A tank can be filled by pipe P in 8 hours and by pipe Q in 4 hours. If they open alternately for one hour each with Q starting first, what is the total time taken to fill the tank?
5 hours
In one hour P fills \(\dfrac{1}{8}\) of the tank and Q fills \(\dfrac{1}{4}\) of the tank.
Since they open alternately with Q first, each 2-hour cycle (Q then P) fills \(\dfrac{1}{4} + \dfrac{1}{8} = \dfrac{3}{8}\).
After 2 full cycles (4 hours) the tank is filled by \(2 \times \dfrac{3}{8} = \dfrac{6}{8} = \dfrac{3}{4}\).
Remaining part \(= 1 - \dfrac{3}{4} = \dfrac{1}{4}\). The 5th hour is Q's turn, and Q fills exactly \(\dfrac{1}{4}\) in one hour.
So the tank fills completely during that 5th hour: total time \(= 4 + 1 = 5\) hours.
Hence, the total time taken to fill the tank is 5 hours.
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