SECTION 5
1. A Subtract 10 from both sides to find 2x = 6, then substitute 6 for 2x. 6 – 10 = –4. You
could also solve for x, and then plug that into the expression. If 2x = 6, then x = 3.
So, 2(3) – 10 = 6 – 10 = –4.
2. C If he bought 3 boxes of muffins at $4.50, he would have 3 × 6 = 18 muffins. That’s
not enough muffins. So, he needs to buy 4 boxes of muffins. Each box costs $1.50;
so, the cost is 4 × $1.50 = $6.00.
3. E From 1 to 100, there are 50 even integers. If you don’t include 2 and 100, then
there are only 48.
4. B Fill in the angles of the small triangle. The lower left angle is 70° and the lower right
angle is 80°. Because the angles in a triangle must add up to 180°, a = 180 – 70 – 80
= 30.
5. E Plug 3 into each answer choice, and try to get 6. Eliminate C and D. Now plug 4 into
each remaining answer choice, and try to get 13. Eliminate A and B.
6. B This is a right triangle with two equal sides, so it is a 45°-45°-90° triangle.
be the hypotenuse, so the legs are each
must
. The legs are also the height and base of
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the triangle. A =
bh, so
.
7. B This question has several steps, so don’t try to do it all at once. Take it one step at a
time. Andrei starts out with 48 baseball caps. In the first step, Andrei gives away 13
caps, so he has 35 left. In the next step, he buys 17 new caps, so now he has 52.
Then Andrei gives Pierre 6 caps (46 left) and gets 8 caps in return. In the end, Andrei
has 54 baseball caps, which is 6 more caps than he had originally. The percent
increase is
. You can also change
to a percent
by typing it into your calculator and multiplying by 100.
8. D Draw a line straight up from D to divide the shape into a rectangle and a triangle.
The area of a 5-by-10 rectangle is 50. To get the area of the triangle, find the length
of the base; you already know the height is 5. Use the Pythagorean theorem, or to
save time, recognize that this is a 30°-60°-90° triangle because the hypotenuse is
twice as long as the shortest side. The base is therefore 5
is therefore
. The area of the triangle
. The total area of ABCD is therefore 50 + 12.5
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.
9.
100
Use a Ratio Box:
You need to work out only the total column to figure out the total number of
garments that the store sold, which is 100.
10.
81
The formula for the volume of a box is length × width × height. But the question
gives you the area of the base of the crate, so you already know that length × width
= 18. The volume of the crate, then, is simply the area of the base times the height:
18 × 4.5 = 81.
11.
.75 or
All you have to do is solve for x:
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12.
8 Using 3b = 2, solve for b by dividing both sides by 3 to get b = . That means
. Fractional exponents tell you to use the denominator as the root and use the
numerator as a regular exponent. So,
. First, cube both sides to find a2 = 43 =
64. Next, take the square root of both sides to find a = 8.
13.
5, 7, or 11 First, think of a prime number that will make 3b greater than 10. How
about 5? To see if that fits the other side of the inequality, you need to find the
value of
which is less than 10, so 5 is one possible value of b. Remember:
You need to find only one possible value of b.
14.
21
Let’s say you have seven dancers: A, B, C, D, E, F, and G. How many different ways
can you pair them up? This is a combination question, because A and B is the same
pair as B and A, and you don’t want to count them twice. Start by finding the
number of possible permutations: 7 × 6 = 42. Finally, divide this number by 2 × 1 =
2 (we’re looking to fill 2 positions) to eliminate redundant combinations. There are
21 possible combinations.
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15.
96
You know that arc BC is a semicircle, which means it’s half a circle. So the
circumference of the entire circle would be 6π × 2 = 12π. Therefore, the diameter of
that circle is 12. Because
is also a side of the rectangle you know that the length
of rectangle ABCD is 12. You can also use the same method to find the width. If the
length of semicircle CD is 4π, then the circumference of the entire circle would be
8π and the diameter is 8. Because
is the width of the rectangle, you can find the
area: length × width = 12 × 8 = 96.
16.
5
f(6) = 62 – 5 = 31. f(4) = 42 – 5 = 11. So f(6) – f(4) = 20. You then find y such that y2 –
5 = 20. y2 = 25, so y = 5 or –5, and the absolute value of y = 5.
17.
282
This is a hard question, so you have to stay on your toes. If the owner buys 3
pounds of each spice, that means she pays the following amounts for each spice:
So she pays a total of 24 + 27 + 21 + 30, or $102 for 12 pounds of spices. She then
sells the spices per ounce, so you have to figure out first how many ounces of spices
she has. If 1 pound is 16 ounces, then 12 pounds is 12 × 16, or 192 ounces. She sells
all the spices at $2 per ounce, so she makes 192 × $2, or $384. To figure out her
profit, subtract the amount she paid for the spices from the amount she made
selling them: $384 – $102 = $282.
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18.
.375 or
Because the question doesn’t give you a figure, you should draw one. Then plug in
some values.
If EG = EF, then you can make EF = 3 and EG = 5. That means FG must be 2. If HF =
5FG, then HF = 5(2) = 10. If HF = 10 and FG = 2, then HG = 8. So
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or .375.
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