One inlet pipe can fill an empty pool in 4 hours, and a drain can empty the pool in 8 hours. How long will it take the pipe to fill the pool if the drain is left open?
Solution
Step 1 ：Let's denote the rate at which the inlet pipe fills the pool as \(inlet\_rate\), which is 1 pool per 4 hours, or \(\frac{1}{4}\) pools per hour.
Step 2 ：Similarly, we denote the rate at which the drain empties the pool as \(drain\_rate\), which is 1 pool per 8 hours, or \(\frac{1}{8}\) pools per hour.
Step 3 ：If both the inlet pipe and the drain are open, the net rate at which the pool is filled is the rate of the inlet pipe minus the rate of the drain. We denote this as \(net\_rate\), so \(net\_rate = inlet\_rate - drain\_rate = \frac{1}{4} - \frac{1}{8} = \frac{1}{8}\).
Step 4 ：We can then find the time it takes to fill the pool by dividing the total volume of the pool (1 pool) by the net rate. We denote this as \(time\_to\_fill\), so \(time\_to\_fill = \frac{1}{net\_rate} = \frac{1}{\frac{1}{8}} = 8\) hours.
Step 5 ：Final Answer: It will take \(\boxed{8}\) hours for the pipe to fill the pool if the drain is left open.
