Shortcut Trick Total Capacity (LCM of 12, 24, 32) = 96 units. Efficiencies: A = 8, B = 4, C = −3 (emptying). Net efficiency (A + B − C) = 8 + 4 − 3 = 9 units/min. Work not emptied by C in the last 2 minutes = 3 × 2 = 6 units. Time taken to fill = (96 − 6) ÷ 9 = 90 ÷ 9 = 10 minutes. ∴ The correct answer is 10 minutes. Alternate Method Given: Time taken by Pipe A = 12 minutes Time taken by Pipe B = 24 minutes Time taken by Pipe C (emptying) = 32 minutes Formula Used: Total Work = Efficiency × Time +--------+------+------------+------------+ | Person | Days | Total Work | Efficiency | +--------+------+------------+------------+ | Pipe A | 12 | 96 | 8 | | Pipe B | 24 | 4 | | | Pipe C | 32 | −3 | | +--------+------+------------+------------+ Calculations: Let the total time to fill the tank be T minutes. Pipe A and B work for the entire duration T. Pipe C is closed 2 minutes before filling, so Pipe C works for (T − 2) minutes. ⇒ Work done by A + Work done by B − Work emptied by C = Total capacity ⇒ (8 × T) + (4 × T) − 3 × (T − 2) = 96 ⇒ 12T − 3T + 6 = 96 ⇒ 9T + 6 = 96 ⇒ 9T = 90 ⇒ T = 10 minutes ∴ The correct answer is 10 minutes. Related Concepts: Pipes and Cisterns Logic Filling pipes have positive efficiency, while emptying pipes (leaks) have negative efficiency. Net efficiency is the algebraic sum of all active pipe efficiencies. Closing Pipes Early If a pipe is closed x minutes before completion, assume it works for (T − x) minutes, where T is the total time. Efficiency Relationship Efficiency is inversely proportional to the time taken to complete the task alone, given constant total work (Efficiency = Total Work ÷ Time). ∴ The correct answer is 10 minutes.