One pipe can fill the tank in 9 minutes while another pipe can empty completely filled tank in 90 minutes. If both the pipes are operated together on empty tank, how long (in minutes) will it take to fill half of the tank?

Given:

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

Time taken by emptying pipe to empty the tank = 90 minutes

Formula used:

Total Capacity (units) = LCM of individual times

Efficiency (units/minute) = Total Capacity / Time taken

Net Efficiency = Efficiency of filling pipe - Efficiency of emptying pipe

Time = Work / Rate = Total Capacity / Net Efficiency

Calculations:

LCM of 9 and 90 = 90 units (Let's assume the tank capacity is 90 units)

Efficiency of filling pipe = \(\frac{90}{9}\) = 10 units/minute

Efficiency of emptying pipe = \(\frac{90}{90}\) = 1 unit/minute

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

⇒ Net efficiency = 10 - 1

⇒ Net efficiency = 9 units/minute

Work to be done = Half of the tank capacity

⇒ Work to be done = \(\frac{90}{2}\) = 45 units

Time taken to fill half of the tank = Work to be done / Net efficiency

⇒ Time taken = \(\frac{45}{9}\)

⇒ Time taken = 5 minutes

∴ It will take 5 minutes to fill half of the tank.