Mathematics: Section 2
Question Number: 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 - 14 - 15 - 16 - 17 - 18 - 19 - 20
1.
Choice (B) is correct. When you substitute 10 for x in the expression, you get 
, which equals 9 + 11, or 20. You could also simplify the original expression first before substituting the value of x. The original expression simplifies to 
, which equals 
, or 2x. Then substituting 10 for x yields 20.
2.
Choice (C) is correct. The information in the question tells you that k is a power of 2. Among the answer choices, 32 is the only one that can be written as 2n where n is a positive integer (32 = 25).
3.
Choice (C) is correct. Since there were none left over when Enrique counted by 5's, it follows that the number of crayons is a multiple of 5. Since there was 1 left over when he counted by 3's and by 4's, it follows that the number of crayons is 1 more than a multiple of 3 and 1 more than a multiple of 4. The least multiple of 5 that is 1 more than a multiple of 3 and 1 more than a multiple of 4 is 25.
4.
Choice (B) is correct. A circle has infinitely many diameters, all equal in length. If you draw the diameter that is parallel to the top of the square, you will see that its length is the same as the length of one side of the square. Since all the diameters are equal in length, 
has the same length as one side of the square. Therefore, AB = 6.
5.
Choice (E) is correct. If n represents Nora's age now, then Kevin's age now would be represented by n – 3, since Kevin is 3 years younger than Nora. Kevin's age 4 years ago was 4 less than n – 3 which is n – 3 – 4, or n – 7.
6.
Choice (D) is correct. Multiplying both sides of the original equation by r + s gives 1 = 2(r + s). Dividing both sides of this equation by 2 gives r + s = 
. You could also have noted that r + s is the reciprocal of 
, so the answer is , which is the reciprocal of 2.
7.
Choice (A) is correct. Since line m bisects the angle with measure x°, it forms two angles, each of measure z°. Since lines 
and p are perpendicular, z + z + y = 90. Solving for y gives y = 90 – 2z.
8.
Choice (A) is correct. From the second equation, 2u + 2v = 60 you can conclude that u + v = 30. Substituting 30 for u + v in the first equation gives t + 30 = 42. Therefore, t = 42 – 30 = 12.
9.
Choice (C) is correct. Some examples of squares of prime numbers are: 72 = 49, 112 = 121, and 132 = 169. The number you are looking for has three digits, so 49 does not qualify. The three-digit squares of primes are 121, 169, etc. The least of these is 121.
10.
Choice (E) is correct. First simplify ((x ÷ 2) ÷ 2) to get 
· = 
. Then substitute into the whole expression to get ÷ 2, which is 
. In order for to be an integer, x must be a multiple of 8. Among the answer choices, the only multiple of 8 is 56.
11.
Choice (B) is correct. The cost of depreciation is given as x%. Since all the percentages in the graph must add up to 100%, it follows that x + 10 + 3 + 10 + 15 + 25 + 2 = 100. Therefore x + 65 = 100, and so x = 35.
12.
Choice (D) is correct. Tracy is not responsible for the insurance, which is 25% of the total cost, but she is responsible for the remaining 75% of the total cost. Of the total cost, 10% is for maintenance. So the fraction of Tracy's costs that will be for maintenance is 
= 
. This fraction is equal to 13
%.
13.
Choice (B) is correct. For any two points (x1,y1) and (x2,y2) on a line, the slope of the line is defined as 
, sometimes referred to as "the change in y over the change in x," or "rise over run." The slope of the line that passes through the points (0,2) and is (2,0) is 
, which equals 
or – 1.
14.
Choice (A) is correct. One way to solve this problem is to substitute the answer choices into the given inequality. By doing this substitution, you find that when x = – 1, then |–1–3| = 4, which is greater than 3. Substituting the other values for x given in the answer choices yields numbers either equal to 3 or less than 3.
You can also solve this inequality by translating it into “The distance between x and 3 is greater than 3.” This means that x is outside of the interval from 0 to 6, so x is either greater than 6 or less than 0. Of the answer choices, only –1 satisfies this condition.
15.
Choice (C) is correct. Since the area of each face is 49 square inches, the length of each edge is 7 inches. There are 12 edges on a cube, so the total length of all the edges is 12 × 7, or 84 inches.
16.
Choice (A) is correct. Expanding the left side of the equation gives (x2 + 2xy + y2) – (x2 – 2xy + y2) = 84, so 4xy = 84; therefore xy = 21. Since x and y are integers, there are four different possibilities for x and y. If x = 1 and y = 21, or if x = 21 and y = 1, then x + y = 22. If x = 3 and y = 7, or if x = 7 and y = 3, then x + y = 10. Since the question asks which of the following could be a value of x + y, and since 22 is not given as a choice but 10 is, the answer is 10.
17.
Choice (C) is correct. Since the two lines are parallel, their slopes are equal. The slope of the line y = ax + 5 is a. The other equation can be rewritten as 8y = –3x + 10, which is equivalent to y = –
x + 
. The slope of this line is –. Therefore, a = –.
18.
Choice (A) is correct. In t seconds, g gallons of gas are pumped into the tank. If x is the number of seconds required to pump 18 gallons (the capacity of the tank), then the rate 18 gallons per x seconds must equal the rate g gallons per t seconds. That is, 
= 
, or 18t = xg. From this, it follows that 
= x.
19.
Choice (D) is correct. The 3-digit integers being formed can be written as XY2, where X and Y stand for digits. Seven digits are available, but one of the digits, 2, has already been used and cannot be repeated. Hence, there are six possibilities for X. Once a digit has been chosen for X, there are five possiblities for Y. The total number of possibilities is 6 × 5, or 30.
20.
Choice (E) is correct. There are two circles (one shown and the other only partially shown). The smaller one, shown completely, has radius r and circumference 2
r. The larger circle, shown only in part, has radius r + 3 and circumference 2(r + 3). The arc of length 6 is a fraction of the smaller circle, and the arc of length x is the same fraction of the larger circle. Therefore, 
= 
, which simplifies to 
= 
= 
. From this, it follows that x = (r + 3), which equals 
.
Another way to solve this is to observe that the two sectors are similar, which means that = . Solving this equation gives the same expression for x.
