SECTION 8
1. D Use the first equation to solve for x: 3x – 5 = 4, so x = 3. Plug in 3 for x: 9x – 15 = 27
– 15 = 12.
2. E This is an excellent calculator question. Here are the costs per unit for each color:
The red buttons are the most expensive.
3. C Plug the answers into the equation starting with C. The (x, y) point is (7, 8), so plug in
7 for x and 8 for y. Because 8 = 2(7) – 6, C is the correct answer.
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4. B Before you reach for your calculator, reduce the expression.
Then simply try each choice; At –1, B has the least value. If you selected E, you
didn’t work out each choice.
5. B One way to solve this problem is to Plug In. First, simplify (a + b)2 = 49 by taking the
square root of both sides to find a + b = 7. Now, brainstorm some values
for a and b that make a + b = 7 and ab = 10. Let’s say a is 2 and b is 5. So, find the
answer that yields 5 when you plug in a = 2. Only B works. The second, more
complicated, way is to FOIL out (a + b)2 = 49 to get a2 + 2ab + b2 = 49. Plug in 10
for ab to get a2 + 2(10) + b2 = 49. That means, a2 + 20 + b2 = 49. Subtract 20 from
both sides to get a2 + b2 = 29. Subtract a2 from both sides to get b2 = 29 – a2. Take
the square root of both sides to find b =
.
6. C First, you can estimate. Because square ABFE has an area of 25, EF equals 5
and EC looks to be a little less than twice EF, or in the 7–9 range. Thus,
since CE = ED, because they are the legs of a 45˚-45˚-90˚ triangle, you can eliminate
D and E. You also know that the area of ΔBCF is 10, and that its base (BF) is 5. Using
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the formula for area, you can calculate FC, the height of the triangle: 10 =
(h), h = 4. So 5(FE) + 4(FC) = 9, which is the length of
and
(5) ×
.
7. C Stack all three equations and add them together to get 3x + 3y + 3z = 51. Factor out
a 3 to get 3(x + y + z) = 51. Divide both sides by 3 to get x + y + z = 17.
8. D Translate into algebra. The sum of two numbers is 10 means x + y = 10. Next, one
number is equal to the sum of 6 and twice the other number means x = 6 + 2y.
Rearrange the second equation into x − 2y = 6. Subtract the second equation from
the first to find 3y = 4. Divide by 3 to find y = . Plug that into the first equation to
get x +
= 10. So, x = 8 . The larger number minus the smaller number is
.
9. D Go through this problem one piece at a time. We know that
of the $45,000 goes
to the sanitation department. Remember that “of” in math questions means to
multiply, so
spends
× 45,000 = $9,000. He now has (45,000 – 9,000) $36,000 left. He
of that $36,000 remaining on the police department, which is ( × 36,000)
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= $24,000. He now has (36,000 – 24,000) $12,000 left to spend, so the answer is D.
10. A Let’s begin by drawing a parallelogram and plugging in a number for y, say 50, and
calling the other two angles x:
Because there are 360 degrees in a quadrilateral, you know that 2x + 100° = 360°,
which means x = 130°. So, you’re looking for the choice that gives you 130
when y = 50°. You simply plug 50 into all of the answer choices to find that A is the
only one that works.
11. D This is a parabola, because one of the two variables is squared. Eliminate A, B, and E,
which are not parabolic graphs. Because the smallest possible value of mx2 is 0, the
smallest possible value of mx2 + b is b, so all of the curve must be above the x-axis.
Only D works.
12. C First, you need to compute all possible values of n:
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Now, be careful! The median value for
is 5, but the median value for n is 25.
13. C Draw a picture! Look at triangle CJK. ∠CJK is 90° because the radius of a circle is
always perpendicular to a line tangent to that circle. Use the Pythagorean theorem,
(CJ)2 + 242 = 262, or your knowledge of right triangles (this is a multiple of a 5:12:13
triangle) to get CJ = 10, which represents the radius of the circle. So the
circumference of the circle = 2π × 10 or 20π.
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14. D This is a tricky question. Let’s draw a picture:
Because 1 centimeter equals 6 kilometers, 4 centimeters equals 24 kilometers:
The area of this region is 242 or 576. In case you were wondering, B is the Joe
Bloggs answer because 16 × 6 = 96.
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15. E Plug In values for m and n and use translation to solve this percent problem. You’re
working with a small percent, so plug in a big number for m. Let’s say m = 2,000.
. Therefore, 10% of n is 2; rewrite this as
0.1% of
.
Solving for n, you get n = 20. Now translate the rest of the problem: m is what
percent of 10n can be written as
. Solving for x, you get x =
1,000, so the answer is 1,000%.
16. E Remember your transformation rules. Whenever a parabola faces down, the
quadratic equation has a negative sign in front of it. It always helps to plug in! Let’s
take an example. If your original equation was (x + 2)2, putting a negative sign in
front, –(x + 2)2, would flip the parabola. If you expand out that equation, you get –
x2 – 4x – 4. Notice that a in this equation is –1. Also, notice that c in the equation is
just the y-intercept, because if you plug in 0 for x you get y = c. On the graph, the yintercept is negative. And a negative number times a negative number is always
positive. Again, plug in if you like it better: (–1)(–4) = +4. The best answer is E.
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