13.423 In a brigade of earthworkers, each works the same number of hours daily. It is known that the labor productivity is the same for all workers in the brigade, and the brigade can dig a trench for laying a cable in 6 days. However, before the work began, it was found that the working day was reduced by 1 hour, and the brigade's composition decreased by 5 people. In this case, the trench could be dug in 9 days. In reality, the trench was dug in 12 days because the working day was reduced not by 1 hour but by 2 hours, and two people did not show up for work due to illness. How many workers were initially in the brigade and how many hours did they work?
Solution. Let $x$ be the original number of workers, $y+2$ be the originally planned duration of their working day. Then the volume of work, according to the first condition of the problem, is $6 x(y+2)$, according to the second condition - $9(x-5)(y+1)$; according to the third condition - $12(x-7) y$. Since the volume of work is constant, then $2 x(y+2)=3(x-5)(y+1)=4(x-7) y \Rightarrow$

$\Rightarrow\left\{\begin{array}{l}2 x y+4 x=4 x y-28 y, \\ 2 x y+4 x=3 x y-15 y+3 x-15\end{array} \Rightarrow\left\{\begin{array}{l}14 y=x(y-2), \\ 15(y+1)=x(y-1)\end{array} \Rightarrow\left\{\begin{array}{l}y=6, \\ x=21 .\end{array}\right.\right.\right.$

Answer: 21 people, 6 hours.