question 7 of 13


In the incomplete table above, the sum of the three integers in each row, column, and diagonal is the same. If the numerical values in four of the blocks are as shown, what is the value of w ?

(A) -6
(B) -5
(C) 2
(D) 5
(E) 8
(from the October 17, 1998 test)

Correct Answer: C
Explanation: Since the sum of the integers in each row, column, and diagonal is the same, it follows that w  - 2 + a  = 3 + a  - 3. Thus w  - 2 = 0 so that w  = 2. The answer to this problem is (C).

question 8 of 13

If n  is an odd integer, which of the following must be an odd integer?


(A) n - 1
(B) n + 1
(C) 2n
(D) 3n + 1
(E) 4n + 1
(from the October 20, 1998 test)

Correct Answer: E
Explanation: If n is an odd integer, both one more and one less than n  will be even integers, eliminating choices (A) and (B). Any even multiple of n  will be an even integer, eliminating choice (C). However, 4n  is even, making 4n +1 an odd integer. The answer to this problem is (E). Note that 3n  + 1 is even if n  is odd and it is odd if n  is even. Since the question asks, "Which of the following MUST be an odd integer," (D) cannot be the correct answer.

question 9 of 13

A 19-liter mixture consists by volume of 1 part juice to 18 parts water. If x liters of juice and y  liters of water are added to this mixture to make a 54-liter mixture consisting by volume of 1 part juice to 2 parts water, what is the value of x ?


(A) 17
(B) 18
(C) 27
(D) 35
(E) 36
(from the October 20, 1998 test)

Correct Answer: A
Explanation:

question 10 of 13

If a and b are integers greater than 100 such that a + b = 300, which of the following could be the exact ratio of a to b ?


(A) 9 to 1
(B) 5 to 2
(C) 5 to 3
(D) 4 to 1
(E) 3 to 2
(from the October 12, 1999 test)

Correct Answer: E
Explanation: To solve this question, you need to look at the answer choices. For any of the answer choices to be the ratio of a to b, some multiple of the sum of the two numbers must evenly divide 300. For example, if the ratio of a to b equaled 9 to 1, then a would equal 9x and b would equal x for some number x. Furthermore, 9x + x would have to equal 300. This is possible since 10x = 300 yields an integer solution, namely x = 30. However, if x = 30, then a would equal 270 and b would equal 30. Although the sum of these numbers equals 300, they do not satisfy the other condition in the problem. That is, both of these numbers are not greater than 100. Therefore, choice (A) can be eliminated.

Answer choices (B) and (C) can be eliminated since neither the sum of the two numbers in (B) nor the sum of the two numbers in (C) evenly divided 300. (5x + 2x = 300 does not yield an integer solution, nor does 5x + 3x = 300.)

Although answer choices (D) and (E) are possible ratios of a to b (both 4x + x = 300 and 3x + 2x = 300 yield integer solutions), (D) results in a = 240 and b = 60 and can be eliminated since 60 is not greater than 100.

Only choice (E) gives a correct ratio of a to b that satisfies all of the conditions in the problem. For (E), a = 180 and b = 120, and both integers are greater than 100.

question 11 of 13


In the figure above, a square is inscribed in a circle with diameter d. What is the sum of the areas of the shaded regions, in terms of d ?

(A)  
(B)  
(C)  
(D)  
(E)  
(from the October 12, 1999 test)

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