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).

_________________

"Please hit +1 Kudos if you like this post"

_________________

Manish

"Only I can change my life. No one can do it for me"