Explanation

The question is clear: we need to find the middle age of three, given that their average is 60. On to the statements, separately first.

Statement (1) tells us that one age is 55. A possible case is that the other ages are 60 and 65, because then the three elements have differences from the average of -5, 0, and +5, cancelling out. Another possible case is that the other ages are 55 and 70, since the differences in that case are -5, -5, and +10, canceling out. The median in the first case is 60 and the median in the second case is 55, so we don't have sufficient information to answer the question definitively. Statement (1) is insufficient.

Statement (2) tells us that one age is 60. Case I from above is allowed by this statement: 55, 60, 65. There the median is 60. Case II from above isn't allowed here. A possible case is that all the ages are 60; in that case, the median is 60, the same as in Case I. Must the median be 60? If one age is 60, the two other elements must balance each other around 60. That means that, if one is higher than 60, the other must be lower, and vice versa. Therefore, the data stipulates that either all ages are 60 or 60 is the middle age. We can definitively respond to the question posed with the answer "60," so this statement is sufficient.

The correct answer is (B).