Question

A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ 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 $3$ hours and $10$ minutes. Find the number of files sorted during the first one and a half hours of sorting.

答案

Solution:

Let $n$ clerks start and $k$ be reassigned at the end of each hour. In the final $10$ minutes each remaining clerk sorts $5$ files, so $30n + 30(n - k) + 30(n - 2k) + 5(n - 3k) = 1775,$ which simplifies to $95n - 105k = 1775,$ or $19n - 21k = 355.$

Modulo $19$ this reads $-2k \equiv 13,$ so $2k \equiv 6$ and $k \equiv 3 \pmod{19}.$ Taking $k = 3$ gives $n = \frac{355 + 63}{19} = 22;$ the next candidate, $k = 22,$ gives $n = 43,$ for which $n - 3k \lt 0.$ So $n = 22$ and $k = 3.$

In the first hour $22$ clerks sort $30 \cdot 22 = 660$ files, and in the next half hour $19$ clerks sort $15 \cdot 19 = 285,$ for a total of $660 + 285 = 945.$

2013 AIME II Problem 7

Solution:

Problem 7 in Other Years