Pipe P alone fills a tank in 50 minutes and pipe Q alone empties a full tank in 75 minutes. Both are opened on an empty tank, and Q is shut the moment the tank is half full. How long in all does the tank take to fill?

Take the tank as 150 units: P gives 3 units a minute, Q takes 2, a net gain of 1. The first 75 units therefore take 75 minutes. With Q shut, P alone covers the remaining 75 units at 3 a minute in 25 minutes. Total 75 + 25 = 100 minutes. 150 minutes leaves Q open throughout. 50 minutes ignores Q altogether. 125 minutes gets the first half right but then adds P's whole 50-minute solo time instead of the 25 minutes the second half actually needs.