Question

If the average (arithmetic mean) of six numbers is 75, how many of the numbers are equal to 75 ?
(1) None of the six numbers is less than 75.
(2) None of the six numbers is greater than 75.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

Answer: (D)

Each statement alone is sufficient.

If the average of six numbers is 75, then of the sum of the numbers is 75. Therefore, the sum of the numbers is (6) (75).

1. If one of the numbers is greater than 75, then we can write that number as 75 + x. for some positive number x. Consequently, the sum of the 6 numbers must be at least (5) * (75) + (75 + x) = (6) * (75) + x, which is greater than (6) (75), contrary to the fact that the sum is equal to (6) (75). Hence, none of the numbers can be greater than 75. Since none of the numbers can be less than 75 (given information) and none of the numbers can be greater than 75, it follows that each of the numbers is equal to 75; SUFFICIENT.

2. If one of the numbers is less than 75, then we can write that number as 75 - x for some positive number x. Consequently, the sum of the 6 numbers must be at most (5) * (75) + (75 - x) = (6) * (75) - x, which is less than (6) (75), contrary to the fact that the sum is equal to (6) (75). Hence, none of the numbers can be less than 75. Since none of the numbers can be less than 75 and none of the numbers can be greater than 75 (given information), it follows that each of the numbers is equal to 75; SUFFICIENT.

• The correct answer is D; each statement alone is sufficient.