Pipes and Cisterns: Concepts, Difference, Formulas, and Solved Examples

Pipes and Cisterns is a fundamental topic in quantitative aptitude that revolves around understanding the dynamics of filling and emptying tanks or cisterns through various pipes. By understanding the core concepts, differences, formulas, and problem-solving techniques, one can efficiently tackle a wide range of questions related to time, work, and efficiency in this pipe and cistern domain. This article will cover important definitions, properties, and examples related to pipes and cisterns. This concept is frequently tested in competitive exams such as SSC, Banking, Railways, and various other entrance exams, making it important for aspirants to understand pipe and cisterns.Continue reading to learn more!

What are Pipe and Cisterns?

Pipes and Cisterns is a mathematical concept that deals with the filling and emptying of tanks or cisterns through pipes. It’s essentially a practical application of the ‘Time and Work’ concept.

Key Components:

• Cistern: A tank or reservoir used to store liquids, typically water.
• Inlet Pipe: A pipe that fills the cistern.
• Outlet Pipe: A pipe that empties the cistern.   

Important Properties:

• Work Done: Filling the cistern is considered positive work, while emptying it is negative work.
• Efficiency: The rate at which a pipe can fill or empty the cistern.   
• Time: The duration taken to fill or empty the cistern.
• Capacity: The total volume of water a cistern can hold.

By understanding these components and properties, we can effectively solve problems related to pipes and cisterns, involving multiple inlets and outlets, leakages, and varying filling/emptying rates.

Also Read: Frequency Polygon

Understanding the Concepts of Pipe and Cisterns

Pipes and Cisterns is a mathematical concept that deals with the filling and emptying of tanks or cisterns through pipes. It involves understanding the interplay between inlet pipes (which fill the tank) and outlet pipes (which empty it). By applying the principles of work and time, we can efficiently solve problems related to the filling and emptying rates of these pipes.

Core Concepts

Some important concepts related to pipe and cisterns are mentioned here:

• Cistern: A container or tank used to store liquid, typically water.
• Inlet Pipe: A pipe that fills the cistern.
• Outlet Pipe: A pipe that empties the cistern.
• Efficiency: The rate at which a pipe can fill or empty the cistern. This is often expressed as a fraction of the cistern filled or emptied per unit time.

Key Ideas

Here in this section we have stated the some key ideas revolving around pipe and cistern :

• Work Done: Filling the cistern is considered positive work, while emptying it is negative work.
• Efficiency and Time: The efficiency of a pipe is inversely proportional to the time it takes to fill or empty the cistern.
• Combined Efficiency: When multiple pipes are working together, their efficiencies are added if they are inlet pipes and subtracted if they are outlet pipes.
• Net Efficiency: The combined efficiency of all pipes determines the net rate of filling or emptying the cistern.
• Leakage: A leak can be considered as an outlet pipe.

Example

If Pipe A can fill a cistern in 6 hours, its efficiency is 1/6 of the cistern per hour. If Pipe B can empty the same cistern in 8 hours, its efficiency is -1/8 of the cistern per hour (negative as it’s emptying).

Solution: If both pipes are opened simultaneously, the net efficiency is 1/6 – 1/8 = 1/24 of the cistern per hour. So, together they can fill the cistern in 24 hours.

Difference between Pipe and Cisterns

Here is the table that shows the difference between pipe and citsterns.

Component	Function	Properties
Pipe	Fills or empties the cistern	Has an efficiency (rate of filling/emptying), Can be of different diameters affecting flow rate
Cistern	A container that stores the liquid (typically water)	Has a capacity (maximum volume it can hold), Can be made of different materials (plastic, concrete, etc.)

Also Read: Real Number Line

Pipe and Cisterns Formulas

Here you find and explore the pipe cistern formulas. By applying the principles of work and time, we can efficiently solve problems related to the filling and emptying rates of these pipes.

Basic Formulas

• Part filled by a pipe in 1 hour: If a pipe fills a tank in ‘x’ hours, then the part filled in 1 hour = 1/x.

