A pipe is connected to a tank or cistern. It is used to fill or empty the tank; accordingly, it is called an inlet or an outlet.

Inlet: A pipe which is connected to fill a tank is known as an inlet.

Outlet: A pipe which is connected to empty a tank is known as an outlet.

Problems on pipes and cisterns are similar to problems on time and work. In pipes and cistern problems, the amount of work done is the part of the tank of filled or emptied. And, the time taken to do a piece of work is the time take to fill or empty a tank completely or to a desired level.

Pipes and Cisterns Points to remember:

1) If an inlet connected to a tank fills it in X hours, part of the tank filled in one hour is = 1/X

2) If an outlet connected to a tank empties it in Y hours, part of the tank emptied in one hour is = 1/Y

3) An inlet can fill a tank in X hours and an outlet can empty the same tank in Y hours. If both the pipes are opened at the same time and Y > X, the net part of the tank filled in one hour is given by;

= (1/X – 1/Y)

Therefore, when both the pipes are open the time taken to fill the whole tank is given by;

= (XY/Y-X) Hours

If X is greater than Y, more water is flowing out of the tank than flowing into the tank. And, the net part of the tank emptied in one hour is given by;

= (1/Y – 1/X)

Therefore, when both the pipes are open the time taken to empty the full tank is given by;

= (YX/X-Y) Hours

4) An inlet can fill a tank in X hours and another inlet can fill the same tank in Y hours. If both the inlets are opened at the same time, the net part of the tank filled in one hour is given by;

= (1/X + 1/Y)

Therefore, the time taken to fill the whole tank is given by;

= (XY/Y+X) Hours

In a similar way, If an outlet can empty a tank in X hours and another outlet can empty the same tank in Y hours, the part of the tank emptied in one hour when both the pipes start working together is given by;

= (1/X + 1/Y)

Type 1: Pipes and Cistern (Tank)

In order to solve questions falling into the first category, imagine each of the pipes as a human doing some work. (Again, we are using a Work and Time reference here). The time taken by a single pipe to fill up the tank by itself, is the time taken by that human to do the entire work. Find out each pipe’s (or human’s) rate of doing work, with respect to each other. For instance, if Pipe A can fill the tank in 8 hours, and Pipe B can fill it in 4 hours, you know that the rate of B is twice that of A.

Once, you have the rates of doing work, all you have to do is add them up to find out the overall rate of filling the tank by all the pipes together. Remember, rate is always inversely proportional to the time taken, so the rates you add up should be reciprocals of the time taken by each pipe.

An easy tip that helps, is to assign any one pipe’s rate as ‘x’, and calculate the others’ rate in terms of ‘x’.

E.g. 1. Pipe A fills a tank in 3½ hours whereas Pipe B fills the same tank in 10½ hours.

Solution:

Since Pipe B takes more time, let us take ‘x’ as the rate at which B fills the tank.
Since time taken by B is 3 times the time taken by A, the rate of A will be 3 times faster than the rate of B. So the rate of A is ‘3x’ .

Another advantage of using the rates, is that rates can be added up, not time. Which means that if two pipes take 4 and 8 hours respectively to fill up a tank, you can add the reciprocals of 4 and 8 to get the net rate. What you cannot do, is add 4 and 8 up, directly. That would result in a wrong answer.

The logic is simple. If you add 4 and 8 up, to find the time taken by both together to fill up the tank, you will get 12. But think, if one pipe takes 4 hours and the other takes 8 hours to fill up the tank, they couldn’t possibly take 12 hours together!
So let us use the rates in the following example:

E.g. 2. Pipes P, Q, R and S can fill a certain tank in 3, 5, 6 and 10 hours, respectively. How much time will it take for the tank to be filled if all pipes are opened together?

Solution:

Let ‘x’ be the rate of Pipe S, which is the slowest. So the rates of all the pipes are as follows. Make sure to make as few calculations as possible – this is the trick to cracking your exam! Try the following method:

So the total efficiency of the four pipes is: x + 2x + 3.33x + 1.66x ≈ 8x = 8 × (1/10)
Thus the total time taken for the four pipes working together to fill the tank would be:
10/8 hours = 1.25 hours = 1 hour 15 mins.

Shorten Your Calculations

By choosing the pipe taking the maximum time first and basing your calculations on that, you effectively bring down the number of reciprocal calculations you make. Of course, you need to pick and choose which pipe’s efficiency to calculate next. Like in the example, after S, we chose Q because its time taken was a direct multiple of the time taken by S. This lets us calculate its reciprocal easily. Next, we were left with Pipes P and R , both of whose times are not multiples of the time taken by S. However, we all know that 10 is 3.33 times 3. So we use this to first find the efficiency of P. Next we know that 6 is two times 3. So clearly the efficiency of R is half of P.

Type 2: Speed of Filling/Emptying

For the second category, take each pipe filling the tank as positive and each hole, drain, outlet or leakage emptying the tank as negative. When you add these negative and positive rates together (being very careful with the signs), you get an equation giving you the net rate by which the tank is being filled up. If this figure is negative, that means the tank is being emptied, rather than filled up.

Once you have the net rate of the tank emptying or filling up, all you have to do is factor in the size of the tank, to find out the time it takes to empty or fill up. The size of the tank can be obtained from other information provided in the question. The question will either tell you how long the tank takes to empty through one hole, or it will give you the capacity of the tank. Either way, once you’re at this point of the solving process, it should be easy to get to the end. Simple math will get you there!

E.g. 3. A pipe fills a tank in 3 hours, whereas an outlet can empty the filled tank completely in 12 hours. In what time will the tank be filled, if both the pipe and the outlet are opened simultaneously?

Solution:

Let ‘x’ be the efficiency of the outlet. However since it is emptying the tank, we always use a negative sign with it.
And efficiency of pipe = (12/3)x = 4x
∴ Total Efficiency = 4x – x = 3x
Since x = 1/12, we can say that the tank will be filled in time = 12/3 = 4 Hours.