Q. 7 Two pipes 𝑷 and 𝑸 can fill a tank in 6 hours and 9 hours respectively, while a third pipe 𝑹 can empty
the tank in 12 hours. Initially, P and R are open for 4 hours. Then P is closed and Q is opened. After 6 more
hours R is closed. The total time taken to fill the tank (in hours) is ____ .
(A) 13.50
(B) 14.50
(C) 15.50
6 hours and 9 hours respectively, while
pipe R empties the tank in 12 hours.
The task is to find the exact total time taken to fill
the tank using step-by-step phases.
Pipes’ Work Rates
- Pipe P fills 1/6 tank per hour
- Pipe Q fills 1/9 tank per hour
- Pipe R empties 1/12 tank per hour
Combined rates:
- P + R =
1/6 − 1/12 = 1/12 tank per hour - Q + R =
1/9 − 1/12 =
(4 − 3) / 36 = 1/36 tank per hour - Q alone = 1/9 tank per hour
Phase 1: P and R for 4 Hours
Rate of P + R = 1/12 tank per hour
Work done in 4 hours =
4 × 1/12 = 1/3 tank
Tank filled after Phase 1: 1/3
Phase 2: Q and R for 6 Hours
Rate of Q + R = 1/36 tank per hour
Work done in 6 hours =
6 × 1/36 = 1/6 tank
Total filled after 10 hours =
1/3 + 1/6 = 1/2 tank
Remaining tank: 1/2
Phase 3: Q Alone to Finish
Q fills at 1/9 tank per hour
Time required to fill remaining 1/2 tank:
(1/2) ÷ (1/9) = 9/2 = 4.5 hours
Total Time Calculation
- Phase 1: 4 hours
- Phase 2: 6 hours
- Phase 3: 4.5 hours
Total Time = 4 + 6 + 4.5 = 14.5 hours
Options Explained
| Option | Time (hours) | Explanation |
|---|---|---|
| (A) | 13.50 | Incorrect. Underestimates the final phase due to wrong Q + R rate. |
| (B) | 14.50 | Correct. Accurate phase-wise calculation and rates. |
| (C) | 15.50 | Incorrect. Overestimates total time due to rate miscalculation. |
Final Answer
✅ Correct Answer: (B) 14.50 hours
