SAT Prep Black Book: The Most Effective SAT Strategies Ever Published (40 page)

For II, f(
a
) * f(
a
) has nothing to do with what the question told us about; the question only told us about an additive property of the functions, not a multiplicative one. So we can’t substitute or manipulate anything further here.

For III, f(2
a
) = f(
a
+
a
) = f(
a
) + f(
a
) = f(
b
) + f(
b
). So this one works too.

So we can see that (C) works because roman numerals I and III
are valid statements given that
f(
x
+
y
) = f(
x
) + f(
y
).

This is a question that won’t allow us to use a concrete example (unless you get
extremely
lucky in making up a function for
f
). The only practical way to approach it is to make substitutions and follow the rules of algebra to see which roman numerals contain valid equations.

For most test-takers, this will be one of the hardest questions. So this is a good time for me to remind you that your primary goal probably shouldn’t be trying to improve your performance on the occasional tough question like this; it’s much more important to make sure you lock down all the questions that seem easier first. Once you get to a point where you never make any ‘careless’ errors, you should feel free to start worrying about questions like these. But if you try to tackle these kinds of questions without first making sure that you’re executing correctly on the questions you can handle more comfortably, you’ll just be wasting your time, and your score won’t improve.

I’m not saying that a question like this can’t be answered, or that if follows different rules. This question can be solved with basic math and careful thinking, just like every other SAT Math question. I’m just saying that it’s important to focus on eliminating mistakes before you focus on figuring out questions that seem more challenging to you.

This question really helps drive home the importance of considering the answer choices along with the rest of the question.

When we look at the answer choices, we can see that they all involve
y
, with no
x
. That means we need a way to express
x
in terms of
y
. The only way to do that is to realize that
x
times
y
is 4000, since that's the area of the rectangle. So
x
is 4000/
y
.

Furthermore, t
he total length of rope needed is
y
+ 4
x
, because there are 4 vertical line segments with length
x
in the diagram.

We can re-write
y
+ 4
x
as
y
+ 4(4000/
y
), which is the same as
y
+ 16000/
y
. So (B) is correct.

When I look back over the other answer choices, I would definitely want to notice that more of the answer choices include 3
y
than just
y
in the denominator, and that would worry me for a second, because I know that elements of the correct answers tend to appear in wrong answers in questions like this. So I would double-check my work again.

Despite my general dislike of algebra for the purposes of SAT Math solutions, sometimes it can’t be avoided, and I think this question represents one of those times for most test-takers. This question is also one of those questions that can be expected to take most test-takers more time than usual. It’s because of questions like this that we have to try to work through other questions as quickly as possible without sacrificing accuracy.

For most people, the obvious first step will be to multiply out the expression on the left (using FOIL), which gives us this:

(
x
-8)(
x
-
k
) =
x
2
-
kx
- 8
x
+ 8
k

In terms of what we would normally get from
FOIL-ing out two binomials, this expression is a little odd, because it has two terms with
x
instead of one. (This is because the original binomials involved more than one unknown value—they had a
k
in addition to the
x
-es.) So let’s try to fix that by combining 8
x
and
kx
:

(
x
-8)(
x
-
k
) =
x
2
-
kx
- 8
x
+ 8
k

(
x
-8)(
x
-
k
) =
x
2
- (8 +
k
)
x
+ 8
k

Now it looks a bit more normal—we’re still stuck with that weird
k
, but at least now we have one term with
x
2
, one term with just
x
, and one term with no
x
at all, which is our normal arrangement after FOIL-ing out two binomials.

So
now our entire equation looks like this:

x
2
- (8 +
k
)
x
+ 8
k
=
x
2
- 5
kx
+
m

Now we realize that (
8 +
k
)
x
must correspond with 5
kx
, and that 8
k
corresponds with
m
. In other words, the two
x
terms on both sides must correspond, and the two terms on both sides with no
x
at all must also correspond. So we can solve for
k
, and then use
k
to solve for
m
:

(
8 +
k
)
x
= 5
kx
              (deal with the
k
terms on both sides first)

(
8 +
k
) = 5
k
              (divide both sides by
x
)

8
+
k
= 5
k
              (simplify)

8 = 4
k
                            (combine like terms)

2 =
k
                            (isolate
k
)

and
, therefore:

m
= 8
k
and
k
= 2

m
= 8(2)              (substitute 2 for
k
)

m
= 16                            (simplify)

So (B) is correct.

