13.337. At a plant that manufactures instant coffee, a batch of coffee beans for processing was delivered in late May. One mechanism, which grinds the beans, was put into operation on Monday, June 1st, and ground $m$ kg daily. Starting from June 6th, a second mechanism was added, which ground $n$ kg daily. By the end of the working day on June 10th, only half of the initial amount of beans remained unground. When was the processing of the entire batch of beans completed, given that both mechanisms ground an equal amount and, apart from Sundays, there were no other breaks in the work?
Solution.

Let the mass of the entire batch of grains be 1. Let $z$ be the number of days required to grind half of all the grains by two mechanisms simultaneously. According to the condition, we have the system

$$
\left\{\begin{array}{l}
(6-1) m+(10-6)(n+m)=\frac{1}{2}, \\
z(n+m)=\frac{1}{2}, \\
(10-1+z) m=(10-6+z) n,
\end{array} \text { from which } z=6,\right. \text { and all the grains were }
$$

ground by $10-1+6=15$ working days, or by $10+1+6=17$ June.

Answer: in 15 working days, or on June 17.