Shortcut Trick
Total work (LCM of 28, 21, 84) = 84 units.
Efficiency of Pipe A = 3, Pipe B = 4, and All pipes together = 1.
Efficiency of Emptying Pipe C = (3 + 4) − 1 = 6 units/hr.
Time taken by Pipe C alone = 84 ÷ 6 = 14 hours.
Capacity of the cistern = 14 hours × 3 gallons/hr = 42 gallons.
∴ The correct answer is 42 gallons.
Alternate Method
Given: Filling time of A = 28 hr, B = 21 hr, Combined (A + B − C) = 84 hr. Emptying rate of C = 3 gallons/hr.
| Person/Pipe | Time (hr) | Total Work (LCM) | Efficiency |
|---|
| Pipe A (Fill) | 28 | 84 units | 3 |
| Pipe B (Fill) | 21 | 4 | |
| Combined (A+B−C) | 84 | 1 | |
⇒ Efficiency of Pipe A + Efficiency of Pipe B − Efficiency of Pipe C = Combined Efficiency
⇒ 3 + 4 − Efficiency of Pipe C = 1
⇒ 7 − Efficiency of Pipe C = 1
⇒ Efficiency of Pipe C = 6 units/hr
⇒ Time taken by Pipe C to empty the cistern = Total Work ÷ Efficiency = 84 ÷ 6 = 14 hours
⇒ Total Capacity = Time × Rate per hour = 14 × 3
⇒ Total Capacity = 42 gallons
∴ The correct answer is 42 gallons.
Additional Information
Pipes and Cisterns Logic
Filling pipes have positive efficiency (+), while emptying pipes (leaks) have negative efficiency (−).
Capacity Calculation
The actual capacity is found by multiplying the time taken to fill/empty the tank by the flow rate (gallons or litres per hour).
Combined Efficiency Formula
1/Ttotal = 1/T1 + 1/T2 − 1/Tleak, where T is the time taken by individual pipes.