How to Solve Cistern-with-a-Leak Problems
Solve cistern-with-a-leak aptitude problems by treating the leak as negative work, with a worked example and practice questions.
Expected Interview Answer
A leak is treated as negative work: if a pipe fills a cistern in 'a' hours and a leak alone would empty it in 'b' hours, their combined one-hour effect is 1/a minus 1/b, and the cistern fills in the reciprocal of that difference.
Filling pipes contribute positive one-hour-work fractions (1/a); a leak or outlet contributes a negative one-hour-work fraction (−1/b) because it removes water instead of adding it. When both act together, sum the signed fractions to get the net one-hour work, and the time to fill is 1 divided by that net fraction — provided the net is positive, otherwise the cistern never fills or net-empties instead. This is the same one-unit-of-work framework as time-and-work problems, just with the leak’s contribution flipped negative, and it extends directly to multiple inlet pipes and one or more leaks combined.
- Leaks are just negative one-hour-work terms in the same work framework
- One signed-sum equation covers pipes and leaks together
- Extends cleanly to multiple pipes plus multiple leaks
AI Mentor Explanation
A team scoring runs at a steady rate is like a filling pipe, adding a fixed fraction of the target total each over; a rain-delay style time penalty that effectively erases part of the total is like a leak, subtracting a fraction each over. If the scoring rate alone would reach the target in 10 overs but the penalty alone would erase the whole innings-equivalent in 15 overs, the net progress per over is 1/10 minus 1/15, and the true time to “fill” the target is the reciprocal of that difference — exactly how cistern-with-leak problems combine positive and negative rates.
Worked example
Pipe rate
- 1/6 per hour
Leak rate
- −1/12 per hour
Net fill time
- 1 / (1/6 − 1/12) = 12 hours
Step-by-Step Explanation
Step 1
Write the pipe's one-hour work
Filling pipe alone: +1/a, where a is the hours to fill alone.
Step 2
Write the leak's one-hour work as negative
Leak alone: −1/b, where b is the hours to empty a full cistern alone.
Step 3
Sum for net one-hour work
Net rate = 1/a − 1/b; a positive net means the cistern eventually fills.
Step 4
Invert for total fill time
Time to fill with the leak open = 1 ÷ net rate.
What Interviewer Expects
- Correctly signing the leak's contribution as negative
- Correct summation of pipe and leak one-hour-work fractions
- Recognizing when the net rate is negative (cistern never fills)
- Extending the method to multiple pipes and multiple leaks
Common Mistakes
- Adding the leak's rate instead of subtracting it
- Forgetting to check whether the net rate is actually positive
- Mixing up which given time belongs to the pipe versus the leak
- Incorrectly inverting the net fraction to get the total time
Best Answer (HR Friendly)
“I treat the leak exactly like negative work in the standard time-and-work framework. The pipe contributes a positive fraction of the cistern per hour, the leak contributes a negative fraction per hour, and I just add those two signed fractions to get the net rate. The time to fill is one divided by that net rate — and if the net rate comes out negative, that tells me the leak actually wins and the cistern never fills.”
Follow-up Questions
- What happens if the leak's rate exceeds the pipe's rate?
- How would you solve this with two filling pipes and one leak together?
- How do you find the leak's emptying time if you know the pipe's time and the combined fill time?
- How does this differ from a problem where the pipe is turned off partway through?
MCQ Practice
1. A pipe fills a tank in 8 hours; a leak empties it in 24 hours. With both open, the tank fills in?
Net rate = 1/8 − 1/24 = 3/24 − 1/24 = 2/24 = 1/12, so fill time = 12 hours.
2. A pipe fills a cistern in 5 hours. With a leak also open, it takes 20 hours to fill. How long would the leak alone take to empty it?
Net rate = 1/20 = 1/5 − 1/b → 1/b = 1/5 − 1/20 = 3/20 → b = 20/3 hours.
3. If a leak's emptying rate equals the filling pipe's rate exactly, what happens?
Equal and opposite rates cancel to a net rate of zero, so the water level never rises.
Flash Cards
How is a leak represented in the work equation? — As a negative one-hour-work fraction, −1/b.
Net rate formula with one pipe and one leak? — Net rate = 1/a − 1/b (pipe rate minus leak rate).
What if net rate is negative? — The cistern never fills — the leak empties it faster than the pipe fills it.
Total fill time formula? — Time = 1 ÷ net one-hour-work rate.