1 hour = 1/x

• Part emptied by a pipe in 1 hour: If a pipe empties a tank in ‘y’ hours, then the part emptied in 1 hour = 1/y.

1 hour = 1/y

Combined Pipes

• Two inlet pipes: If Pipe A fills a tank in ‘a’ hours and Pipe B fills it in ‘b’ hours, then the time taken to fill the tank when both pipes are opened together = ab / (a + b) hours.

Both pipes are opened together = ab / (a + b) hours

• One inlet and one outlet pipe: If Pipe A fills a tank in ‘a’ hours and Pipe B empties it in ‘b’ hours (where b > a), then the net part filled in 1 hour = 1/a – 1/b.

Time taken to fill the tank = ab / (b – a) hours.

• Multiple pipes: If there are multiple inlet and outlet pipes, calculate the net part filled in 1 hour by adding the efficiencies of inlet pipes and subtracting the efficiencies of outlet pipes.

Multiple Pipes

• If there are multiple inlet pipes with efficiencies 1/a, 1/b, 1/c, … and an outlet pipe with efficiency 1/d, then the net part filled in 1 hour = (1/a + 1/b + 1/c + …) – 1/d.

Net part filled in 1 hour = (1/a + 1/b + 1/c + …) – 1/d

Additional Formulas

• Leakage: If a pipe can fill a tank in ‘x’ hours and due to a leak, it takes ‘y’ hours to fill the tank, then the time taken by the leak to empty the full tank = xy / (y – x) hours.

Time taken by the leak to empty the full tank = xy / (y – x) hours

Note: Efficiency is often represented as the part of the tank filled or emptied in one hour.

Also Read: Area of Triangles

Pipe and Cisterns Solved Examples

Here are some solved examples related to pipes and cisterns:

Example 1:Pipe A can fill a tank in 6 hours and pipe B can fill the same tank in 10 hours. How long will it take to fill the tank if both pipes are opened together?

Solution:

• Pipe A’s work in 1 hour = 1/6
• Pipe B’s work in 1 hour = 1/10
• Work done by both pipes in 1 hour = 1/6 + 1/10 = 8/30 = 4/15
• Time taken to fill the tank by both pipes = 15/4 hours = 3 hours and 45 minutes.

Example 2:A pipe can fill a tank in 6 hours. Due to a leakage, it takes 8 hours to fill the tank. How long will the leak take to empty the full tank?

Solution:

• Work done by the pipe in 1 hour = 1/6
• Net work done (pipe + leak) in 1 hour = 1/8
• Work done by the leak in 1 hour = 1/6 – 1/8 = 1/24
• Time taken by the leak to empty the tank = 24 hours.

Example 3:Two pipes A and B can fill a tank in 20 and 30 hours respectively. If both pipes are opened together for 5 hours and then pipe B is closed, how long will pipe A take to fill the remaining tank?

Solution:

• Work done by A and B together in 1 hour = 1/20 + 1/30 = 1/12
• Work done by A and B together in 5 hours = 5/12
• Remaining work = 1 – 5/12 = 7/12
• Time taken by A to fill 7/12 part = (7/12) * 20 = 35/3 hours = 11 hours and 40 minutes.

Example 4:A pipe can fill a tank in 12 hours. Another pipe can empty the full tank in 15 hours. If both pipes are opened together, how long will it take to fill the tank?

Solution:

• Pipe A’s work in 1 hour = 1/12
• Pipe B’s work in 1 hour = -1/15 (as it’s emptying)
• Net work done in 1 hour = 1/12 – 1/15 = 1/60
• Time taken to fill the tank = 60 hours.

Example 5:Three pipes A, B, and C can fill a tank in 12, 15, and 20 hours respectively. If all three pipes are opened together, how long will it take to fill the tank?

Solution:

• Pipe A’s work in 1 hour = 1/12
• Pipe B’s work in 1 hour = 1/15
• Pipe C’s work in 1 hour = 1/20
• Total work done in 1 hour = 1/12 + 1/15 + 1/20 = 1/5
• Time taken to fill the tank = 5 hours.

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