A pipe can fill a tank in 9 hours. Another pipe can empty the filled tank in 27 hours. If both the pipes are opened simultaneously, then the time (in hours) in which the tank will be two-third filled, is:

Given:

Time taken by the filling pipe to fill the tank = 9 hours

Time taken by the emptying pipe to empty the tank = 27 hours

We need to find the time taken to fill two-third of the tank when both pipes are open.

Formula Used:

Efficiency = Total Work / Time taken

Net efficiency when both pipes are open = Efficiency of filling pipe - Efficiency of emptying pipe

Time = Total Work / Net Efficiency

Calculation:

LCM of 9 and 27 = 27. Let the Total Work (capacity of the tank) = 27 units.

Efficiency of the filling pipe = 27 / 9 = 3 units/hour (positive as it fills the tank).

Efficiency of the emptying pipe = 27 / 27 = 1 unit/hour (negative as it empties the tank).

Net efficiency when both pipes are open = Efficiency of filling pipe - Efficiency of emptying pipe = 3 - 1 = 2 units/hour.

Amount of work to be done (two-third of the tank) = (2/3) × Total Work = (2/3) × 27 = 18 units.

Time taken to fill 18 units with a net efficiency of 2 units/hour = Amount of work / Net efficiency = 18 / 2 = 9 hours.

The tank will be two-third filled in 9 hours when both pipes are opened simultaneously.