Alheri has three hoses. Water flows out of each hose at the same constant rate. Using all three hoses, she starts to fill a swimming pool at 6:00 a.m. and calculates that the pool will be full at exactly 6:00 p.m. on the same day. At 11:00 a.m., one of the hoses unexpectedly stops working. Assuming water still flows out of the other two hoses at the same rate as before, at what time will the pool be full?
Solution 1

Let $V$ equal the volume of the pool in litres and let $x$ be the rate at which each hose outputs water in litres per hour.

With three hoses, the pool will fill in 12 hours at a rate of $3 x$ litres per hour. This means $3 x=\frac{V}{12}$ or $x=\frac{V}{36}$.

One of the hoses stops working after 5 of the 12 hours, which means that the hose stops working when there are $\frac{5}{12} \mathrm{~V}$ litres in the pool, or $\frac{7}{12} \mathrm{~V}$ more litres to be added to the pool.

Starting at 11:00 a.m., the new rate at which water is entering the pool is $2 x$, and $\frac{7}{12} V$ litres need to be added to the pool.

If we let $t$ be the amount of time in hours remaining to fill the pool with two hoses, then we
have $2 x=\frac{\frac{7}{12} \mathrm{~V}}{t}$ and so $t=\frac{7 \mathrm{~V}}{24 x}$.

Substituting $x=\frac{V}{36}$, we get

$$
t=\frac{7 V}{24 \frac{V}{36}}=\frac{21}{2}=10.5
$$

Therefore, another 10.5 hours are required, so the pool will be full 10.5 hours after 11:00 a.m., or at 9:30 p.m.

## Solution 2

With three hoses working, the pool will fill in 12 hours, which means that one hose could fill the pool in 36 hours. We will say that the pool takes 36 hose-hours to fill.

At the time when one of the hoses stops working, three hoses have been working for 5 hours. In other words, which is the same as if one hose had been working for $3 \times 5=15$ hours. In other words, 15 hose hours have been spent putting water in the pool.

This means $36-15=21$ hose hours are still required. There are two hoses still working, so it will take and additional $\frac{21}{2}=10.5$ hours to fill the pool.

Since 10.5 hours after 11:00 a.m. is 9:30 p.m., the pool will be filled at 9:30 pm.

ANSWER: $9: 30$ p.m.