13.287. Two dairy factories together should process an equal amount of a certain number of liters of milk. The second workshop started working on the task $a$ working days later, but processed $m$ liters of milk more daily than the first. After another $5a/9$ working days from the start of the joint work of these workshops, $1/3$ of the entire task remained uncompleted. How many working days were required to complete the task, if the work was completed simultaneously and each workshop processed half of the assigned amount of liters of milk?
Solution.

Let's take the entire volume of work as 1. Let $x$ be the labor productivity of the first workshop per 1 hour, and $n$ be the number of days required to complete the task. According to the conditions (1) $n x=(n-a)(x+m)$, (2) $a x+\frac{5 a}{9} x+\frac{5 a}{9}(x+m)=1-\frac{1}{3}$ (hours). From (1), $m=\frac{a x}{n-a}$, substituting into (2) we find $n=\frac{14 a^{2} x-6 a}{19 a x-6}$ (3). Since each workshop processed half of the assigned amount of milk, then $n x=\frac{1}{2}$, from which $x=\frac{1}{2 n}$ (4). Substituting (4) into (3), we find $n=2 a$.

Answer: $2 a$.