Pipes and Cisterns – Two inlets and one outlet together from empty: Pipe A fills the tank in 30 minutes, pipe B fills it in 10 minutes, and pipe C empties a full tank in 40 minutes. If all three pipes are opened simultaneously starting with an empty tank, how long will it take to fill the tank completely?

Introduction / Context:
This problem involves concurrent filling and emptying. The net rate equals the sum of the two inlet rates minus the outlet rate. Once the net rate is found, the time to fill is its reciprocal.

Given Data / Assumptions:

• A alone: 30 min ⇒ rate = 1/30 per min.
• B alone: 10 min ⇒ rate = 1/10 per min.
• C empties: 40 min ⇒ rate = 1/40 per min (negative for filling).
• Tank starts empty and is considered 1 unit capacity.

Concept / Approach:
Net rate = 1/30 + 1/10 − 1/40. Compute with a common denominator and invert to get time.

Step-by-Step Solution:
LCM 120 ⇒ 1/30 = 4/120, 1/10 = 12/120, 1/40 = 3/120.Net rate = (4 + 12 − 3)/120 = 13/120 per min.Time to fill = 1 / (13/120) = 120/13 min ≈ 9 min 14 sec.

Verification / Alternative check:
Multiply net rate by 120/13 to check: (13/120) * (120/13) = 1 exactly, so the timing is correct.

Why Other Options Are Wrong:
10 or 11 minutes ignore the outlet; any value not equal to 120/13 does not invert the correct net rate.

Common Pitfalls:
Adding times instead of rates, or treating the outlet as a positive contributor.

Final Answer:
120/13 min