ANSWERS

Drill 1

  1.      C

  2.      B

  3.      B

  4.      C

  5.      C

  6.      C

  7.      A

  8.      D

  9.      E

10.      D

11.      A

12.      25

Drill 2

  1.      A

  2.      E

  3.      C

  4.      C

  5.      24

  6.      1

  7.      8

  8.      C

  9.      E

10.      C

11.      B

12.      E

EXPLANATIONS

Drill 1

  1.  C  If f(x) = 2x + 5, then f(4) = 2(4) + 5 = 13.

  2.  B  Draw it. The rectangle in Quantity A has two sides of 2, for a total of 4. The remaining 20 units in the perimeter are divided evenly into two sides of length 10; the area of this rectangle is lw = (2)(10) = 20, and the “par” is one half that, or 10. Quantity B is greater.

  3.  B  Notice that each term in the sequence is 1.5 greater than the last (i.e., k = 1.5). So the second term is 3 + 1.5 = 4.5, the third term is 4.5 + 1.5 = 6, and so forth. So the fifth term is 7.5 + 1.5 = 9, the sixth term is 9 + 1.5 = 10.5, the seventh term is 10.5 + 1.5 = 12, and finally, the eighth term is 12 + 1.5 = 13.5. So, Quantity A is 13.5, and the answer is choice (B). Another way to attack this problem is to use the sequence formula of 3 + 1.5(n – 1), where the 3 is the first term, the 1.5 is the increase, and you are looking for the nth term. So, the 8th term is 3 + 1.5(8 – 1) = 13.5.

  4.  C  To follow the order of operations, first evaluate the expression in parentheses. 3 → (2→4)= 3 → 2=3→4=3=6. The answer is choice (C).

  5.  C  Write out sequences until you see the pattern. The second term in the sequence is 3(–1) = –3. Adding 3 gives you the third term, 0. Multiplying by –1 gives you the fourth term, also 0. Adding 3 gives you 3, the fifth term. So the sequence repeats every four terms: 3, –3, 0, 0, 3, –3, 0, 0, and so forth. Dividing 168 by 4 gives you a remainder of zero, and the fourth, eighth, twelfth, and every other nth term where n is a multiple of 4 (including the 168th term) will all be the same value, 0. The answer is choice (C).

  6.  C  Even if you know the summation formula, you can avoid a lot of time-consuming calculation by disregarding the numbers that are common to both sums—the even integers from 22 to 36, ­inclusive. That leaves 18 + 20 = 38 as the sum of the unique terms in Quantity A, and 38 as the only unique term in Quantity B. The quantities are equal.

  7.  A  Set up a group grid and fill in what you have:

            Use this to find that there are 65 – 30 = 35 total standard members. So there are 35 – 20 = 15 standard male members, thus, Quantity A is 15. The answer is choice (A).

  8.  D  Whenever you have a function within another function, you have to first calculate the value of the function on the inside, and then plug that value into the function on the outside. Plug in to test the values. If you plug in 2 for x, then, in Quantity A, 2§ = 2² + 2(2) = 8 and 8# = 8² + 3(8) = 88; in Quantity B, 2# = 2² + 3(2) = 10 and 10§ = 10² + 2(10) = 120; eliminate choices (A) and (C). However, if you plug in 0 for x, then both quantities are 0; eliminate choice (B), and you’re left with choice (D), the correct answer.

  9.  E  The top row contains 1 can, the second row contains 1 + 1(6) = 7 cans, the third row contains 1 + 2(6) = 13 cans, and so forth, so that the sixteenth row contains 1 + 15(6) = 91 cans. But you need to find the total number of cans, which is 1 + 7 + 13 +…+ 79 + 85 + 91. Notice that adding the first and last term in the sequence gives you 92. Adding the second and second to last term also gives you 92: As you move to the next term at the beginning of the sequence, you are adding 6, while as you move to the previous term at the end of the sequence, you are subtracting 6, so the sum will remain constant. Thus, for each pair of rows, the sum is 92. Sixteen rows represents eight pairs of rows, so the total number of cans is (8)(92) = 736. The answer is choice (E).

10.  D  Decoding the definition of the sequence tells you that, to find the value of each term, you take the previous term, and raise it to the power of the term before it. You know , and . So s₄ is the digit 1 followed by twenty zeroes, which is a total of 21 digits, and s₅ is the digit 1 followed by 2,000 zeroes, for a total of 2,001 digits. So the fourth term is the one that meets the condition set forth in the question, and the answer is choice (D).

11.  A  For Quantity A, start with the innermost parentheses: g(–1) = (–1) – 2 = –3. So f(–3) = (–3)² + 1 = 9 + 1 = 10. For Quantity B, f(–1) = (–1)² + 1 = 1 + 1 = 2. So, g(2) = 2 – 2 = 0. Thus, Quantity A is greater.

