SECTION 3
1. E If you divide 12 by 4, you’ll see that x = 3. When you plug x = 3 into the term,
you’ll see that 4(3) + 2 = 12 + 2 = 14.
2. B The shaded region lies in the quadrant where x is positive and y is negative. Given
this, you can get rid of A, C, and E. If you plot answers B and D, you’ll find that (1, –
2) is inside the shaded region, while (5, –4) is not.
3. A You need to take this question one step at a time. First, figure out how many batches
there are in 12 jars of cookies. If one jar holds 1.5 batches, then 12 jars will hold 12
× 1.5, or 18 batches. Now you need to figure out how much flour is needed for 18
batches. If you need 6 cups of flour for 1 batch, then for 18 batches you will need 18
× 6, or 108 cups.
4. B There are variables in the answers, so this is a question for the technique of Plugging
In. If n = 2, then answer choice B is the only answer choice that gives you an odd
integer: 3(n + 1) = 3(2 + 1) = 3(3) = 9. Plugging In makes this problem much easier.
5. A Use the midpoint formula
: Take the average of the x-coordinates of
the two points to get the x-coordinate of the midpoint, and do the same for the y-
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coordinates. The midpoint between (3, 4) and (0, 0) is therefore
2).
= (1.5,
6. C This question is much easier if you work out the square roots first. You know that
= 2 and
= 3, so you can rewrite the question like this: 2x – 3x = –x.
7. B Use average pies:
In the first pie you have the number of red parrots sold, which is 5, and the average
weight, which is 2. That gives you a total weight of 10 pounds. In the second pie you
have the number of blue parrots, which is 4, and the average weight, which is 3. That
gives you a total weight of 12. To find the average weight of all the parrots, you
need to find the total weight of all the parrots. This is simply the total of the red plus
the total of the blue. In the last pie you have the total number of parrots, which is 9,
and the total weight of all the parrots, which is 22. This gives you an average weight
of
.
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8. C Remember that the perimeter is the sum of all the sides. FCDE is a square, so all the
sides are equal. Because
= 3, each side of the square is 3, so you know that
,
, and
are all 3. ABDE is a rectangle, which means that the opposite sides are
equal.
= 2, so
= 2 also. Along the same lines
equals 3, so that means the
opposite side,
, also equals 3. Add up all the sides of ABDE to find the perimeter:
2 + 2 + 3 + 3 + 3 + 3 = 16. To find out how much smaller the perimeter of FCDE is,
just subtract: 16 – 12 = 4.
9. D The sides of a 45°-45°-90° triangle have a special pattern, which you can find in the
gray box at the beginning of every Math section. Each leg of a 45°-45°-90° triangle
is equal to the hypotenuse divided by
8,
must be equal to
. Because the hypotenuse in triangle ABC is
. You can’t have a square root on the bottom of a
fraction; so multiply the top and the bottom by
. That gives you
.
Meanwhile, you can use Ballparking to eliminate A and B. Because the hypotenuse
of a right triangle is always the longest of the three sides,
10. D Multiply both sides by 2 to get
must be less than 8.
. Square both sides to get x = 162.
Therefore, x = 32. You could also have plugged in the answer choices. For D, put 32
in for x and ask, does
? Then does
? Square both sides to find
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that it does, so D is correct.
11. D If there are variables in the answer choices you should … plug in! First, cross out that
phrase in terms of b, because you don’t need it. Next, let’s plug in a number
for a. This is a percent question, so let a = 100. Because b is 40% of a, that
means b = 40. If a = 100, then 4a = 400. Use your calculator to find 40% of 400,
which is 160. That’s your target answer. When b = 40, D gives you 160.
12. D According to the function, f(x) = 2x2 + 4. To find the value of f(4), just substitute 4
for x: f(4) = 2(4)2 + 4 = 32 + 4 = 36.
13. B To find the value of f(3) + f(5), find the values of f(3) and f(5) separately: f(3) = 2(3)2 +
4 = 22 and f(5) = 2(5)2 + 4 = 54. So f(3) + f(5) = 76. You already know that f(4) = 36
from question 12, so you can cross out A. C is the Joe Bloggs answer because Joe
simply adds 3 and 5, and it can’t be that easy. If you ballpark D and E, putting 10 or
15 in the function will give you a number bigger than 100, and you’re looking for
76, so D and E are too big. That means the answer is B by POE.
14. C The formula for the area of a circle is , so you need to find the radius, r, of the circle.
You know that the diameter is 9, and the radius is half of the diameter, so the radius
is . Because there are fractions in the answer choices, you might as well keep the
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radius as a fraction. Now you replace r with
in the formula for area of a circle:
.
15. D Lines that are parallel have the same slope. In the form y = mx + b, m is the slope.
So, the slope is −3. Find the line that has a slope of −3 when you rewrite it in the
form y = mx + b. Only D works when you rewrite it: Subtract x from both sides to
get y = 5 − x, then multiply both sides by 3 to get y = 15 – 3x or y = –3x + 15.
16. C If |x| = |2x – 1| this means that either x = 2x – 1 or –x = 2x – 1. The solutions to these
equations are 1 and , respectively. However, the only thing you need to recognize
is that the equation has two different solutions to establish that the answer is C.
17. A Variables in the answer choices? Plug in! This is a percent question, so make k = 100
and d = 40. If 40 out of the 100 gallons have been pumped, that equals 40%. So
40% is your target answer. When you plug k = 100 and d = 40 into the answers, only
A gives you 40. Plugging In turns a hard question into a much easier question.
18. C For this question you need to know the distance formula: d = r × t. There are two
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good ways to solve this question. One is Plugging In The Answers. The question
asks how far Ray and Jane will be from Ray’s house when they meet. Start with C: If
they are 56 miles away from Ray’s house, and Ray traveled from home at 30 miles
per hour, then you can figure out the time he traveled using the d = r × t formula
(and your calculator): 56 = 30 × t, t = 1 . In this case Ray has traveled for 1
hours. If Jane has traveled 93
miles and Ray has traveled 56 miles, then they
have traveled a total of 150 miles when they meet. Bingo! You’re done. An even
easier way is to think about how fast Ray and Jane are traveling put together. You
can simply add the rates. Together they are traveling at 80 miles per hour. Therefore
you can figure out the time by setting 150 miles = 80 × t. The time is 1
find how far Ray has traveled, use the formula one last time:
hours. To
.
19. D To figure out probability, you need to work with fractions; the total number of
possible outcomes goes on the bottom, and the number of desired outcomes goes
on the top. To figure out the probability of selecting three blue hammers, you need
to figure out the probability of getting a blue hammer each time a hammer is
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selected. The first time, there are a total of 20 hammers and 10 of them are blue, so
the probability of getting a blue hammer is
. When the second hammer is
selected, there are only 19 hammers left, and only 9 of them are blue. So the
probability of getting a blue hammer the second time is
. When the third hammer
is selected, there are a total of 18 hammers left and 8 are blue, so the probability of
getting a blue hammer on the third try is
. To find the probability of selecting
three blue hammers, you need to multiply the three separate probabilities:
. By the way, A and C are Joe Bloggs answers because he doesn’t
consider how the probability changes with each event.
20. D Don’t forget that you can plug in on geometry questions. Let’s make b = 70°
and a = 30°. So the third angle in the triangle is 80°. You know that c would be 80°,
because it is opposite an 80° angle. Your target answer is a = 30°, so plug in 80° and
70° to find it. The only possible answer is D.
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