Two inlet pipes and one outlet pipe act on the same cistern. Pipe A fills the cistern in 12 minutes; Pipe B fills it in 15 minutes; Pipe C (outlet) can empty a full cistern in 6 minutes. First, only A and B are opened for exactly 5 minutes. After that, Pipe C is also opened along with A and B. From the moment all three pipes are open together, how long will it take to empty the cistern completely (i.e., bring the water level from its then state down to empty)?

Introduction / Context:
When pipes fill and drain simultaneously, we work with rates (fractions of a tank per minute). We will compute how much water has accumulated after the first phase (only inlets), then use the net rate with the outlet added to determine the time to reach empty from that intermediate level.

Given Data / Assumptions:

• A fills in 12 min ⇒ rate = 1/12 tank/min.
• B fills in 15 min ⇒ rate = 1/15 tank/min.
• C empties in 6 min ⇒ rate = −1/6 tank/min.
• Phase 1: A and B for 5 minutes.
• Phase 2: A, B, and C together until empty.

Concept / Approach:
Add inlet rates for Phase 1 to find the filled fraction. Then, for Phase 2, compute the net rate including the outlet (inlet rates minus outlet rate). Time = (remaining volume) / (absolute value of net outflow rate).

Step-by-Step Solution:

Verification / Alternative check:
The negative net indicates draining. Magnitude 1/60 means 1 minute removes 1/60 of capacity; 45 minutes remove 45/60 = 3/4, exactly the amount present after Phase 1.

Why Other Options Are Wrong:
38/22/42 min do not match the exact fraction arithmetic; only 45 min empties exactly 3/4 at the given net rate.

Common Pitfalls:
Forgetting the outlet subtracts from fill rate; mixing “from the start” vs “from when outlet opens.” The question explicitly asks for the emptying time from the moment all three are open.

Final Answer:
45 min