| Quantity A | Quantity B |
| The sum of all the even integers from 18 to 36 inclusive | The sum of all the even integers from 22 to 38 inclusive |
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Solution
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.
Each of the even-numbered terms in a certain sequence is formed by multiplying the preceding term by –1. Each of the odd-numbered terms in the sequence is formed by adding 3 to the preceding term. If the first term in the sequence is 3, then what is the 168th term?
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Solution
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).
The operation denoted by the symbol →is defined for all real numbers a and b as a → b=a√b. What is the value of 3 →(2→4)?
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Solution
To follow the order of operations, first evaluate the expression in parentheses. 3 → (2→4)= 3 → 2√4=3→4=3√4=6.The answer is choice (C).
3, 4.5, 6, 7.5,…
Each term in the sequence above is formed by adding the positive number k to the preceding term.
| Quantity A | Quantity B |
| The eighth term in the sequence above | 14 |
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Solution
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.
Let the “par” of a rectangle be defined as one half the area of that rectangle.
| Quantity A | Quantity B |
| The par of a rectangle with a perimeter of 24 and a width of 2 | 11 |
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Solution
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 1w = (2)(10) = 20, and the “par” is one half that, or 10. Quantity B is greater.
If the function f is defined by f(x) = 2x + 5, what is the value of f(4) ?
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Solution
If f(x) = 2x + 5, then f(4) = 2(4) + 5 = 13.
Three dice are rolled simultaneously. What is the probability that exactly two of the dice will come up as the same number?
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Solution
There are a total of 63 = 216 total possible rolls for the three dice. First figure out the probability of getting exactly two 1’s.There are 5 × 3 = 15 ways this could happen: 112, 113, 114, 115, 116; 121, 131, 141, 151, 161; or 211, 311, 411, 511, 611.You could repeat this list of 15 possibilities in the obvious way for exactly two 2’s, exactly two 3’s, and so on.Thus, the total number of favorable rolls is 6 × 15 = 90.Because there are 216 possible rolls, 90 of which are favorable, the probability of getting exactly two of the three dice to show the same number is \(\frac{90}{216}=\frac{5}{12}\), choice (A).
In both rural and urban areas of country G, the average annual number of holidays taken by citizens is 8, and the annual number of holidays follows a normal distribution. In rural areas, 2% of the citizens take more than 12 holidays per year. In urban areas, 2% of the citizens take more than 16 holidays per year. How much greater is the standard deviation of the annual number of holidays taken by urban citizens than that of rural citizens?
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Solution
2
Draw two bell curves: one for rural areas, and one for urban areas.The three standard deviations above the mean each represent 34%, 14% and 2% of the population, respectively.The mean in both cases is 8. In rural areas, 2% of the citizens take more than 12 holidays a year, so 12 is two standard deviations above 8; the standard deviation is thus the difference between 8 and 12 divided by 2, or 2. In urban areas, similarly, the standard deviation is 16 – 8 divided by 2, or 4.The difference between the two standard deviations is thus 4 – 2 = 2.
| Quantity A | Quantity B |
| The average (arithmetic mean) cost per hinge for 16 hinges that cost a total of 2p cents | The average (arithmetic mean) cost per hinge for 4 hinges that cost a total of p⁄2 cents |
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Solution
Plug in a value for p.Try p = 16: In Quantity A, then 16 hinges cost a total of 32 cents, for an average cost of 2 cents per hinge; in Quantity B, 4 hinges cost a total of 8 cents, for, again, an average cost of 2 cents per hinge.The quantities are equal, so eliminate choices (A) and (B). Any value for p will yield the same results:The quantities will always equal; the answer is choice (C).
A photocopier can copy r pages per hour. How many pages can it copy in s seconds?
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Solution
Try plugging in a number for s that divides easily by 60, such as 7,200. So, if s = 7,200 seconds, that’s 120 minutes or 2 hours. Plug in a nice number for r such as 5. So, if the copier makes 5 pages per hour for 2 hours, your target is 10 pages. Plug s = 7,200 and r = 5 into the answers.Ballpark:Choice (A) is too large, choice (B) too small, choice (C) too small, and choice (E) far too large. Only choice (D) yields your target of 10.