A pump can fill a tank with water in 2 hours when there is no leak. Because of a leak in the tank, it now takes 2 hours 20 minutes to fill the tank. In how much time will the leak alone empty a full tank?

Introduction / Context:
This is a pipes and cisterns problem with one inlet (the pump) and one outlet (the leak). The presence of the leak reduces the effective filling rate, causing the tank to take longer to fill. From the difference in times, we can calculate the leak rate and then find how long the leak alone would take to empty the tank.

Given Data / Assumptions:

The pump alone fills the tank in 2 hours.

With the leak active, the net filling time is 2 hours 20 minutes.

2 hours 20 minutes equals 7 / 3 hours.

We assume the tank capacity is 1 full tank unit for easier calculation.

Concept / Approach:
Let the capacity of the tank be 1 unit. Then the pump rate and the net rate with the leak can be written as fractions of the tank per hour. The leak rate is the difference between the pump rate and the net rate. Once we know the leak rate, we can invert it to get the time taken by the leak alone to empty the full tank.

Step-by-Step Solution:
Step 1: Pump rate alone = 1 / 2 tank per hour.Step 2: Net rate with leak = 1 / (7 / 3) = 3 / 7 tank per hour.Step 3: Let leak rate (emptying) be L tank per hour. Then pump rate - leak rate = net rate.Step 4: So 1 / 2 - L = 3 / 7.Step 5: Solve for L: L = 1 / 2 - 3 / 7 = (7 / 14) - (6 / 14) = 1 / 14 tank per hour.Step 6: If the leak alone is open, it empties 1 / 14 of the tank per hour.Step 7: Therefore, time taken by the leak alone to empty the full tank = 1 / (1 / 14) = 14 hours.

Verification / Alternative check:
Check the combined rate: pump rate minus leak rate = 1 / 2 - 1 / 14 = (7 / 14) - (1 / 14) = 6 / 14 = 3 / 7, which matches the earlier net rate. Filling at 3 / 7 tank per hour, the tank will indeed take 7 / 3 hours or 2 hours 20 minutes to fill, confirming our calculations.

Why Other Options Are Wrong:
7 hours and 8 hours are too small and correspond to larger leak rates, which would cause much more delay than 20 extra minutes.

12 hours is closer but still incorrect and results from arithmetic mistakes when subtracting fractions or converting mixed times to fractional hours.

Common Pitfalls:
Students often forget to convert 2 hours 20 minutes into a single fraction (7 / 3 hours) and instead use inconsistent units. Others mistakenly add the leak rate instead of subtracting it, as if the leak also filled the tank. Remember that outlet rates reduce net filling speed. Carefully convert all times into hours or minutes before proceeding with algebra.

Final Answer:
The leak alone will empty a full tank in 14 hours.