Set S contains exactly four distinct positive integers. Is t

Question

Set S contains exactly four distinct positive integers. Is the mean of S equal to the median of S?

(1) The smallest term in S is equal to the sum of the two middle terms minus the largest term in S.
(2) When the range of S is added to the sum of all the terms in S, the resulting sum is equal to the smallest term in S plus three times the largest term in S.

Marcab · Answer

Let the 4 distinct integers be w,x,y and z, where these are arranged in ascending order.
Mean of S=(w+x+y+z)/4
Median of S= (x+y)/2
The question is asking whether (w+x+y+z)/4=(x+y)/2? or is (w+x+y+z)=2x +2y or Is  w+z=x+y?

Statement 1)
 w=x+y-z  or  w+z=x+y .
Sufficient

Statement 2)
Range=z-w
When this range is added to sum of all the terms in S, then the resulting sum is equal to the smallest term in S plus three times the largest term in S.
(z-w)+(w+x+y+z)= w+3z 
The above can be rewritten as (x+y+2z)=w+3z or  x+y=w+z .
This is what statement 1 is saying.
Sufficient.

IMO the answer has to be D and not C.

daviesj · Answer

oh yes...D is the answer...

...can't edit via mobile though..

Posted from my mobile device

unceldolan · Answer

If mean = median then you have an evenly spaced set. So question is "is S an evenly spaced set?"

(1) This a property of an evenly spaced set. (you can test if you like). Suff.
(2) Also a property of an evenly spaced set. (Test once more..) Suff.

Ans. D.

CAMANISHPARMAR · Answer

Set S contains exactly four distinct positive integers. Is the mean of S equal to the median of S?
(1) The smallest term in S is equal to the sum of the two middle terms minus the largest term in S.
(2) When the range of S is added to the sum of all the terms in S, the resulting sum is equal to the smallest term in S plus three times the largest term in S.

CAMANISHPARMAR · Answer

MANHATTAN GMAT OFFICIAL SOLUTION:

Since the four integers in the set are distinct positive integers, give them distinct ordered
variables to represent their relative sizes: a, b, c, and d are the terms of set S, such that a < b < c < d.
The mean of S is  \frac{(a+b+c+d)}{4} . The median of S is the average of the two middle terms,  \frac{(b+c)}{2}  .
Express the question algebraically:
 \frac{(a+b+c+d)}{4}  =  \frac{(b+c)}{2} 
2(a + b + c + d) = 4(b + c)
(a + b + c + d) = 2(b + c)
a + b + c + d = 2b + 2
Is a + d = b + c?
In words, this question has been rephrased as, “Is the sum of the largest and smallest terms in S equal
to the sum of the two middle terms?”
(1) SUFFICIENT: If the smallest term is equal to the sum of the two middle terms minus the largest
term, then the question is satisfied:
a = b + c − d
a + d = b + c
The answer to the rephrased question is a definite “Yes.”
(2) SUFFICIENT: If the sum of the range of S and all the terms in S is equal to the smallest term in S
plus three times the largest term in S, then:
(d − a) + (a + b + c + d) = a + 3d
b + c + 2d = a + 3d
b + c = a + d
The answer to the rephrased question is a definite “Yes.”
The correct answer is (D).

Bunuel · Answer

________________________
Merging topics. Hope it helps.

bumpbot · Answer

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Set S contains exactly four distinct positive integers. Is t

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