## Task 2 - 100832

Pump $P_{1}$ fills a basin in exactly $4 \mathrm{~h} 30 \mathrm{~min}$. A second pump $P_{2}$ fills the same basin in exactly $6 \mathrm{~h} 45 \mathrm{~min}$. One day, when filling this basin, pump $P_{1}$ was initially used alone for exactly $30 \mathrm{~min}$. Subsequently, both pumps were used together until the basin was filled.

Calculate how long it took in total to fill the basin under these conditions! (It is assumed that both pumps operated at a constant performance during their operation.)
Since $P_{1}$ fills the entire basin in exactly 4 h 30 min, this pump filled exactly $\frac{1}{9}$ of the basin in 30 min. Thus, $P_{1}$ filled exactly $\frac{1}{270}$ of the basin each minute.

Since $P_{2}$ fills the entire basin in exactly 6 h 45 min, which is 405 min, this pump filled $\frac{1}{405}$ of the basin each minute.

During the time both pumps worked together, they filled exactly $\frac{1}{162}$ of the basin each minute because

$$
\frac{1}{270}+\frac{1}{405}=\frac{15}{2430}=\frac{1}{162}
$$

In total, both pumps together filled $\frac{8}{9}$ of the basin.

Since $\frac{8}{9}=\frac{144}{162}$, this happened in exactly 144 min. Consequently, the basin was filled in the manner specified in the problem in exactly 174 min, which is 2 h 54 min.