If 0

Question

If 0 < a < b < c, does the median of a, b, and c equal the mean of a, b, and c?

(1) The range of a, b, and c equals 2(c - b).
(2) The sum of a, b, and c is three times one of the numbers a, b, or c.

amanvermagmat · Answer

Here a, b, c are unequal positive numbers such that a < b < c. So their median is definitely 'b'. We need to know whether their mean is also equal to 'b' or not.

(1) Range of a, b, c = c - a. This is given as 2(c-b). So c - a = 2c - 2b or c = 2b - a or c + a = 2b. 
Their mean = (a+b+c)/3 = (2b+b)/3 = b. So the mean is same as median. Sufficient.

(2) Sum = a+b+c. Since the numbers are unequal and sum is three times one of these, it could only be equal to three times b (since b is the middle number). 
So a+b+c = 3b or a+c = 2b. Same as in first statement, so mean = b, same as median. Sufficient.

Hence  D answer

push12345 · Answer

Mean would be equal to median if sequence is in AP

I)c-a=2(c-b)
Solving this we get 
2b=c+a
Which is basic condition of AP
Sufficient

II)a+b+c=3a or 3b or 3c
Which will yield to a+c=2b or a+b=2c or b+c=2a
In every case basic condition of AP is fulfilled
Sufficient

Answer D

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kanishaksharma · Answer

1) c-a=2(c-b)
 (c-a)/2 = b , which means b is average mean of a and c, and also is median.
2) first case: a+b+c=3a (Not possible because for that to happen b and c either need to be equal to a or less than a)
 second case: a+b+c=3b
 a+c=2b (again, it means b is average mean of a and c)
 third case: a+b+c=3c (Not possible since for that to happen a and b needs to be at least equal to c, which is not true)
Hence either of the statement is sufficient.

andrenoble · Answer

Hi  Bunuel ! THank you for the question, it's a really interesting one! My only concern is that (2) in 2 cases contradicts the initial statement in the question, because if a < b < c, then a + b + c != 3a or 3c. Is there a way to rephrase the question to eliminate this confusion?

Bunuel · Answer

Yes, because 0 < a < b < c, then from (2) we get that a + b + c = 3b only but where is the contradiction here?

If 0 < a < b < c, does the median of a, b, and c equal the mean of a,

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