A tank is normally filled by an inlet pipe in 9 hours, but due to a leak at the bottom it now takes 4 hours longer to fill. If the tank is full, in how many hours will the leak alone empty the tank?

Introduction / Context:
This is another leak and inlet combination problem. The leak slows down the filling process, increasing the time from 9 hours to 13 hours. From this information we can find the leak rate and then determine how long it would take the leak alone to empty a full tank.

Given Data / Assumptions:
- Inlet alone fills the tank in 9 hours
- With the leak present, the tank fills in 13 hours
- The tank is initially full when we later consider the leak acting alone
- There are no other pipes or leaks

Concept / Approach:
Treat the inlet rate as positive and the leak rate as negative. The net rate with both acting equals 1/13 of the tank per hour. The inlet rate is 1/9. Subtracting these gives the leak rate. The time taken for the leak alone to empty the tank is the reciprocal of its magnitude.

Step-by-Step Solution:
Step 1: Inlet rate = 1/9 tank per hour. Step 2: Net rate with leak present = 1/13 tank per hour. Step 3: Let leak rate be L tanks per hour (negative). Then 1/9 + L = 1/13. Step 4: Solve for L: L = 1/13 - 1/9. Step 5: LCM of 9 and 13 is 117, so L = (9 - 13) / 117 = -4/117. Step 6: Magnitude of leak rate = 4/117 tank per hour. Step 7: Time taken by leak alone to empty the full tank = 1 / (4/117) = 117/4 hours = 29.25 hours.

Verification / Alternative check:
Check net rate: inlet 1/9 minus leak 4/117 equals (13 - 4) / 117 = 9/117 = 1/13 tank per hour, which corresponds to 13 hours to fill. This confirms the correctness of the leak rate and the derived emptying time of 29.25 hours.

Why Other Options Are Wrong:
32.5, 30.30 and 31 hours: These values do not correspond to a leak rate that produces a net filling time of 13 hours when combined with an inlet rate of 1/9. Substituting them back into the rate equations fails to yield the correct net rate.

Common Pitfalls:
Some learners may subtract 9 and 13 directly or confuse which time is associated with which scenario. Another mistake is to forget to use fractions for rates and end up subtracting times instead. As always, rate equations are more reliable than working directly with times.

Final Answer:
The leak alone will empty a full tank in 29.25 hours.