Given:
Pipe A can fill the tank in 19 hours.
Pipe B can fill the tank in 27 hours.
Both pipes are opened simultaneously.
We need to find the time taken to fill half of the tank.
Formula Used:
Work done per hour by A = 1 / Time taken by A.
Work done per hour by B = 1 / Time taken by B.
Total work per hour = Work done by A + Work done by B.
Time to complete required work = Required work / Total work per hour.
Calculation:
Work done per hour by A = 1 / 19.
Work done per hour by B = 1 / 27.
⇒ Total work per hour = (1 / 19) + (1 / 27).
⇒ Total work per hour = (27 + 19) / (19 × 27).
⇒ Total work per hour = 46 / 513.
Required work = Half of the tank = 1/2.
Time to fill half of the tank = (1/2) / (46 / 513).
⇒ Time to fill half of the tank = (1/2) × (513 / 46).
⇒ Time to fill half of the tank = 513 / (2 × 46).
⇒ Time to fill half of the tank = 513 / 92.
⇒ Time to fill half of the tank = 5 (53 / 92) hours.
∴ The correct answer is 5 (53 / 92) hours.