An inlet pipe on a swimming pool can be used to fill the pool in 40 hours. The drain pipe can be used to empty the pool in 42 hours. If the pool is $\frac{2}{3}$ filled and then the inlet pipe and drain pipe are opened, how long from that time will it take to fill the pool?

Answer
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Step 1 ：The problem involves rates of work. The rate of work of the inlet pipe is 1 pool per 40 hours, and the rate of work of the drain pipe is -1 pool per 42 hours (negative because it's emptying the pool). When both pipes are open, their rates of work add together.

Step 2 ：The pool is initially \( \frac{2}{3} \) filled, so we need to find out how long it takes for the combined pipes to fill the remaining \( \frac{1}{3} \) of the pool.

Step 3 ：We can set up the equation as follows: \( \frac{1}{40} - \frac{1}{42} \) * t = \( \frac{1}{3} \), where t is the time it takes to fill the remaining \( \frac{1}{3} \) of the pool.

Step 4 ：Solving this equation for t, we get t = 280 hours.

Step 5 ：This means it will take 280 hours to fill the remaining \( \frac{1}{3} \) of the pool when both the inlet and drain pipes are open.

Step 6 ：Final Answer: \( \boxed{280} \) hours.