Question

Is the number of members of Club X greater than the number of members of Club Y?
(1) Of the members of Club X, 20 percent are also members of Club Y.
(2) Of the members of Club Y, 30 percent are also members of Club X.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

The correct option is C Both statements together are sufficient, but neither statement alone is sufficient.

• Let a be the number of members in Club X that do not belong to Club Y, let b be the number of members in Club Y that do not belong to Club X, and let c be the number of members that belong to both Club X and to Club Y. Determine whether a + c > b + c, or equivalently, whether a > b.

1. If a = 80, b = 79, and c= 20, then 20 percent of the members of Club X are also members of Club Y (because c = 20 is 20 percent of a + c = 100) and a > b is true. However, if a = 80, b = 80, and c = 20, then 20 percent of the members of Club X are also members of Club Y (because c = 20 is 20 percent of a + c =100) and a > b is false. Therefore, it cannot be determined whether a > b; NOT sufficient.
2. If a = 71, b = 70, and c =30, then 30 percent of the members of Club Y are also members of Club X (because c = 30 is 30 percent of b + c = 100) and a > b is true. However, if a = 70, b = 70, and c = 30, then 30 percent of the members of Club Y are also members of Club X (because c
   

= 30 is 30 percent of b + c =100) and a > b is false. Therefore, it cannot be determined whether a > b; NOT sufficient.

• Now assume both (1) and (2). From (1) it follows that , or 5c = a + c, and so a = 4c. From (2) it follows that , or 10c = 3b + 3c, and so 7c = 3b and . Since (from the statements it can be deduced that c > 0), it follows that a > b. Therefore, (1) and (2) together are sufficient.
• The correct answer is C; both statements together are sufficient.