12.  25  Ignore the unfamiliar terminology and follow directions. The area of a circle with radius 5 is πr²=5²π=25π. Dividing the area by π gives you 25.

Drill 2

  1.  A  The second positive integer multiple of 5 is 10. The third positive integer multiple of 5 is 15. The sum of 10 and 15 is 25, so Quantity A is larger.

  2.  E  Rather than listing out the actual times, figure out the pattern. The vent releases steam at 25, 45, and then 5 minutes after 6 p.m., and repeats this pattern every hour thereafter. Only choice (E) fits the pattern.

  3.  C  When a problem gives you a relationship signified by an unfamiliar symbol: Just plug in the given values into the given “function” and solve. If l § m=, then 3§= –|1|= –1. The quantities are equal, so select choice (C).

  4.  C  Use the group formula and fill in what you know. So Total = Group1 + Group2 – Both + Neither becomes 30 = 14 + 13 – Both + 9. So Both = 6, and the answer is choice (C).

  5.  24  Find the measure of an angle in a regular 10-sided polygon by plugging 10 into the given formula: =144°. Then do the same for a regular 6-sided polygon by plugging 6 into the given formula: =120°. Finally, 144 – 120 = 24.

  6.  1  To follow the order of operations, first evaluate the expression in parentheses: 3 ⊕ 1 = 2(3) – 1 = 5, so now the first equation can be written as 5 ⊕ 2 = 2(5) – 2 = 8. Next, rewrite the second equation so that you have 6 ⊕ 3 = 2(6) – 3 = 9. Finally,|8-9|=|-1|=1, so the answer is 1.

  7.  8  Use brute force to solve this one: Write down the 2 given terms, find half the sum of the previous 2 terms, and repeat the process until you have a non-integer. When you work it out, Sequence S should begin 64, 32, 48, 40, 44, 42, 43, 42.5; the first non-integer term is the 8th term, so n = 8.

  8.  C  To find the pattern in each sequence, write out the units digit of the first few terms in the sequence. The pattern for the units digit of powers of 7 is: 7, 9, 3, 1. The pattern for the units digit of powers of 3 is: 3, 9, 7, 1. For both numbers, 1 repeats as the units digit every 4 powers, so the 4th power will have a units digit of 1, as will the 8th, the 12th, and so on. Because 28 is a multiple of 4, you know that 7²⁸ will have a units digit of 1. So moving forward one in the pattern, 7²⁹ will have a units digit of 7. Similarly, 3²⁸ will have a units digit of 1, so moving backward one in the pattern, 3²⁷ must have a units digit of 7. The quantities are equal, so the answer is choice (C).

  9.  E  Set up a group grid and, because you are dealing with percents and fractions, plug in 100 for the total number of employees at the company. There will be 60 men, of whom 6 are still employed after the restructuring. Subtracting 60 from 100 gives you 40, the total number of women. Five times the 6 men who are still employed gives you 30, the number of women still employed. After filling in this information, the group grid looks like the figure below.

           There are 30 women, but the question asks you what percent this represents of the total number of women. 30 out of 40 is 75 percent, so the answer is choice (E).

10.  C  Rather than trying to remember a bunch of rules about even and odd numbers, Plug In for a and b. If a is 2, then 6a is 12, and #12 = –2. Because b – 3 is odd, make b = 6, and #b = –2 as well. The two quantities are equal, so eliminate choices (A) and (B). Any set of values gives the same outcome, so select choice (C).

11.  B  Plugging 10 values into two compound functions is going to involve lots of arithmetic and will take a long time, so it is better to do this one algebraically. Working from the inside out, find Quantity A: f(g(x))=f(x+1)=3(x+1)²=3x²+6x+3; remember to FOIL the (x + 1) when you square it. Similarly, find Quantity B: g(f(x)) = g(3x²) = 3x² + 1. You can add or subtract the same value from both quantities without affecting which is bigger; doing so with 3x² + 1 leaves you with 6x + 2 in Quantity A and 0 in Quantity B. Because 6x + 2 is a linear function whose graph is a line with positive slope, you know that the values of the function will increase as the values of x increase. So you only need to plug in the endpoints of the given range of x-values to see what happens to the function: 6(–10) + 2 = –58 and 6(–1) + 2 = –4. So all possible values of Quantity A are still less than 0, and the answer is choice (B).

12.  E  The equation 3 × 2^n–1 follows a pattern. When n = 1, the result is 3. When n = 2, the result is 6. When n = 3, the result is 12. When n = 4, the result is 24. When n = 5, the result is 48. When n = 6, the result is 96. Beginning with the second term, the final digit in each result follows the pattern: 6, 2, 4, 8, 6, 2, 4, 8, etc. The 25th term will thus end in the same digit as all the other kth terms, where k is one greater than a multiple of 4. Thus, the (4 + 1)th, (8+1)th, (12+1)th.…(24+1)th terms all have a final digit of 8, and the only answer in which that is true is choice (E).