A cistern is normally filled by an inlet pipe in 8 hours, but because of a leak at the bottom it now takes 10 hours to fill. If the cistern is full and only the leak is left open, in how many hours will the cistern be emptied?

Introduction / Context:
This is another inlet outlet problem where a leak slows down the filling of a cistern. We know how long the inlet takes alone and how long the cistern takes to fill when both inlet and leak are active. From that information, we can determine the leak rate and then find the time for the leak alone to empty a full cistern.

Given Data / Assumptions:

Inlet alone fills the cistern in 8 hours.

With the leak also open, the cistern fills in 10 hours.

We take the cistern capacity as 1 unit.

The leak continuously drains water at a constant rate.

Concept / Approach:
We represent the inlet rate and the net filling rate as fractions of the cistern per hour. The leak rate is simply the difference between the inlet rate and the net rate. Once we know the leak rate, the time to empty the cistern is the reciprocal of that rate.

Step-by-Step Solution:
Step 1: Inlet rate alone = 1 / 8 cistern per hour.Step 2: Net rate with leak = 1 / 10 cistern per hour.Step 3: Let leak rate be L cistern per hour (this is an emptying rate).Step 4: Inlet rate minus leak rate equals net rate, so 1 / 8 - L = 1 / 10.Step 5: Solve for L: L = 1 / 8 - 1 / 10 = (5 / 40) - (4 / 40) = 1 / 40 cistern per hour.Step 6: Leak alone empties 1 / 40 of the cistern per hour.Step 7: Therefore, time taken by the leak alone to empty a full cistern = 1 / (1 / 40) = 40 hours.

Verification / Alternative check:
If the leak rate is 1 / 40, then the net filling rate with inlet and leak together is 1 / 8 - 1 / 40 = (5 - 1) / 40 = 4 / 40 = 1 / 10 cistern per hour, which means the cistern is filled in 10 hours. This matches the given information, so our leak rate and emptying time are consistent.

Why Other Options Are Wrong:
20 hours and 28 hours are too small and correspond to leak rates of 1 / 20 or 1 / 28, which would slow down filling more than observed.

36 hours is closer but still incorrect and comes from miscalculating the difference between 1 / 8 and 1 / 10.

Common Pitfalls:
Some students incorrectly add the rates instead of subtracting, as if both pipe and leak were filling. Others subtract the times instead of subtracting the rates, which is not valid. Always convert times into rates (1 / time) and work with those when combining effects of multiple pipes or leaks.

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