Pipes and Cisterns – Closing one pipe after a short interval: Two pipes P and Q can fill a cistern in 12 minutes and 15 minutes, respectively. Both are opened together; after 3 minutes, pipe P is closed. How much additional time will be required to fill the cistern completely?

Introduction / Context:
When one pipe is shut after a while, split the process into phases: (1) both pipes operate; (2) the remaining pipe continues alone. Add the volumes (fractions) done in each phase to reach the full tank.

Given Data / Assumptions:

• P alone: 12 min ⇒ rate = 1/12 per min.
• Q alone: 15 min ⇒ rate = 1/15 per min.
• Both open for first 3 min; then only Q continues.

Concept / Approach:
Compute work done in 3 minutes by both pipes, subtract from 1 to get the remainder, then divide by Q’s rate to obtain the extra time.

Step-by-Step Solution:
In 3 min, both pipes do: 3*(1/12 + 1/15) = 3*( (5 + 4)/60 ) = 3*(9/60) = 27/60 = 9/20.Remaining = 1 − 9/20 = 11/20.Q’s rate = 1/15 per min ⇒ extra time = (11/20) / (1/15) = (11/20)*15 = 165/20 = 8.25 min = 8 min 15 sec.

Verification / Alternative check:
Convert to seconds if preferred: 0.25 min = 15 sec; total is 8 min 15 sec.

Why Other Options Are Wrong:
5, 7:30, 9 minutes fail to match the computed remainder at Q’s solo rate.

Common Pitfalls:
Multiplying times directly or forgetting to split into phases properly.

Final Answer:
8 min 15 sec