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27 Nov 2009, 15:39
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The follwing question seems fairly easy to me, but my answer differs from the OA. I believe the OA is wrong as the authors make an unstated assumption in the detailed solution. I would like to have your opinion.
The crew of the Voelkle Mountain Observatory traks the outdoor temperature for x days. If the temperature increases of exactly 1 degree per day, and the sum of all the temperatures is 0, then which of the following must be true?
I. x represents an even number of days.
II. x represents an odd number of days.
II. the average temperature for the x days is 0
(A) I only
(B) II only
(C) I and III
(D) II and III
(E) None of the above
OA
My answer
The crew of the Voelkle Mountain Observatory traks the outdoor temperature for x days. If the temperature increases of exactly 1 degree per day, and the sum of all the temperatures is 0, then which of the following must be true?
I. x represents an even number of days.
II. x represents an odd number of days.
II. the average temperature for the x days is 0
(A) I only
(B) II only
(C) I and III
(D) II and III
(E) None of the above
OA
My answer
E. The solution makes the unstated assumption that the temperature on day 1 is an integer value. The only option that is certainly true is III, but this is not one of the answers.
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27 Nov 2009, 16:57
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Let's deal with the easy statement first.
III. the average temperature for the x days is 0.
We are told that the sum of all temperatures for x days is 0. Average=Sum/x=0/x=0. Hence this statement is always true no matter temperatures are integers or not.
Now, if the temperatures per x days have integer values it means that we have consecutive integers and the number of terms is x. In consecutive integers average=median, and we know that the average=0=median, which means that integers are symmetric to 0. Hence x is odd. So II is always true, again if we assume that temperatures have integer values.
But if we are not told that temperatures have integer values, the above won't be always true. We could have the following temperatures:
-1.5, -0.5, 0.5, 1.5.
The sum is 0, but x=4, so II must not be always true.
I would say that if stem doesn't specify that temperatures have integer values answer should be E. As only III is always true and we don't have this option.
III. the average temperature for the x days is 0.
We are told that the sum of all temperatures for x days is 0. Average=Sum/x=0/x=0. Hence this statement is always true no matter temperatures are integers or not.
Now, if the temperatures per x days have integer values it means that we have consecutive integers and the number of terms is x. In consecutive integers average=median, and we know that the average=0=median, which means that integers are symmetric to 0. Hence x is odd. So II is always true, again if we assume that temperatures have integer values.
But if we are not told that temperatures have integer values, the above won't be always true. We could have the following temperatures:
-1.5, -0.5, 0.5, 1.5.
The sum is 0, but x=4, so II must not be always true.
I would say that if stem doesn't specify that temperatures have integer values answer should be E. As only III is always true and we don't have this option.
28 Nov 2009, 07:35
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As the temperature increases uniformly every day by 1 degree and the sum is zero, implies there are equal number of numbers on either side of the zero. So the total number is always odd.
As the sum is zero => mean is also zero (mean=sum/n, where n is the total number of elements).
Hence the answer D
As the sum is zero => mean is also zero (mean=sum/n, where n is the total number of elements).
Hence the answer D
