Pipes & Cisterns

The topic of pipes and cisterns is often asked in competitive exams. All government exams allot 2 to 4 marks for questions on pipes and cisterns. The concept deals with the time taken or the work done in filling or emptying the tank or cistern through an inlet or an outlet pipe. The time taken for the outlet or the inlet pipe to fill the cistern depends on the capacity of the cistern. 

Concept of pipes and cisterns

1. Inlet: The pipe that fills the cistern is the inlet pipe. This is called positive work.

2. Outlet: The pipe that empties the cistern is the outlet pipe. This is called negative work.

The part of the cistern which is filled or emptied at a particular time is measured as work. Time and work are directly related to each other.  

Formulas of Pipes & Cistern

Work done = time taken ✕ rate of work

Depending on the capacity of a cistern, if an inlet connected to it can fill the whole cistern in x’ hours, part of the cistern getting filled by the same inlet in 1 hour will be 1/x. Similarly, if an outlet connected to it can empty the whole cistern in y hours, part of the cistern being emptied by the same outlet in 1 hour will be 1/x.

In the first case, if the pipes attached to a cistern can fill and empty it completely in x and y hours, and both pipes are opened simultaneously, then the net part of the cistern filled in one hour will be xy/(y – x) 

Here, the time taken to empty is more than the time taken to fill the complete tank (y > x).

In the second case, if the pipes attached to a cistern can fill and empty it completely in x and y hours, and when both pipes are opened simultaneously, then the net part of the cistern filled in one hour will be xy/(x – y) 

Here the time taken to fill the cistern is more than the time taken to empty (x > y).

Net work done by both the pipes = (Sum of the work done by inlets) – (Sum of the work done by outlets)

When a cistern has two inlets – where one inlet takes x hours to fill the cistern and the other takes y hours to do the same – if both the inlets are kept open at the same time, the time taken to fill the whole cistern is xy/(y + x)

Now, assume a cistern has three pipes, two for inlets and one for an outlet. One inlet takes x hours to completely fill the cistern and the other takes y hours. The third outlet takes z hours to empty the cistern. When all these three pipes are opened together for an hour then the level of water in that time is 1x+1y +1z

When all three inlets take x, y and z hours to fill the tank, with all opened simultaneously, the time taken will be  x+y+z/xy + yz + zx hours.

If there is a leak in the cistern, and the pipe which took x hours to fill the cistern now takes y hours, then the time taken to empty the cistern will be xy/y – x

Consider inlet A and B where A fills the cistern faster than B, and thus takes less time than B. The time taken to fill the cistern when both inlets are opened simultaneously will be xy/(x-1)(x-1)

A inlet will fill the cistern in y/x-1 minutes

B inlet will fill the cistern in xy/x-1

How to solve questions on pipes and cisterns?

Familiarise with the terms related to the concept, such as inlet, outlet, cistern, tank, leak, emptying, etc. Practice and work with different types of questions. Memorise the formula but know when to apply them. There are two methods to solve the questions.

1. Unitary method as per the formulas

In this method, depending on the question, you just have to use the formula and find the answer.

2. LCM method

Let’s understand this method with the help of an example.

Q: Two pipes fill a cistern in 20 min and 60 min. How much time will both the pipes take to fill the cistern if they are opened simultaneously.

A: LCM of 20 and 60 is 60

Rate of the cistern filling from the first pipe = 60/20 = 3 litres per min

Rate of the cistern filling from the second pipe = 60/60 = 1 litre per min

Now in a minute, 3 +1 = 4 litres of water fill the cistern

Thus, 60 litres of water will be filled in 60/4 = 15 mins

Conclusion

Questions on the pipes and cistern problems are present in syllabuses for bank exams, SSB, RRB, etc. Since competitive exams require time management skills, it is important to practise these problems during preparation.