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.
I got that:
Let $x$ be the number of initial clerks, and let $y$ be the number reduced.
$$1775 = 30x + 30(x - y) + 30(x - 2y) + 30(x - 3y)$$
But then, $x$ and $y$ cant be integers?
The equation that you have written above should read: \begin{eqnarray} 1775 &=& 30(x) + 30(x-y) + 30(x-2y) + 30(10/60)(x-3y) \\ \Rightarrow 1775 &=& 95x - 105 y \\ \Rightarrow 355 &=& 19x - 21 y. \end{eqnarray} Since we must also have $(x-3y) > 0$ (i.e., number of clerks working after 3 hours must be positive), we have \begin{eqnarray} x &>& 3y \\ 19x &>& 57y \\ 355 + 21y &>& 57y \\ 36y &<& 355 \\ y &\leq& 9. \end{eqnarray} Since we also need $x = \frac{355 + 21y}{19}$ to be an integer, we can plug in integer values of $y \leq 9$ to determine $x$. This recovers $y = 3, x = 22$.
To answer the original question, in 1.5 hours, the clerks sort $$30x + 30(30/60)(x-y) = 30(22) + 15(19) = 945 \mbox{ files}.$$