Carcass wrote:
100 jellybeans were distributed to a group of 9 people such that the 3 people with the most jellybeans have 60 jellybeans among them, and no one has fewer than 5 jellybeans. What is the maximum possible ratio of the number of jellybeans held by the person with the most to number of jellybeans held by the person with the least?
Give your answer as a fraction.
Let a, b, c, d, e, f, g, h, i represent the nine people receiving jelly beansLet's also say that, when we arrange the values in
ascending order we get:
a ≤ b ≤ c ≤ d ≤ e ≤ f ≤ g ≤ h ≤ i Our goal is to MAXIMIZE the value of
i/aTo do this, we must MAXIMIZE the value of i and MINIMIZE the value of a
GIVEN: no one receives fewer than 5 jellybeansSo, 5 is the MINIMUM value of a
We get:
5 ≤ b ≤ c ≤ d ≤ e ≤ f ≤ g ≤ h ≤ i The 3 people with the most jellybeans have 60 jellybeans among themWe can write:
g + h + i = 60This also means that the 6 people with fewer jelly beans must have 40 jellybeans among them.
We can write:
5 + b + c + d + e = 40 IMPORTANT: We need values of b, c, d, e, and f that both satisfy the above equation AND minimize the value of f.
We need to minimize the value of f, so that we can minimize the values of g, h and i
The best we can do is as follows:
5 ≤ 7 ≤ 7 ≤ 7 ≤ 7 ≤ 7 ≤ g ≤ h ≤ i When we minimize the values of g and h, we get:
5 ≤ 7 ≤ 7 ≤ 7 ≤ 7 ≤ 7 ≤ 7 ≤ 7 ≤ i Finally, since the sum of all 9 numbers is 100, i must equal
46We get:
5 ≤ 7 ≤ 7 ≤ 7 ≤ 7 ≤ 7 ≤ 7 ≤ 7 ≤ 46 So the maximum value of
i/a =
46/5Answer: 46/5
Cheers,
Brent