In the department of Mathematics at Green University, more than half

Lets Say there are 8 professors (for Eg)
Exactly half is 4 people
4*80,000 = 320,000
Average is 320,000/8 = 40,000

Since more than half of the professors earn more than 80,000, The average cannot be 40,000

Answer is E

On what basis did we assume that there are 8 professors ??

This logic seems faulty

Say the number of all the professors is X => the number of professors with income more than $80,000 is X/2
Say the amount of income of the rest is A (A<80,000)
=> Total average income = [(80,000)(X/2) + A(X/2)]/X = 40,000 + A/2 > 40,000
not to mention that half of the professors may even earn more than $80,000, so the actual Total average income equals [(more than 40,000) + A/2]
Hence E

The question can be answered by making two simple observations:

1) The question gives us a lower limit for the income of some of the professors, but no upper limit. Therefore, if some value in the answer choices is not a possibility for the average annual income, it must be because it is too low.

2) If one value is not a possibility for the average annual income, than any value smaller than that is also not possible.

Since there can be only one answer choice, it must be the smallest value provided in the answer choices, which is $40,000.

Alternate Solution:

Let’s consider an extreme case where there are 20 professors, 11 of which earn $80,001 and 9 of which earn $0. In this case, the average salary will be (80,001 x 11)/20 = 880,011 / 20 ≈ 44,000. As we can see, anything greater than or equal to $45,000 is possible, which leaves $40,000 as the only remaining answer choice.

Answer: E
_________________

Bunuel,

Please elaborate the solution.


Thanks