PIPE & CISTERN

Pipe and Cistern problems are analogous to those in the 'Time and Work' category. In this context, the time required to fill a cistern corresponds to the time needed to complete a task. The volume of the cistern is equivalent to the total work, and the rate of filling the cistern is analogous to the efficiency of the work.

Within this article, we will explore the fundamental principles of pipe and cistern problems, encompassing various question types, as well as offering valuable tips and strategies. Additionally, we have included solved examples to aid candidates in their exam preparation. It is recommended to thoroughly read the article to clarify any uncertainties related to this topic

Pipes and Cistern – Topic and Concept

"Pipes and Cisterns" is a topic in mathematics, specifically in quantitative aptitude, that deals with the filling or emptying of tanks or cisterns through pipes. It is a common topic found in competitive exams, including those conducted by UPSC, as it assesses the candidate's ability to understand and apply concepts related to time, speed, and work.

Concepts in Pipes and Cisterns:

Inlet and Outlet Pipes:
- Inlet pipes are pipes through which a cistern is filled.
- Outlet pipes are pipes through which a cistern is emptied.
Rate of Filling or Emptying:
- The rate at which a pipe can fill a cistern is generally given in terms of units per hour.
- Similarly, the rate at which a pipe can empty a cistern is given in terms of units per hour.
Efficiency of Pipes:
- The efficiency of an inlet pipe is positive, as it contributes to filling the cistern.
- The efficiency of an outlet pipe is negative, as it leads to emptying the cistern.
Time and Work Formulas:
- The basic formula connecting time, rate, and work is: $Work = Rate \times Time$
- For pipes and cisterns, if a pipe can fill a cistern in $x$ hours, its filling rate is $x 1$ of the cistern per hour.
- If a pipe can empty a cistern in $y$ hours, its emptying rate is $y 1$ of the cistern per hour.
Combined Efficiency:
- When two or more pipes are working simultaneously, their rates are added if they are inlet pipes and subtracted if they are outlet pipes.
- For example, if $A$ and $B$ are two pipes filling a cistern simultaneously, their combined rate is $Rate_{A + B} = Rate_{A} + Rate_{B}$ .
Time Taken to Fill/Empty a Cistern:
- The time taken to fill or empty a cistern can be calculated using the formula:

Important Formula on Pipes and Cistern

1.Filling rate:

If a pipe can fill a tank in x hours, then the part filled in 1 hour is 1/x.
Similarly, if a cistern is emptied in y hours, then the part emptied in 1 hour is 1/y.

2. Combined working:

Two pipes filling together:
- Net part filled in 1 hour = 1/a + 1/b (where a and b are the individual filling times)
- Time taken to fill the tank = ab/(a + b)
Two pipes, one filling and one emptying (filling rate > emptying rate):
- Net part filled in 1 hour = 1/a - 1/b (where a is the filling time and b is the emptying time)
- Time taken to fill the tank = ab/(b - a)
Multiple pipes and cisterns:
- Net part filled/emptied in 1 hour = sum of filling rates - sum of emptying rates
- Time taken to fill/empty = total volume / net part filled/emptied per hour

3. Leaks and combined working:

If a leak empties a full tank in c hours while a pipe fills it in a hours, the time taken to empty the tank with both open = ab/(b - a)

Pipes and Cistern – Sample Questions

Question 1: Pipe

A

can fill a cistern in 8 hours, and pipe

B

can empty the cistern in 12 hours. If both pipes are opened simultaneously, how long will it take to fill the cistern?

Solution 1: $Rate_{A} = 1/8 cistern/hour$

$Rate_{B} = - 1/12 cistern/hour$

$Rate_{A + B} =3/24 - 2/24 = 1/24 cistern/hour$

$Time = 24 hours$

Therefore, it takes 24 hours to fill the cistern when both pipes $A$ and $B$ are opened simultaneously

Question 2: A pipe can fill a cistern in 10 hours. Due to a leak, it takes an additional 5 hours to fill the cistern. Find the time taken by the leak alone to empty the full cistern.

Solution 2: Let $x$ be the rate of filling when there is no leak. Then, the rate of the leak is (1/10-1/x).

Given that it takes an additional 5 hours to fill the cistern with the leak, the equation is:

$5 \times (1/10 - 1) = 1$

Solving this equation gives $x = 20$ .

Therefore, the rate of the leak is and the time taken by the leak alone to empty the full cistern is 20 hours.

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PIPE & CISTERN

PIPE & CISTERN

Important Formula on Pipes and Cistern

Pipes and Cistern – Sample Questions

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