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

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In the department of Mathematics at Green University, more than half  [#permalink]

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New post 02 Jan 2018, 01:20
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In the department of Mathematics at Green University, more than half of the professors have an income in excess of $80,000 a year. Which of the following cannot be the average (arithmetic mean) annual income of the professors in the department?

A. $160,000
B. $120,000
C. $50,000
D. $45,000
E. $40,000

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Re: In the department of Mathematics at Green University, more than half  [#permalink]

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New post 02 Jan 2018, 10:35
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
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Re: In the department of Mathematics at Green University, more than half  [#permalink]

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New post 20 Mar 2018, 01:44
On what basis did we assume that there are 8 professors ??

This logic seems faulty
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Re: In the department of Mathematics at Green University, more than half  [#permalink]

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New post 29 Mar 2018, 03:52
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
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Re: In the department of Mathematics at Green University, more than half  [#permalink]

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New post 05 Apr 2018, 11:06
Bunuel wrote:
In the department of Mathematics at Green University, more than half of the professors have an income in excess of $80,000 a year. Which of the following cannot be the average (arithmetic mean) annual income of the professors in the department?

A. $160,000
B. $120,000
C. $50,000
D. $45,000
E. $40,000


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
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Re: In the department of Mathematics at Green University, more than half &nbs [#permalink] 05 Apr 2018, 11:06
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