Data Sufficiency

treker wrote: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.

Since the average of the six numbers is 75, the sum of the six numbers is 6 x 75 = 450. We need to determine how many of the six numbers are equal to 75.

Statement One Alone:

None of the six numbers are less than 75.

Recall that the sum of the 6 numbers is 6 x 75 = 450.

Using the information in statement one, we know that all the numbers in the list must be greater than or equal to 75.

However, if any of the numbers in our list are greater than 75, we would need another number to be less than 75 for the 6 numbers to sum to 450. Let's test this theory.

Let's say we had the following 6 numbers:

75, 75, 75, 75, 75, 76

The sum of these numbers is 451, which is greater than the required sum of 450.

The only way to get our list to sum to 450 would be to reduce one of the 75s to 74. However, since we know that none of the six numbers are less than 75, we cannot have a value of 74. Thus, all the numbers in the list must equal 75. Statement one is sufficient to answer the question. We can eliminate answer choices B, C, and E.

Statement Two Alone:

None of the six numbers are greater than 75.

We can apply similar logic to the information in statement two as we did in statement one.

Using the information in statement two, we know that all the numbers in the list must be less than or equal to 75.

Thus, if any of the numbers in our list were less than 75, we would need another number to be greater than 75 for the 6 numbers to sum to 450. Let's test this theory.

Let's say we had the following 6 numbers:

75, 75, 75, 75, 75, 74

The sum of these numbers is 449, which is less than the required sum of 450.

The only way to get our list to sum to 450 would be to increase one of the 75s to 76. However, since we know that none of the six numbers are greater than 75, we cannot have a value of 76. Thus, all the numbers in the list must equal 75. Statement two is sufficient to answer the question.

Answer: D