Bunuel wrote:
Five logs of wood have an average length of 100 cm and a median length of 116 cm. What is the maximum possible length, in cm, of the shortest piece of wood?
(A) 50
(B) 76
(C) 84
(D) 96
(E) 100
Kudos for a correct solution.
VERITAS PREP OFFICIAL SOLUTION:First thing that comes to mind – median is the 3rd term out of 5 so the lengths arranged in increasing order must look like this:
___ ___ 116 ___ ___
The mean is given and we need to maximize the smallest number. Basically, the smallest number should be as close to the mean as possible. This means the greatest number should be as close to the mean as possible too (if the shortfall deviation is small, the excess deviation should by equally small).
If this doesn’t make sense, think of a set with mean 20:
19, 20, 21 (smallest number is very close to mean; greatest number is very close to the mean too)
1, 20, 39 (smallest number is far away from the mean, greatest number is far away too)
Using the same logic, let’s make the greater numbers as small as possible (so the smallest number can be as large as possible). The two greatest numbers should both be at least 116 (since 116 is the median). Now the lengths arranged look like this:
___ ___ 116 116 116
Since the mean is 100 and each of the 3 large numbers are already 16 more than 100 i.e. total 16*3 = 48 more than the mean (excess deviation is 48), the deviations of the two small numbers should be a total of 48 less than the mean. To make the smallest number as great as possible, each of the small numbers should be 48/2 = 24 less than the mean i.e. they both should be 76.
Answer (B).