Pipes and Cisterns

Filling and draining problems

101 practice questionsPage updated 27 Mar 2026Review metadata pendingPractice questions

Pipes and Cisterns problems deal with the filling and emptying of tanks or cisterns using pipes. These problems are analogous to Time and Work problems, where inlet pipes fill the tank (positive work) and outlet pipes empty it (negative work). Understanding rates of work and combined operations is essential for solving these efficiently.

Key formulas and rules

If a pipe fills a tank in 'x' hours, its filling rate = 1/x per hour

If a pipe empties a tank in 'y' hours, its emptying rate = 1/y per hour

Net rate when both work together = (1/x - 1/y) per hour (when y > x, tank fills; when x > y, tank empties)

Time taken to fill/empty = 1 / Net Rate

Part of tank filled in 't' hours = t x Filling Rate

Part of tank emptied in 't' hours = t x Emptying Rate

If pipe A is open for 'a' hours and pipe B for 'b' hours: Part filled = a/x +/- b/y

For multiple pipes: Combined rate = 1/x +/- 1/x +/- 1/x +/- ...

Tank capacity = LCM of individual times (useful for whole number calculations)

Key concepts

Inlet and Outlet Pipes

An inlet pipe fills the tank and contributes positive work. An outlet pipe (or leak) empties the tank and contributes negative work. When calculating net work, inlet pipes are added and outlet pipes are subtracted. The sign convention is crucial - always verify whether the final answer represents filling or emptying.

Work Rate Concept

The fundamental principle is that work done = rate x time. For pipes, we consider the tank as 1 unit of work. A pipe's rate is the reciprocal of the time it takes to complete the work alone. For example, a pipe that fills in 5 hours works at 1/5 of the tank per hour. This rate remains constant unless stated otherwise.

Worked examples

Example 1

Pipe A can fill a tank in 6 hours and Pipe B can fill the same tank in 8 hours. If both pipes are opened together, how long will it take to fill the tank?

Solution

Step 1: Identify individual rates
Pipe A's rate = 1/6 per hour
Pipe B's rate = 1/8 per hour

Step 2: Calculate combined rate
Combined rate = 1/6 + 1/8
= (4 + 3)/24 = 7/24 per hour

Step 3: Calculate time to fill
Time = 1 / (7/24) = 24/7 hours

Step 4: Convert to mixed number
24/7 = 3 3/7 hours = 3 hours and (3/7 x 60) minutes
= 3 hours and 180/7 minutes
~= 3 hours and 25.7 minutes

Answer: 24/7 hours or approximately 3 hours 26 minutes

Example 2

A pipe can fill a tank in 12 hours. Due to a leak at the bottom, it takes 15 hours to fill the tank. How long will the leak take to empty the full tank?

Solution

Step 1: Identify rates
Filling pipe rate = 1/12 per hour
Let leak empty rate = 1/x per hour

Step 2: Set up equation for combined operation
Net rate = 1/12 - 1/x

Step 3: Use given combined time
Time with leak = 15 hours
Net rate = 1/15

Step 4: Solve for leak rate
1/12 - 1/x = 1/15
1/x = 1/12 - 1/15
1/x = (5 - 4)/60 = 1/60

Step 5: Find leak time
x = 60 hours

Answer: The leak will empty the full tank in 60 hours

Example 3

Two pipes A and B can fill a tank in 20 and 30 minutes respectively. If both pipes are opened together but pipe B is turned off after 10 minutes, how long will it take to fill the tank completely?

Solution

Step 1: Convert to consistent units (minutes)
Pipe A rate = 1/20 per minute
Pipe B rate = 1/30 per minute

Step 2: Calculate combined rate for first 10 minutes
Combined rate = 1/20 + 1/30 = (3 + 2)/60 = 5/60 = 1/12 per minute

Step 3: Work done in first 10 minutes
Work = 10 x (1/12) = 10/12 = 5/6 of tank

Step 4: Remaining work
Remaining = 1 - 5/6 = 1/6 of tank

Step 5: Time for A to complete remaining work
Time = (1/6) / (1/20) = 20/6 = 10/3 minutes = 3 1/3 minutes

Step 6: Total time
Total = 10 + 10/3 = 40/3 = 13 1/3 minutes

Answer: 13 1/3 minutes or 13 minutes 20 seconds

Example 4

Three pipes A, B, and C are attached to a tank. A and B can fill it in 4 and 6 hours respectively, while C can empty it in 12 hours. If all three pipes are opened together when the tank is empty, how long will it take to fill the tank?

Solution

Step 1: Identify rates
Pipe A (filling) = 1/4 per hour
Pipe B (filling) = 1/6 per hour
Pipe C (emptying) = 1/12 per hour

Step 2: Calculate net rate
Net rate = 1/4 + 1/6 - 1/12

Step 3: Find common denominator (12)
= 3/12 + 2/12 - 1/12
= (3 + 2 - 1)/12 = 4/12 = 1/3 per hour

Step 4: Calculate time to fill
Time = 1 / (1/3) = 3 hours

Answer: The tank will be filled in 3 hours

Pipes and Cisterns

Key formulas and rules

Key concepts

Inlet and Outlet Pipes

Work Rate Concept

Combined Operations

Pipes Operating in Sequence

Tank Capacity Method (LCM Approach)

Partial Operations and Leaks

Worked examples

Example 1

Example 2

Example 3

Example 4

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Key formulas and rules

Key concepts

Inlet and Outlet Pipes

Work Rate Concept

Combined Operations

Pipes Operating in Sequence

Tank Capacity Method (LCM Approach)

Partial Operations and Leaks

Worked examples

Example 1

Example 2

Example 3

Example 4

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

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