If the average (arithmetic mean) of 13, 17, x, 12 and 16 is greater

We are given that the average of 13, 17, x, 12, and 16 is greater than 15. Let’s use the formula for calculating the average (average = sum/number of items) to express this.
(13 + 17 + x + 12 + 16)/5 > 15

(58 + x)/5 > 15

58 + x > 75

x > 17

Let’s now analyze each Roman numeral.

I. x is the median of the numbers.

Since x > 17, we see that x must be the largest number in the list when we order the numbers from least to greatest:

12, 13, 15, 16, 17, x

Since x is the largest number, Roman numeral I is not true.

II. x is equal to 15.

Since x > 17, x cannot be 15. Roman numeral II is not true.

III. x is greater than the average of the numbers.

Since we’ve shown that x is the largest number in the list, it has to be greater than the average of the numbers. Roman numeral III is true.

Answer: D