Problem:
A group of clerks is assigned the task of sorting files. Each clerk sorts at a constant rate of files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar reassignment occurs at the end of the third hour. The group finishes the sorting in hours and minutes. Find the number of files sorted during the first one and a half hours of sorting.
Solution:
Let represent the original number of clerks and the number of clerks reassigned at the end of each hour. Because a clerk sorts one file every minutes, in minutes each clerk sorts files, so . Then and . Thus , and because and must both be positive integers, must be a multiple of . The only such value of for which , and are also positive integers is . Then , and the number of files sorted in an hour and a half is .
The problems on this page are the property of the MAA's American Mathematics Competitions