A water tank is three-fourth full. Pipe P can fill the empty tank in 20 minutes and pipe Q can empty the full tank in 10 minutes. If both the pipes are open, how long will it take to empty or fill the tank completely?

To solve this problem, we need to determine the rate at which each pipe fills or empties the tank.

Let's start by finding the rate at which pipe P fills the tank. We know that pipe P can fill the empty tank in 20 minutes. Therefore, the rate at which pipe P fills the tank is 1/20 of the tank per minute.

Next, let's find the rate at which pipe Q empties the tank. We know that pipe Q can empty the full tank in 10 minutes. Therefore, the rate at which pipe Q empties the tank is 1/10 of the tank per minute.

Now, let's consider both pipes being open at the same time. Since pipe P fills the tank and pipe Q empties the tank, we need to subtract the rate at which pipe Q empties from the rate at which pipe P fills.

The combined rate of both pipes is (1/20 - 1/10) = (1/20 - 2/20) = -1/20 of the tank per minute.

Since the combined rate is negative, it means that the tank is being emptied. To find out how long it will take to completely empty the tank, we need to invert the rate and multiply it by the tank's capacity.

Since the tank is three-fourth full, it means it is already 3/4 of its capacity. Therefore, the time it will take to completely empty the tank is (3/4) / (1/20) = (3/4) * (20/1) = 60/4 = 15 minutes.

So, if both pipes P and Q are open, it will take 15 minutes to completely empty the tank.