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.
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a)none is less than 75 so all numbers are 75 as averagege is 75
b) Noe is greater than 75 so all numbers are 75 as average is 75. If any number become less than 75 then average will not 75 it will less than 75.
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If the average (arithmetic mean) of six numbers is 75, how many of the numbers are equal to 75?

\(Sum = 75 * 6\)

(1) None of the six numbers is less than 75.

If none of the six numbers is less than 75, then we can have only one possibility that all the 6 numbers are 75

Hence, (1) ===== is SUFFICIENT

(2) None of the six numbers is greater than 75.

If none of the six numbers is greater than 75, then we can have only one possibility that all the 6 numbers are 75

Hence, (2) ===== is SUFFICIENT

Hence, Answer is D
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We are given that the average of 6 numbers in a list is 75. Using this information we see that the sum of the 6 numbers is 6 x 75 = 450. We need to determine how many numbers in the list are equal to 75.

Statement One Alone:

None of the six numbers is 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 in order for the 6 numbers to sum to 450.

For example, 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 is if one of the 75s were reduced to 74. However, this is not possible, according to the information provided in statement one. Thus, all the numbers in the list must equal 75. Statement one is sufficient to answer the question.

Statement Two Alone:

None of the six numbers is greater than 75.

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

Once again, we know that the sum of the 6 numbers is 6 x 75 = 450.

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 are less than 75, we would need another number to be greater than 75 in order for the 6 numbers to sum to 450.

For instance, 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 is if one of the 75s were increased to 76. However, this is not possible, according to the information provided in statement two. Thus, all the numbers in the list must equal 75. Statement two is sufficient to answer the question.

Answer: D
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