06/06/2020 (2011: National Sprint Round, Problem 18)

Q: The median k of a set of five numbers is three times the mean of the five numbers. If the median is removed from the set, the mean of the remaining numbers is what fraction of k? Express your answer as a common fraction.

A: 1/6

If the numbers in the set are: a, b, k, c, d, then k = 3(a + b + k + c + d) ÷ 5 -->

k = (3a + 3b + 3k + 3c + 3d) ÷ 5 --> (2/5)k = (3/5)(a + b + c + d)

That simplifies to: (2/3)k = (a + b + c + d) --> (a + b + c + d) ÷ 4 numbers = (2/3)k ÷ 4 --> average of (a + b + c + d) = 2/12 = 1/6