Manhattan challenge problem of the week -- June 26

If set S = {7, y, 12, 8, x, 9}, is x + y less than 18?

(1) The range of set S is less than 9.

(2) The average of x and y is less than the average of set S.

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Answer - B

(1) INSUFFICIENT: Statement (1) tells us that the range of S is less than 9. The range of a set is the positive difference between the smallest term and the largest term of the set. In this case, knowing that the range of set S is less than 9, we can answer only MAYBE to the question "Is (x + y) <>

Consider the following two examples:Let x = 7 and y = 7. The range of S is less than 9 and x + y <>

Let x = 10 and y = 10. The range of S is less than 9 and x + y > 18, so we conclude NO.

Because this statement does not allow us to answer definitively Yes or No, it is insufficient.

(2) SUFFICIENT: Statement (2) tells us that the average of x and y is less than the average of the set S. Writing this as an inequality:

(x + y)/2 < (7 + 8 + 9 + 12 + x + y)/6

(x + y)/2 < (36 + x + y)/6

3(x + y) <>

2(x + y) <>

x + y <>

Therefore, statement (2) is SUFFICIENT to determine whether x + y <>