Question

In the pipe can fill a swimming pool in nine hours and an outlet pipe can empty pool in 12 hours through an error. Those pipes are left open. How long will it take to build a pool?

Answer 1

To solve the problem, we can determine the rate at which each pipe fills and empties the pool and then combine those rates to find the effective rate.

1. **Filling Pipe Rate**: The filling pipe can fill the pool in 9 hours. Therefore, its rate is:
\[
\text{Filling rate} = \frac{1}{9} \text{ pools per hour}
\]

2. **Outlet Pipe Rate**: The outlet pipe can empty the pool in 12 hours. Therefore, its rate is:
\[
\text{Emptying rate} = \frac{1}{12} \text{ pools per hour}
\]

3. **Combined Rate**: When both pipes are open, the combined rate is:
\[
\text{Combined rate} = \text{Filling rate} - \text{Emptying rate} = \frac{1}{9} - \frac{1}{12}
\]

To combine these fractions, we need a common denominator, which in this case is 36:
\[
\frac{1}{9} = \frac{4}{36} \quad \text{and} \quad \frac{1}{12} = \frac{3}{36}
\]
Now we can subtract:
\[
\text{Combined rate} = \frac{4}{36} - \frac{3}{36} = \frac{1}{36} \text{ pools per hour}
\]

4. **Time to fill the pool**: To find the time it takes to fill one pool at the combined rate of \(\frac{1}{36}\) pools per hour, we can calculate:
\[
\text{Time} = \frac{1 \text{ pool}}{\frac{1}{36} \text{ pools per hour}} = 36 \text{ hours}
\]

Thus, if both pipes are left open, it will take **36 hours** to fill the swimming pool.