SAT Prep Black Book: The Most Effective SAT Strategies Ever Published (46 page)

Many test-takers try to approach this question by figuring out individual values for
p
and
n
, but it literally can’t be done. It is possible, however, to figure out the value of
p
/
n
.

We'll figure out
the ratio by using the definition of the term "average," which tells us that if
p
students have an average of 70, then the total number of points scored by those
p
students is 70
p
.

Similarly, the total number of points scored by the
n
students is 92
n
.

Using the definition of average, again, we find that the average of all the
n
students’ scores plus the
p
students’ scores must be (70
p
+ 92
n
)/(
n
+
p
) = 86. Then we solve:
70
p
+ 92
n
= 86
p
+ 86
n
                            (multiply both sides by
n
+
p
)

6
n
= 16
p
                                          (combine like terms)

6
n
/16 =
p
                                          (isolate
p
)

6/16 =
p
/
n
                                          (isolate
p
/
n
since the question asks for it)

3
/8 =
p
/
n
                                          (simplify)

Many test-takers miss this question because they mistakenly think that a square yard is the same thing as 3 square feet. But a square yard is 9 square feet, because a square yard is 1 yard x 1 yard, or 3 feet x 3 feet, or 9 feet
2
.

Once we know that, it becomes easier to realize the correct answer is 12 * 18 / 9, or 24.

Another way to approach the situation is to convert the dimensions of the floor to yards right from the beginning, so that we calculate 4 yards by 6 yards, again getting 24 yards
2
.

So (C) is correct.
Note that (A) is there for us in case we divide by 3 one time too many, and (E) is there in case we accidentally find the square footage instead of the square yardage. Things like this are why it’s always so important to read carefully on the SAT Math section.

This question often frustrates people, but it’s actually not as hard as people often think. The question asks for the shortest distance between the center of the cube and the base of the cube. If we visualize the cube, we'll see that the distance from its center to its base is the same as the distance from the center of one of its sides to the base—or, in other words, if you looked at the cube straight on it would look like this:

where
C
is the center of the square/cube and the bottom line of the square/cube is the base.

If the
volume of the cube is 8, then the sides are all length 2, which means the distance from the center to the base is 1 (because it's half the distance from the base to the top side of the cube). So the answer is (A).

This question provides one more terrific example of the difficulty that the College Board can manage to infuse into an SAT Math question without actually using any math.

This question throws a lot of people. But, as always, all we really need to do
here is follow what the text says, alternately switching the left or the right wire with the center wire. So the pattern goes like this:

step 1: BAC
step 2: BCA
step 3: CBA
step 4: CAB
step 5: ACB
step 6: ABC

So the answer is (D).
Notice that it’s possible to end up with the wrong answer of 7 if we accidentally count the “Start” as a step. One more moment when critical reading skills really come in handy on the math part of the SAT.

T
his is also a great example of an SAT Math question for which no formula exists. There’s no way you could use a calculator on this question; there’s not even any real way you could use basic arithmetic. This question is literally about braiding, but untrained test-takers in advanced calculus classes miss it frequently in practice.

Page
852, Question 19
Like the previous question, this one ends up involving no actual math. All we need to do is think carefully about the definition and properties of the term “median.”

If you double the value of each number or increase each number by 10
, you must end up changing the median number (because you either double it or increase it by 10, as the case may be).

In other words, if the 11 numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, then doubling them all makes the median 12 and adding
ten makes it 16. The median is the value in the middle--that's the definition of the term. Changing the number that's in the middle of the range of numbers changes the median.

(C) or (D) might also end up affecting the median, if we increase the smallest number by such an amount that it becomes greater than the original median, or if we decrease the largest number by such an amount that it becomes less than the original median.

Only (E) has no possibility of changing the median. Increasing the largest number still causes it to remain the largest number, so the number in the middle of the series can’t change.

Page 861, Question 16
Many students try to use some kind of formula here, but that’s a mistake, of course. For one thing, the SAT can't ask us to find the area of a trapezoid because it doesn't give that formula in the beginning of the math section or in the question itself. So there must be some other way to solve the problem that is easier and more direct.

If you draw out the rectangle and the trapezoid
, we'll see that the trapezoid takes up 3/4 of the area of the rectangle, like so (I dashed in some other lines so you can see that
ABED
takes up 3/4 of
ABCD
more clearly):

So that means that we can do this:

ABED
= 3/4(
ABCD
)
2/3 = 3/4(
ABCD
)
8/9 =
ABCD

This means that the answer is (C).

Notice that one of the wrong answers is 3/4. This is a good sign for us, because it indicates that we were probably on the right track when we figured out that
ABED
was 3/4 of
ABCD
.

This question is often frustrating for untrained test-takers because it seems at first as though it will be impossible to figure out the product of the slopes if we don’t know the slopes themselves, and the question doesn’t tell us the slopes.

But if we remember the definition of the term “rectangle” and the properties of perpendicular lines in a coordinate plane, then we can realize that t
he slopes of lines that are perpendicular to one another are opposite reciprocals (like, for instance, 2 and -1/2), so they'll have to multiply together to give us -1.

There are two pairs of perpendicular sides,
in a rectangle, so the final product of all of these slopes is 1, because -1 * -1 is 1. (For instance, if the slopes were 2, -1/2, 2, -1/2, they would multiply together to make 1.) So (D) is the right answer.

So, in the final analysis, this question really just involved knowing that rectangles have 2 pairs of perpendicular sides, knowing that perpendicular lines have slopes that are opposite reciprocals, and knowing that -1 * -1 is 1. Each of those facts is pretty basic on its own, but most people who look at this question never realize that those facts are involved in answering it. This is why it’s so important to develop the instincts for taking these questions apart by reading carefully, thinking about definitions and properties, and ignoring formulas and calculators for the most part.

This question basically hangs on the definition of the term “directly proportional.” We need to know that w
hen
n
is directly proportional to
q
,
n
=
kq
, where
k
is some unknown proportional constant that we usually need to figure out.

So
, in this case, if
y
is directly proportional to
x
2
, then
y
=
k
(
x
2
).

So let’s substitute and see what happens:

1/8 =
k
(1/2)
2
                            (plug in the given values for
x
and
y
)

1/8 =
k
(1/4)                            (simplify on the right)

4(1/8) =
k
                            (isolate
k
)

1/2 =
k
                                          (simplify)

Now that we know what
k
equals, we can plug it in with a
y
value of 9/2, to find our new
x
:

9/2 = 1/2(
x
)
2
                            (plug in values for
y
and
k
)

9 =
x
2
                                          (combine like terms)

3 =
x
                                          (isolate
x
)

So the answer is (D).
Notice that our other answer choices include 9, which is what we’d get if we accidentally solved for
x
2
instead of
x
, which is a mistake people often make on this question.

It’s not common for the SAT to test a math concept as
obviously as it’s testing direct proportionality in this question—the only wrinkle in this question is the idea of being proportional to
x
2
instead of
x
. I think this particular question is so direct because direct proportionality is a concept that many algebra teachers no longer cover for some reason. (This is just a theory on my part, but I think it matches the evidence. Math concepts that are more commonly encountered on the SAT aren’t usually tested this directly.)