Shortcut Trick Let efficiency of Pipe C = 1 unit/hr. Efficiency of Pipe A = 6 × 1 = 6 units/hr. Efficiency of Pipe B = 6 ÷ 2 = 3 units/hr. Total efficiency (A+B+C) = 6 + 3 + 1 = 10 units/hr. Total Work = 10 × 3 hours = 30 units. Time for B and C = 30 ÷ (3 + 1) = 30/4 = 7.5 hours. ∴ The correct answer is 7(1)/(2) hours. Alternate Method Given: Time taken by (A + B + C) to fill the tank = 3 hours. Efficiency relation: A = 6C and A = 2B. Formula Used: Total Work = Combined Efficiency × Total Time Time Taken = Total Work ÷ Efficiency of required pipes +--------+------------------+-----------------------+ | Pipe | Efficiency Ratio | Efficiency (units/hr) | +--------+------------------+-----------------------+ | Pipe A | 6 × C | 6 | | Pipe B | A ÷ 2 | 3 | | Pipe C | Base | 1 | +--------+------------------+-----------------------+ Calculations: ⇒ Combined efficiency of A, B, and C = 6 + 3 + 1 = 10 units/hr. ⇒ Total capacity of the tank = 10 × 3 = 30 units. ⇒ Combined efficiency of B and C = 3 + 1 = 4 units/hr. ⇒ Time taken by B and C = 30 ÷ 4 hours. ⇒ Time = 15/2 = 7.5 hours. ⇒ 7.5 hours = 7(1)/(2) hours. ∴ The correct answer is 7(1)/(2) hours. Related Concepts: Efficiency and Time Relationship Efficiency is inversely proportional to the time taken to complete a work. If the ratio of efficiency is x : y, the ratio of time taken is y : x. Pipes and Cisterns Rule If a pipe can fill a tank in 'x' hours and another can empty it in 'y' hours, the net part filled in 1 hour = (1/x − 1/y). Combined Work Formula If A, B, and C can do a work in x, y, and z days respectively, they together can finish it in (xyz) / (xy + yz + zx) days.