one inlet of pipe can fill an empty poolin 12 hours,and a drain can empty the pool in 15 hours. How long will it take to fill the pool if the drain is left open?

Question

mww · Answer

This one you will need to write a series of equations to represent the in flow and out flow.
Consider the pool to be filled. Then volume at any time is given by rate of flow x time. Assuming this rate is constant,
$$V(t ) = R _{flow IN} t$$
The full volume occurs when t = 12 hours so the volume of the pool is given by
$$V = 12R _{flow IN}$$

What about flow out? 
Do the same thing. You R will be a different constant and it will take 15 hours. 
$$V = R _{flow out} t = 15R _{flow out} $$

But the volume has to be the same, thus you can solve these equations simultaneously to get a ratio of the rate constants (or simply express one in terms of the other). $$\frac{ R _{flow IN} }{ R _{flow out} }$$

Second part of this question is about putting these two together.
Water comes in and out at the same time. So the volume at any time must be volume in - volume out. Use the starting equations for V(t) in and out and sub in your rate constants and you should be able to get a precise number.