About Pipes and Cistern
Pipes and cistern questions are work and time questions wearing different clothes. A pipe filling a tank is doing work, the tank is the whole of that work, and a leak is negative work. Every question in the topic is solved by adding and subtracting rates, never times.
What you need to understand
- A pipe that fills a tank in x hours fills one x-th of the tank in a single hour. It is that hourly rate, not the time, that gets added.
- An inlet adds to the filling rate. An outlet or a leak subtracts from it.
- The net rate is the sum of the filling rates minus the sum of the emptying rates.
- If the net rate is one x-th of the tank per hour, the tank fills in x hours.
- In a leak question, the rate of the leak is the rate of the pipe alone minus the net rate achieved with the leak open.
- Two pipes together always fill the tank faster than either one on its own.
Formulas to remember
How to work through these questions
- Convert every time in the question into a rate per hour before combining anything at all.
- Decide which pipes fill and which empty, and give the emptying ones a minus sign.
- Add the rates, then invert the total to turn a rate back into a time.
- For a leak question, subtract the net rate from the rate of the pipe alone to isolate the leak.
- Check the answer, because a combined time must be smaller than the smallest individual filling time.
Mistakes that cost marks
- Adding the times instead of the rates, which produces an answer larger than either pipe alone.
- Averaging the two times, which is correct only when the two pipes are identical.
- Subtracting rates when the question describes two pipes both filling the tank.
- Giving the time taken with the leak still running when the question asks about the leak by itself.
- Forgetting to invert the net rate and reporting a rate where a time was asked for.
Worked example
Pipe A fills a tank in 12 hours and pipe B fills it in 6 hours. How long will the two pipes take to fill it together?
- In one hour pipe A fills 1/12 of the tank, and pipe B fills 1/6 of it.
- Together they fill 1/12 + 1/6 = 1/12 + 2/12 = 3/12 = 1/4 of the tank in an hour.
- A rate of one quarter per hour means the tank is full in 4 hours.
- Check: 4 hours is less than the 6 hours pipe B needs alone, which it must be.
Answer: 4 hours
Practice questions with answers
A few Pipes and Cistern questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
With a leak in the tank, a pipe that normally fills it in 27 hours now needs 54 hours. Find the time in which the leak alone empties the full tank.
Answer: Option B — with explanation
The pipe fills 1/27 of the tank in an hour, but with the leak only 1/54 is actually gained.
So the leak empties 1/27 - 1/54 = 1/54 of the tank in an hour.
The leak alone would empty the full tank in 54 hours.
Common mistakes
- averaged the two times: 40.5
- gave the extra time the leak causes: 27
- added the two rates instead of subtracting them: 18
Question 2
A pipe can fill a tank in 108 hours. Because of a leak in the bottom it takes 216 hours to fill. If the tank is full, how long would the leak alone take to empty it?
Answer: Option C — with explanation
The pipe fills 1/108 of the tank in an hour, but with the leak only 1/216 is actually gained.
So the leak empties 1/108 - 1/216 = 1/216 of the tank in an hour.
The leak alone would empty the full tank in 216 hours.
Common mistakes
- gave the extra time the leak causes: 108
- added the two rates instead of subtracting them: 72
- averaged the two times: 162
Question 3
A tap fills a tank in 12 hours, but with a leak at the bottom it takes 6 hours more. Find the time the leak alone would take to empty the full tank.
Answer: Option D — with explanation
With the leak the tank fills in 12 + 6 = 18 hours.
The pipe supplies 1/12 of the tank per hour and the leak removes 1/36 of it.
1/12 - 1/36 = 1/18, so the leak alone empties the full tank in 36 hours.
Common mistakes
- averaged the two times: 15
- gave the extra time the leak causes: 6
- gave the pipe's own filling time: 12
Question 4
A tap fills a tank in 18 hours, but with a leak present the filling takes 45 hours. How long would the leak take to empty a full tank on its own?
Answer: Option C — with explanation
The pipe fills 1/18 of the tank in an hour, but with the leak only 1/45 is actually gained.
So the leak empties 1/18 - 1/45 = 1/30 of the tank in an hour.
The leak alone would empty the full tank in 30 hours.
Common mistakes
- gave the pipe's own filling time: 18
- averaged the two times: 31.5
- gave the extra time the leak causes: 27
Question 5
Filling a tank normally takes 15 hours; with a leak it takes 3 hours more. In how many hours would the leak empty the full tank?
Answer: Option A — with explanation
With the leak the tank fills in 15 + 3 = 18 hours.
The pipe supplies 1/15 of the tank per hour and the leak removes 1/90 of it.
1/15 - 1/90 = 1/18, so the leak alone empties the full tank in 90 hours.
Common mistakes
- gave the extra time the leak causes: 3
- gave the pipe's own filling time: 15
- gave the time taken with the leak still running: 18
Frequently asked questions
Why are rates added instead of times?
Because the pipes work at the same time. In any one hour each pipe contributes its own fraction of the tank, so the fractions are added. Times would be added only if the pipes worked one after the other.
How do I deal with a leak in the tank?
Treat the leak as a pipe with a negative rate. Subtract its rate from the filling rate and invert whatever is left to turn it back into a time.
What if one pipe is being used as an outlet?
Give it a minus sign in the same equation. If the outlet empties the tank faster than the inlet fills it, the tank will never fill.
Why must the combined time be less than either time alone?
Because together the pipes put more water into the tank each hour than either pipe manages by itself, so the tank reaches full sooner.
