Browse · MATH
Printjmc
prealgebra intermediate
Problem
A list of five positive integers has all of the following properties:
The only integer in the list that occurs more than once is
its median is and
its average (mean) is
What is the largest possible integer that could appear in the list?
The only integer in the list that occurs more than once is
its median is and
its average (mean) is
What is the largest possible integer that could appear in the list?
Solution
We write the list of five numbers in increasing order. We know that the number occurs at least twice in the list. Since the median of the list is then the middle number (that is, the third number) in the list is Thus, the list can be written as
Since occurs more than once and the middle number is then must occur twice only with Thus, the list can be written as
Since the average is and there are numbers in the list, then the sum of the numbers in the list is Therefore, or or
Since is the only integer that occurs more than once in the list, then Thus, and To make as large as possible, we make as small as possible, so we make and so
The list has the desired properties, so the largest possible integer that could appear in the list is
Since occurs more than once and the middle number is then must occur twice only with Thus, the list can be written as
Since the average is and there are numbers in the list, then the sum of the numbers in the list is Therefore, or or
Since is the only integer that occurs more than once in the list, then Thus, and To make as large as possible, we make as small as possible, so we make and so
The list has the desired properties, so the largest possible integer that could appear in the list is
Final answer
15