Pipes X, Y, and Z can fill a tank in 20, 40, and 80 minutes, respectively. Pipes Y and Z are kept open for 15 minutes. Then, pipe Y is shut while pipe X is opened, and pipe Z is closed 5 minutes before the tank overflows. How long does it take to fill the tank (in minutes)?

In this problem, we need to understand the concept of rates. If a pipe can fill a tank in 'x' minutes, then in one minute, the pipe can fill 1/x of the tank. Therefore, pipe x can fill 1/20 of the tank in one minute, pipe y can fill 1/40 of the tank in one minute, and pipe z can fill 1/80 of the tank in one minute.

According to the problem, pipes y and z are open for 15 minutes, hence they can fill 15*(1/40 + 1/80) = 15/40 = 3/8 of the tank. Now, pipe y is shut but pipes x and z are open, but z is closed 5 minutes before the tank overflows. Therefore, x and z together fill the remaining (1-3/8) = 5/8 of the tank. Pipe x can fill (1/20)*5 = 1/4 of the tank in 5 minutes. As the tank is not yet full, remaining part of the tank has to be filled by pipe x alone. It takes 20*1/4 = 5 more minutes. Hence, the total time taken to fill the tank is 15 (y and z) + 5 (x and z) + 5 (x alone) = 25 minutes.

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