4. Early in the morning, the pump was turned on and they started filling the pool. At 10 am, another pump was connected and by 12 pm the pool was half full. By 5 pm, the pool was full. What could be the latest time the first pump was turned on?
Solution. Let the volume of the pool be $V$. Denote by $x$ and $y$ the capacities of the pumps, and by $t$ the time the first pump operates before the second pump is turned on.

Then $t x + 2 x + 2 y = V / 2.5 x + 5 y = V / 2$.

From this, $t x + 2 x + 2 y = 5 x + 5 y$ or $t x = 3 x + 3 y$.

In the end, $t = 3 + 3 y / x$. Since $x > 0$ and $y > 0$, then $t \geqslant 3$.

Answer. The first pump was turned on no later than 7 a.m.