Set S contains seven distinct integers. The median of set S is the integer m, and all values in set S are equal to or less than 2m. What is the highest possible average (arithmetic mean) of all values in set S ?
A)m
B)10m/7
C)10m/7 - 9/7
D)5m/7 + 3/7
E)5m
Here's an algebraic approach:
When we arrange all 7 values in ASCENDING order, with m as the MEDIAN, we get: _ _ _ m _ _ _
All values in set S are equal to or less than 2m
Since we are trying to MAXIMIZE the average, we'll take 2m as a value in set S
So, we get: _ _ _ m _ _ 2m
At this point, we are tying to MAXIMIZE the other values AND make sure all are DISTINCT.
The 2nd biggest value is 2m - 1. So, we get: _ _ _ m _ 2m-1, 2m
The 3rd biggest value is 2m - 2. So, we get: _ _ _ m, 2m-2, 2m-1, 2m
The remaining values must be less than m.
When MAXIMIZING these values, we get: m-3, m-2, m-1, m, 2m-2, 2m-1, 2m
The average = [(m-3)+(m-2)+(m-1)+(m)+(2m-2)+(2m-1)+(2m)]/7 =
= (10m - 9)/7
= [spoiler]10m/7 - 9/7[/spoiler]
Answer:
C
Cheers,
Brent
