SAT Prep Black Book: The Most Effective SAT Strategies Ever Published (47 page)

At any rate, if you saw this question and realized that you had forgotten (or never been taught) the idea of direct proportionality, it would be a good idea to review it. The odds of it coming up on test day for you are relatively slim, but it’s such a simple concept that it can’t hurt to take a few moments to memorize it.

For this question, as for most SAT Math questions, we simply need to read carefully and look for ways to apply basic math ideas.

If there are 18 arcs of each length in the circle, and if the circle has a circumference of 45 units, then we can write this:

18(2 +
b
) = 45

That means that
36 + 18
b
= 45, which means that 18
b
= 9, so
b
= 1/2 unit.

If
b
is 1/2 unit and there are 45 units in the circumference of the circle, then 45 units corresponds to 360 degrees of arc. This means that each of the 45 units in the circle's circumference accounts for 45/360 degrees, or 8 degrees; since
b
is 1/2 of a unit,
b
must be 4 degrees, and the answer is (A).

Notice, as is often the case, that the answer choices represent mistakes that can be easily made. 16 degrees is the arc measurement of the 2-unit arcs, and 20 degrees is the arc measurement of a (2 +
b
) combination.

This question, like many questions near the end of an SAT Math section, is actually much easier than it might look. It’s also an especially clear example of a few of the SAT Math rules and patterns from this book, all wrapped into one test item.

One of the first things I’d want to notice is that the diagram isn’t drawn to scale. That means I should think about re-drawing it to scale. To figure out how to do that, I have to look at the text of the question, which mentions that each of the five line segments are congruent (which is just a fancy way to say they’re all the same length). That means that this diagram should show two equilateral triangles that share a side, like this:

The text of the question also includes another piece of information that was left out of the original diagram, which is the existence of line segment AC. So I’ll add that in:

Another fundamental thing that we should pay attention to is the set of a
nswer choices. Notice that they all involve either √3 or √2 (choice (E) actually involves both).

We should ask ourselves what kinds of math concepts are related to the concepts we’ve
encountered so far. We know the diagram involves two equilateral triangles, and we know equilateral triangles are 60
o
-60
o
-60
o
triangles. We also know that the answer choices involve √3 and √2; √3 is related to 30
o
-60
o
-90
o
triangles and √2 is related to 45
o
-45
o
-90
o
triangles.

At this point, we’d want to realize that line AC cuts
the two 60
o
-60
o
-60
o
triangles in half, creating 4 new triangles, and
each of these new triangles is a 30
o
-60
o
-90
o
triangle.

We know that the ratio of the legs of a 30
o
-60
o
-90
o
triangle is 1: √3, with √3 representing the long leg. (This is something we’d want to have memorized, but even if you forget it momentarily you can always look at the given information at the beginning of any SAT Math section).

The question asks for the ratio of AC to BD. AC includes two long legs of 30
o
-60
o
-90
o
triangles, while BD includes two short legs of 30
o
-60
o
-90
o
triangles. That means the ratio is 2√3:2, which is the same thing as √3:1, which is what (B) says, so (B) is correct.

Surveying all the answer choices, we’ll see that most of them start with
√3, which is a good sign that starting with √3 is probably the correct option. But more of them end with something related to 2, not 1, so I would double-check myself to make sure that I hadn’t made a mistake in simplifying the ratio.

Most untrained test-takers will see this question and assume that they need to figure out the actual measurements of each angle in the answer choices, because that kind of solution might seem similar to something we would do in a trigonometry class. But, as trained test-takers, we’d want to realize that it’s possible to figure out which angle is smallest without figuring out their actual measurements—and, anyway, trig isn’t allowed on the SAT, so there has to be another way.

Since the two endpoints of each angle are the same (always
X
and
Y
), the farther away the vertex is, the smaller the angle will end up being. Since
D
is the farthest point from
X
and
Y
,
XDY
is the smallest angle, even if we never figure out its actual measurement. So (D) is the correct answer.

