GRE
Numeric Test 79
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Given an alphabet of 26 letters, with 21 consonants, and 5 vowels, approximately how many three-letter words can be formed with a vowel as the middle letter and a consonant as the last letter?

  • Solution

    When you see the word approximately, you are being told to Ballpark. In this case, you have 26 possibile letters for that slot, 5 for the second slot, and 21 for the third slot.Estimate and call this 25 × 20 × 5.The total is 2,500, so select choice (C).

A club consists of 8 women and 8 men.
The club has a president and a vice president.
No club member can hold more than one position.

Quantity A Quantity B
The number of possible assignments such that a woman is president and a man is vice president The number of possible assignments such that both the president and vice president positions are filled by women
  • Solution

    For Quantity A, there are 8 options for president and 8 options for vice president, giving you 8 × 8 = 64 total assignments. For Quantity B, once you pick a woman to be president, there are only 7 women left to be vice president, giving you 8 × 7 = 56 assignments.The answer is choice (A).

Three digits have been removed from each of the following numbers. If n = 25, which of the numbers is equal to 3 × 2n–1?

  • Solution

    The equation 3 × 2n–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).

f(x) = 3x2

g(x) = x + 1

x is an integer such that –10 ≤ x ≤ –1.

Quantity A Quantity B
f(g(x)) g(f(x))
  • Solution

    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).

If q is even, then #q = –2;

If q is odd, then #q = –4.

a and b are integers such that b – 3 is odd.

Quantity A Quantity B
#(6a) #b
  • Solution

    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).

Of the employees at a company, 60 percent were men and, of these,\(\frac{1}{10}\) were still employed after a recent corporate restructuring. If the number of women who were still employed after the restructuring was five times the number of men who were employed after it, what percent of the women were still employed after the restructuring?

  • Solution

    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).

Quantity A Quantity B
The units digit of 729 The units digit of 327
  • Solution

    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).

Starting with the third term, each term in Sequence S is one-half the sum of the previous 2 terms. If the first 2 terms of Sequence S are 64 and 32, respectively, and the nth term is the first non-integer term of Sequence S, then n =

  • Solution

    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.

For all real numbers a and b, the operation ⊕ is defined by a ⊕ b = 2a – b. What is the absolute value of the difference between (3 ⊕ 1) ⊕ 2 and 6 ⊕ 3?

  • Solution

    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.

In a regular n-sided polygon, the measure of a single angle is \(\frac{(n-2)180^{\circ}}{n}\). The degree measure of an angle in a regular 10-sided polygon is how much greater than the degree measure of an angle in a regular 6-sided polygon?

  • Solution

    24

    Find the measure of an angle in a regular 10-sided polygon by plugging 10 into the given formula: \(\frac{(n-2)180^{\circ}}{n}\) = 144°.Then do the same for a regular 6-sided polygon by plugging 6 into the given formula:\(\frac{(n-2)180^{\circ}}{n}\) =120°. Finally, 144 – 120 = 24.

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