13.427 After passing through the filter, the liquid is evenly poured into a 40-bucket barrel and can be drained through a tap at the bottom of the barrel. If this tap is open, the inflow and outflow of the liquid are such that every 4 minutes, one bucket is reduced in the barrel. How long will it take for the filtered liquid to fill an empty barrel with the lower tap closed, if it is known that this will take 3 minutes less than the time it takes for the open lower tap to pass 66 buckets?
Solution. Let $x$ and $y$ be the rates (bucket/min) of liquid passing through the filter and the tap, respectively. Then, according to the conditions of the problem,

$$
\begin{aligned}
& \left\{\begin{array} { l } 
{ 4 ( y - x ) = 1 , } \\
{ \frac { 6 6 } { y } - \frac { 4 0 } { x } = 3 }
\end{array} \Rightarrow \left\{\begin{array} { l } 
{ 4 ( y - x ) = 1 , } \\
{ 2 6 x - 4 0 ( y - x ) = 3 x y }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
4(y-x)=1, \\
26 x-10=3 x y
\end{array} \Rightarrow\right.\right.\right. \\
& \Rightarrow 12 x^{2}-101 x+40=0 \Leftrightarrow\left[\begin{array} { l } 
{ x = 8 , } \\
{ x = \frac { 5 } { 1 2 } }
\end{array} \Rightarrow \left[\begin{array}{l}
\frac{40}{x}=5, \\
\frac{40}{x}=96
\end{array}\right.\right.
\end{aligned}
$$

Answer: 96 min or 5 min.