A water tank in a village is normally filled in 8 hours by an inlet pipe, but due to a leak at the bottom it now takes 2 hours longer to fill. If the tank is full, in how many hours will the leak alone empty the tank?

Introduction / Context:
This question explores the effect of a leak on the filling process. The inlet pipe alone can fill the tank in a known time, but with the leak present the filling time increases. From the difference in effective rates, we deduce the rate of the leak and then compute the time taken by the leak alone to empty a full tank.

Given Data / Assumptions:
- Inlet alone fills the tank in 8 hours
- With the leak present, the tank fills in 10 hours
- The tank is initially full when we later consider the leak acting alone
- There are no other inlets or outlets

Concept / Approach:
We treat the inlet rate as positive and the leak rate as negative. The effective net rate when both are acting equals 1/10 of the tank per hour. The inlet rate is 1/8 per hour. The leak rate is then inlet rate minus net rate. Once we have the leak rate, its time to empty a full tank is the reciprocal of its magnitude.

Step-by-Step Solution:
Step 1: Rate of inlet = 1/8 tank per hour. Step 2: Net rate with leak present (longer filling time) = 1/10 tank per hour. Step 3: Let leak rate be L tanks per hour (negative number). Then 1/8 + L = 1/10. Step 4: Solve for L: L = 1/10 - 1/8 = (4 - 5) / 40 = -1/40. Step 5: Magnitude of leak rate = 1/40 tank per hour. Step 6: Time taken by leak alone to empty the full tank = 1 / (1/40) = 40 hours.

Verification / Alternative check:
Check the net rate: Inlet rate 1/8 minus leak rate 1/40 equals (5 - 1) / 40 = 4/40 = 1/10 tank per hour, which indeed corresponds to a filling time of 10 hours. This confirms that the leak empties at 1/40 tank per hour and needs 40 hours to empty the full tank.

Why Other Options Are Wrong:
34, 36 and 38 hours: These times would correspond to leak rates that do not satisfy both the original 8 hour filling time without a leak and the 10 hour filling time with the leak. Substituting them back into the rate equation would not yield a consistent net rate of 1/10 tank per hour.

Common Pitfalls:
Students sometimes subtract the times instead of the rates, or misinterpret the direction of the leak rate. It is also easy to confuse which time corresponds to which scenario. Always write separate rate equations for the normal case and the leak case, and then subtract to isolate the leak rate.

Final Answer:
The leak alone will empty a full tank in 40 hours.