1997 AJHSME Problems/Problem 14
Problem
There is a set of five positive integers whose average (mean) is 5, whose median is 5, and whose only mode is 8. What is the difference between the largest and smallest integers in the set?
Solution
Call the set , with .
Since the median, or middle number, is , we have .
Since the mode, or most common number, is , we have .
Thus, the set is now . Additionally, since there is only one mode, and are both less than and are distinct from each other.
Since the mean of the set is , and there are five numbers in the set, the sum of the numbers in the set is .
Thus, we have , which leads to
Since and are distinct positive integers, they must equal and .
The smallest number is , and the largest number is , giving a difference of , leading to answer
See also
1997 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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