That’s the algebraic approach. It’s kind of ugly, at least by SAT standards, but it works.

There’s another approach we can use that will probably be a little easier on our brains, though it might not be any faster. That would be to look at each answer choice and try to solve backwards; only one of the choices will make this possible.

For instance, if we want to test out choice (A), we would see what has to happen if
m
is 8. Since
m
is the product of 8 and
k
, according to FOIL, then we know that
k
would have to be 1 if we made
m
be 8. But if we FOIL out the expression on the left with
k
equal to 1, the rest of the expression doesn’t end up matching with the rest of the expression on the right.

Then we could try with (B), (C), and so on. We would see that the process can be made
to work when
m
equals 16, but not for any other choice.

Another way to go, which is probably faster, but which requires a bit more awareness of algebra, is to notice the relationship between
m
and 8
k
, and the relationship between 8
x
,
kx
and 5
k
. Basically, we can see that
k
must be a value such that 8+
k
and 5
k
are equal, which means
k
must be 2, as we figured out before. And that would mean that
m
has to be 16.

Again, this is one of the ugliest questions in the book, but we can work out the answer if we stick to the fundamentals of algebra, and if we’re willing to play around with an unfamiliar presentation of those fundamentals.

Depending on the approach you take, this question might eat up a little (or a lot) more time than the average SAT Math question. While we should always try to find the most efficient solutions during practice sessions, remember that it’s normal for some questions to take longer than others on test day. Questions like this one are a big part of the reason why it’s so important to try to save time on questions that seem easier to you. That way, you have enough time left to play around with the harder questions, or to go back over your work and check your answers.

This question is a little complicated for an SAT Math question, but, as trained test-takers, we expect that we might see a little more complication in the last question or two of the grid-in section. As always, we’ll pay close attention to what the question is asking and see what we can figure out.

This question asks about the value of
a
, and the only place in the whole question where we can find any reference to
a
is in the expression
y
=
ax
2
. So that means we’re going to need values for
y
and
x
that we can plug in to that expression, so we can solve for
a
.

Apart from (0,0), which won’t help us figure out
a
, there are only two points on the graph that have (
x
,
y
) coordinates we can probably figure out: points
Q
and
R
, which are also in square
PQRS
.

We’re told that
PQRS
has an area of 64. We know this is an important dimension to be aware of because it was included in the text of the question after being omitted from the diagram itself. If the area is 64, that means each of the square’s sides is 8 units long. And that, in turn, means that point
R
is located at (4,8), because OS is half of PS.

Now we have values for
x
and
y
that we can plug in for point R, which is on the graph of
y
=
ax
2
:

y
=
ax
2
                            (given equation for the graph)

8 =
a
(4
2
)              (substitute
x
and
y
values for the point (4,8))

8 = 16
a
                            (simplify expression on the right)

8/16 =
a
              (divide both sides by 16 to isolate
a
)

1/2 =
a
                            (simplify expression on the left)

Notice that all of the ideas in this question are relatively simple ideas on their own; the trick was to trace the proper approach back through the wording of the question to figure out how to string together the relevant ideas.

Notice, also, that this question is actually very similar to question 18 on page 717 of the College Board’s Blue Book. (The two questions aren’t identical, but the only real difference between them is that
ABCD
isn’t a square in that other question, and the function in this question isn’t a parabola.) Do not be misled by this coincidence into assuming that the College Board frequently repeats a limited number of question types in each SAT Math section. On the contrary, in the entire Blue Book these two questions are among the only examples of question material being repeated so closely on the SAT Math section.