\section*{Task 2 - 330922}

To mill a quantity of grain, two milling machines \(A\) and \(B\) can be used. Each milling machine processes the same amount in the same time.

If machine \(A\) is used for 8 hours first and then only machine \(B\) is used, \(B\) would need exactly 18 more hours to complete the entire grain quantity. If, however, machine \(A\) is used for 10 hours first and then only machine \(B\) is used, \(B\) would need exactly 15 more hours to complete the entire quantity.

How long will it take to complete the entire quantity if \(A\) and \(B\) are used together from the beginning?
Let \(x\) and \(y\) be the hourly performance percentages of the mills. Then the system is given by

\[
\frac{8}{x}+\frac{18}{y}=1 \quad \text { and } \quad \frac{10}{x}+\frac{15}{y}=1
\]

with the solution \(x=20, y=30\). If both work together for \(t\) hours, then \(\frac{t}{20}+\frac{t}{30}=1\) with the solution \(t=12\). Both mills need 12 hours if they work together from the start.