Pipes and Cisterns – Staggered opening times, including a strong outlet later: A cistern has three pipes A, B, and C. Pipe A fills the cistern in 3 hours and pipe B in 4 hours. Pipe C empties a full cistern in 1 hour. The pipes are opened in order at 3:00 pm (A), 4:00 pm (B), and 5:00 pm (C). At what time will the cistern be empty?

Introduction / Context:When pipes open at different times, break the timeline into intervals with constant rates. First, only A runs; then A and B; finally A, B, and the strong outlet C act together, potentially reversing the net flow.

Given Data / Assumptions:

• A fills in 3 h ⇒ 1/3 per h.
• B fills in 4 h ⇒ 1/4 per h.
• C empties in 1 h ⇒ 1 per h (negative).
• Intervals: [3–4 pm] A only; [4–5 pm] A + B; [after 5 pm] A + B − C.

Concept / Approach:Accumulate the filled fraction at each stage, then after C opens the net rate becomes negative and the cistern empties from its then-current level.

Step-by-Step Solution:3–4 pm: A adds 1/3 ⇒ level = 1/3.4–5 pm: A + B add 1/3 + 1/4 = 7/12 ⇒ new level = 1/3 + 7/12 = 11/12.After 5 pm: net rate = 1/3 + 1/4 − 1 = 7/12 − 1 = −5/12 per h.Time to empty from 11/12 at rate 5/12 per h = (11/12) / (5/12) = 11/5 = 2.2 h = 2 h 12 min.Empty time = 5:00 pm + 2 h 12 min = 7:12 pm.

Verification / Alternative check:Reasonable since a strong outlet opens late, quickly draining near-full cistern.

Why Other Options Are Wrong:6:15 pm is too early; 8:12 pm and 8:35 pm are too late given the high emptying rate after 5 pm.

Common Pitfalls:Forgetting that after 5 pm the net flow is outward (negative rate).

Final Answer:7:12 pm