Most untrained test-takers will try to identify the actual values of
x
and
y
, because that’s what we would typically have to do in school if we saw a math question with multiple variables. But on the SAT, we’ll frequently be asked questions in which the variables don’t need to be identified, and this is one of those questions. (One of the major clues that we don’t need to identify the variables is that we weren’t asked to find either variable’s individual value; we were only asked to find the value of a complex expression that involves both variables.)

As always, it’s important to think about the answer choices, and how they relate to the concepts and quantities in the question. Notice that every answer choice has a clear relationship to the quantities 7 and 5 from the question:

(A) is the difference of 7 and 5, so it would make sense as the right answer if
x
2
y
-
y
2
x were the same thing as the difference between
xy
and
x
-
y

(B
) is the sum of 7 and 5, so it would make sense if
x
2
y
-
y
2
x were the sum of
xy
and
x
-
y
.

(C) is twice the sum or 7 and 5, which also doesn’t make any sense here.

(D) is the product of 7 and 5, so it will be the right answer if
x
2
y
-
y
2
x is the same thing as the product of
xy
and
x
-
y
. Which is exactly what it is, so (D) is correct.

(E) is twice the product of 7 and 5.

As I’ve mentioned many times, we really want to get in the habit of thinking of the answer choices as part of the question itself, not as an afterthought. Looking for ways that the choices relate back to the text of the question can really help you increase both your speed and your accuracy. This question is one of the most frequently missed questions in this section (we can tell this from the College Board’s difficulty ranking), but it can literally be answered without even picking up a pencil if we notice what’s going on in the answer choices. This question ultimately boils down to paying attention, knowing what 7 * 5 is, and knowing how to multiply simple expressions in algebra.

This is a grid-in question that asks us to find “one possible value of the slope” of a line that intersects line segment
AB
. Remember that when a question like this asks us to find “one possible value,” then there must be more than one value possible—whenever a question refers to the possibility of multiple values, there are multiple values.

In this case, any slope value of any line whatsoever that intersects
AB
will be fine. If a line (8,3) hits point
A
exactly, then a line through (9,3) must intersect
AB
. The line through (9,3) will have a slope of 3/9, or 1/3. So that’s one possible value.

Another way might be to try to cut
AB
in half. To do that, we could imagine a line through (8,1.5), or through (16,3) (that would be the same line, determined at two different, collinear, points). Either one would create a line with a slope of 3/16.

Yet another way to create a valid answer might be to realize that a line with a slope just barely over horizontal would have to intersect
AB
since point
B
lies on the
x
-axis. So we could pick a slope of 1/10 or .001 or something.

Any of these approaches could work, as could a wide variety of others.

I once saw a YouTube video of a guy who explained how to attack this question in the most complicated way I could possibly imagine. He began by identifying the curve in the graph as a parabola, and then tried to figure out what the equation for the parabola would be (after trying to figure out the coordinates of various points on the parabola, like the vertex, the intercepts, and so on). Once he had the equation of the parabola worked out, he set it equal to 0, solved it, and then found his answer.

It worked, of course, but it created a ridiculous amount of extra work for no reason—and let’s not even mention the extra opportunities for error it created by drawing the solution out unnecessarily.

Unfortunately, that kind of approach is what most test-takers would try on this question, because they’re programmed to answer questions on the SAT Math section the same way they would answer questions in school.

There’s a much easier way to approach this question, which relies on simply understanding the properties and definitions of concepts relating to the graphs of functions.

We should recognize that the places where
h
(
a
) = 0 will be the places where the graph of
h
(
a
) crosses the
x
-axis.

We should also recognize that the graph is drawn to scale, which means we can eyeball it relative to the provided values and figure out where the graph crosses the axis from there.

If we look at the answer choices, we’ll see that (A) must be correct, because where
x
= -1 on the given graph, we find
h
(
a
) crossing the
x
-axis.