it's asking for the area of something else without giving you another area formula to use, then that something else can always be expressed in terms of rectangles, triangles, and circles. Always.

In this case
, it’s probably pretty clear that we can’t use triangles and rectangles, because there are no corners in this figure. So we’ll have to figure out the area of these figures as though they were circles.
But how can we do that?

I think the easiest way is to
imagine reversing the bottom half of the big circle from left to right, so that the bottom half of the circle becomes just a reflection of the top half, and we're left with 3 circles, all tangent to each other on their left-most points. It would look something like this:

So now we have to find the areas of those circles, and add and subtract them appropriately. If AD is 6, then each of those dots is 1 unit from the dot before it or after it. That means the radius of the smallest circle is 1, so its area is pi. The radius of the biggest circle is 3 units, so its area is 9pi. The radius of the medium circle, the unshaded one,
is 2 units, so its area is 4pi.

So we want the amount equal to the area of the biggest circle minus the area of the medium circle plus the area of
the smallest. So it's 9pi – 4pi + 1pi, or 6pi. Which means (C) is correct.

This is just one more example of a question that seems a lot more exotic than it is. If you remember the rules the College Board has to play by, then you’ll find a lot of things much easier than most people will. An untrained test-taker will throw up his hands in frustration over this question, but a  trained test-taker knows how to turn it into a simple question about circles, and then solve it using the basic formula provided in the beginning of each SAT Math section.

For this question, I would just draw 6
points out so that no 3 are on a line together, and then try connecting them: basically, each of the 6 points connects to the other 5. We can count the lines up after drawing them out, or we can try doing a little multiplication.

If we multiply, it might seem
like there should be 30 lines, since 6 * 5 = 30, but we have to remember that each line touches two of the points. So there aren’t actually 30 lines, because each line counts as a connection for both of the points. In other words, if we call the points A, B, C, D, E, and F, then the line from A to B is the same line as the one from B to A.

So we want to divide the 30
apparent connections by 2 in order to compensate for the fact that each line serves as one connection between 2 points, so we don’t double-count the lines. That gives us 15 for our final answer, so (A) is correct.

Note the patterns in the answer choices: 15 is half
of 30, and there's also 36 and 18 (36 would be 6 x 6 instead of 6 x 5, and 18 is half of that). Once more, the answer choices help point us in the right direction and make us aware of potential mistakes that could be easily made.

This question often blows people away because it seems much more complicated than it actually is—which, as we’ve said many times, is typical for SAT Math questions in general. As usual, we’ll approach this by reading carefully and thinking carefully.

First, it’s important to realize that the question tells us that
a
and
b
are equal, so f(
a
) and f(
b
) must also be equal.

We also want to remember that questions with roman numerals for answer choices are often based on abstract properties.

Finally, we want to remember that we don’t need to know what the actual function is. All we need to know is that it has the property that f(
x
+
y
) = f(
x
) + f(
y
).

So
now let’s try to reframe each Roman numeral in terms of that property, and see if we can do it:

F
or I, 2f(
a
) = f(
a
) + f(
a
) = f(
a
+
a
) = f(
a
+
b
). We can substitute
b
for
a
at the end to arrive at f(
a
+
b
).