Explanation

The six rates listed are a set of numbers, which we can order and write without the percentage signs:

{11, 12, 13, 14, 16, 17}

Each of the Roman numerals listed in the question is a different case - a different value of the seventh number that goes in the set. Case I: the number that we add into the set is 17. So then, the ordered set is:

{11, 12, 13, 14, 16, 17, 17}

We are asked whether the average, or mean, equals the median. The median is easier to find, so we can start there. The middle number is 14, so that's the median. That will be the average if all the differences of the numbers from 14 cancel out. For example, the 11 and the last 17 cancel out, because they are a -3 and a +3. But the others don't cancel out. They are -2, -1, and +2, +3 so they net to +2, not zero. Therefore Case I is out. The median and mean are different, so Roman numeral I is not in the correct answer.

Case II: the final number is a 15, giving us this set:

{11, 12, 13, 14, 15, 16, 17}

We can see pretty quickly here that the median equals the mean, because 14 is the median, and the other numbers balance out around 14 in pairs - 11 with 17, 12, with 16, and so on. So II is in the correct answer. We have narrowed to answer choices (A) and (C), but we still must look at Case III.

Case III: the last number is 14. That gives us the set

{11, 12, 13, 14, 14, 16, 17}

Again, 14 is the median. So again, the 11 and the 17 cancel each other, as do the 12 and the 16. But the differences from the would-be mean of 14 don't cancel for 13 and 14, so the mean isn't 14. Therefore III is out, and the answer is II only. The correct answer is (A).