There are 30 students in Mr. Peterson’s gym class. 14 of them play basketball, 13 play baseball, and 9 play neither basketball nor baseball.
| Quantity A | Quantity B |
| The number of students who play both basketball and baseball | 6 |
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Solution
Use the group formula and fill in what you know. So Total = Group 1 + Group 2 – Both + Neither becomes 30 = 14 + 13 – Both + 9. So Both = 6, and the answer is choice (C).
For all nonzero integers l and m, let the operation § be defined by l § m = – \(\left | \frac{l+m}{lm} \right |\)
| Quantity A | Quantity B |
| 3§(3⁄2) | -1 |
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Solution
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 = \(-\left | \frac{l+m}{lm} \right |\), then 3§(3⁄2) =\(\left | \frac{3+\frac{3}{2}}{3\left ( \frac{3}{2} \right )} \right |=-\left | \frac{\frac{9}{2}}{\frac{9}{2}} \right |\)= –|1|= –1.The quantities are equal, so select choice (C).
A certain vent releases steam every 20 minutes. If the vent releases steam at 6:25 p.m., which of the following could be a time at which the vent releases steam?
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Solution
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.
a is the sum of the second and third positive integer multiples of a.
| Quantity A | Quantity B |
| 5 | The daily rent of a man who pays $11,650 rent per year |
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Solution
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.
The “pluck” of a circle is defined as the area of the circle divided byπ. What is the pluck of a circle with radius 5 ?
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Solution
25
Ignore the unfamiliar terminology and follow directions.The area of a circle with radius 5 is πr2=52π=25π. Dividing the area byπ gives you 25.
f(x) = x2 + 1
g(x) = x – 2
| Quantity A | Quantity B |
| f(g(–1)) | g(f(–1)) |
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Solution
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.
The sequence of numbers S={s1,s2,s3…} is defined by s1 = 2, s2 = 10, and sn = sn-1sn-2 for each positive integer n greater than or equal to 3. For example, s3 = 102. What is the greatest value of n for which sn has 2,000 or fewer digits?
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Solution
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 s3 = 102 = 100,s4 = (102)10 = 1020, and s5 = (1020)100 = 102000. 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).
Mary is building a pyramid out of stacked rows of soup cans. When completed, the top row of the pyramid contains a single soup can, and each row below the top row contains 6 more cans than the one above it. If the completed pyramid contains 16 rows, then how many soup cans did Mary use to build it?
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Solution
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).
x# = x2 + 3x
x§ = x2 + 2x
| Quantity A | Quantity B |
| (x§)# | (x#)§ |
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Solution
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.
A club of 65 people includes only standard members and gold members. Of the club’s 30 gold members, 18 are men. Exactly 20 women are standard members.
| Quantity A | Quantity B |
| The number of standard members who are men | 13 |
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Solution
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).
