daftypatty wrote:
Which of the following CANNOT be the median of the four positive integers a, b, c, and d ?
(A) c
(B) d
(C) (a+d)/2
(D) (a+b+c)/2
(E) a + b + d
Where is this question from? It has two correct answers. a+b+d cannot be the median, as Karishma explains above, but (a+b+c)/2 cannot be the median either, if our numbers are all positive.
I won't go through a complete proof, since it's not the kind of thing you need to do on the GMAT, but if you wanted to prove it you'd consider two cases:
- first assume that two out of the three letters a, b and c are the two "middle numbers". Set the median equal to (a+b+c)/2. You'll find one letter must equal zero, which we know is impossible, so answer D cannot be the median in this case;
- only one possibility remains: assume that the fourth letter, d, is one of the two "middle numbers", and assume some other letter, say c, is the other "middle number" (it won't matter which one - you could repeat the proof identically using any letter). Again set the median equal to (a + b + c)/2. You'll find that d = a + b. But that means, if a and b are positive, that d is larger than both a and b, and if that's true, d and c cannot be the two "middle numbers" in the set. So we can't make answer choice D the median in this case either.
That second case gets a bit technical to explain fully, so I've just outlined it in enough detail to hopefully help anyone attempting it on their own. I feel like there's probably an easier way to prove this that I might not be seeing so early in the morning, but the question is flawed unless there's a typo among the answer choices.
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