Problem:
There is a set of five positive integers whose average (mean) is , whose median is , and whose only mode is . What is the difference between the largest and smallest integers in the set?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Since is the median, there must be two integers greater than and two less than . The five integers may be arranged this way: . Since the mean is , the sum of all five integers is . The numbers , and total , leaving for the sum of the first two. They can't both be since is the only mode, so they are and . The difference between and is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions