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Pipes and Cisterns (Inflow–Outflow)

Introduction

Problems on Pipes and Cisterns are an application of Time and Work concepts. Here, pipes (or taps) can either fill a tank (inflow) or empty it (outflow). The aim is to determine how long it will take to fill or empty a tank under given conditions.

This pattern is important because it extends the concept of work rate to simultaneous opposite actions - where inflow increases water and outflow decreases it.

Pattern: Pipes and Cisterns (Inflow–Outflow)

Pattern

The key idea: Filling and emptying rates are treated as positive and negative works respectively.

If one pipe fills a tank in x hours, its rate = 1/x (per hour). If another pipe empties it in y hours, its rate = -1/y. When both are open together, their combined rate = (1/x - 1/y).

Formula: Time to fill (or empty) = 1 ÷ (Sum of rates of all pipes)

Step-by-Step Example

Question

Pipe A can fill a tank in 12 hours. Pipe B can fill the same tank in 15 hours. Pipe C can empty it in 20 hours. If all three pipes are opened together, how long will it take to fill the tank?

Solution

  1. Step 1: Compute individual rates (fill / empty)

    Rate of A = 1/12 (filling), B = 1/15 (filling), C = -1/20 (emptying).

  2. Step 2: Calculate combined rate

    Combined rate = 1/12 + 1/15 - 1/20.

  3. Step 3: Find common denominator and sum rates

    LCM of 12, 15, 20 = 60.
    So, (1/12 + 1/15 - 1/20) = (5 + 4 - 3)/60 = 6/60 = 1/10.

  4. Step 4: Compute time to fill

    Time = 1 ÷ (1/10) = 10 hours.

  5. Final Answer:

    10 hours

  6. Quick Check:

    In 10 hours, A does 10×(1/12)=5/6, B does 10×(1/15)=2/3, C empties 10×(1/20)=1/2 → net = 5/6+2/3-1/2=10/12+8/12-6/12=12/12=1 ✅

Quick Variations

1. Two or more filling pipes (inflow) and one emptying pipe (outflow).

2. Outflow pipes open after a certain delay (partial operation).

3. Finding rate or time for partial filling (e.g., half tank).

4. Situations where tank overflows or drains out if outflow > inflow.

Trick to Always Use

  • Step 1: Treat inflow (+) and outflow (-) separately.
  • Step 2: Combine their rates (sum or difference).
  • Step 3: Apply Time = 1 ÷ (Net rate).
  • Step 4: For partial filling, multiply time by required fraction.

Summary

Summary

In Pipes and Cisterns (Inflow-Outflow) problems:

  • Filling → positive rate; Emptying → negative rate.
  • Use combined rate = sum of all inflow and outflow rates.
  • Time = 1 ÷ (net rate).
  • For half/partial tanks, adjust time proportionally.

Practice

(1/5)
1. Pipe A can fill a tank in 10 hours while Pipe B can fill it in 15 hours. If both are opened together, in how many hours will the tank be filled?
easy
A. 6 hours
B. 7 hours
C. 8 hours
D. 9 hours

Solution

  1. Step 1: Identify individual rates

    Rate of A = 1/10 per hour; Rate of B = 1/15 per hour.

  2. Step 2: Compute combined rate

    Combined rate = 1/10 + 1/15 = (3 + 2)/30 = 5/30 = 1/6 per hour.

  3. Step 3: Compute total time

    Time to fill = 1 ÷ (1/6) = 6 hours.

  4. Final Answer:

    6 hours → Option A

  5. Quick Check:

    6×(1/10 + 1/15) = 6×(1/6) = 1 ✅
Hint: Add filling rates directly; invert to get time.
Common Mistakes: Averaging times instead of summing rates.
2. Pipe A can fill a tank in 12 hours and Pipe B can empty it in 24 hours. If both are opened together, how long will it take to fill the tank?
easy
A. 18 hours
B. 20 hours
C. 24 hours
D. 16 hours

Solution

  1. Step 1: Assign signs to inflow and outflow

    A’s filling rate = +1/12 per hour; B’s emptying rate = -1/24 per hour.

  2. Step 2: Compute net rate

    Net rate = 1/12 - 1/24 = (2 - 1)/24 = 1/24 per hour.

  3. Step 3: Compute required time

    Time to fill = 1 ÷ (1/24) = 24 hours.

  4. Final Answer:

    24 hours → Option C

  5. Quick Check:

    In 24 hours A fills 2 units; B empties 1 unit → net = 1 unit (full tank) ✅
Hint: Subtract outflow rate from inflow rate; invert net rate.
Common Mistakes: Adding outflow instead of subtracting it from inflow.
3. Two pipes A and B can fill a tank in 8 hours and 12 hours respectively. If both are opened together and after 4 hours Pipe A is closed, how much more time will B take to fill the tank?
easy
A. 4 hours
B. 2 hours
C. 5 hours
D. 3 hours

Solution

  1. Step 1: Identify individual rates

    A’s rate = 1/8; B’s rate = 1/12.

  2. Step 2: Find combined rate

    Combined = (3 + 2)/24 = 5/24 per hour.

  3. Step 3: Work done in first 4 hours

    Work = 4 × (5/24) = 20/24 = 5/6.

  4. Step 4: Remaining work and B's time

    Remaining = 1/6. B alone does 1/12 per hour → time = (1/6) ÷ (1/12) = 2 hours.

  5. Final Answer:

    2 hours → Option B

  6. Quick Check:

    5/6 + 1/6 = 1 → complete tank ✅
Hint: Compute work done together, then finish with solo rate.
Common Mistakes: Using combined rate even after A stops.
4. Pipe A can fill a tank in 16 hours. Pipe B can empty it in 24 hours. If both are opened together, in how many hours will the tank be filled?
medium
A. 48 hours
B. 64 hours
C. 96 hours
D. 72 hours

Solution

  1. Step 1: Assign individual rates

    A = +1/16; B = -1/24.

  2. Step 2: Compute net rate

    Net = (3 - 2)/48 = 1/48 per hour.

  3. Step 3: Compute filling time

    Time = 1 ÷ (1/48) = 48 hours.

  4. Final Answer:

    48 hours → Option A

  5. Quick Check:

    A fills 3 units; B empties 2 units → net = 1 unit (full tank) in 48 hours ✅
Hint: Use common denominator to combine rates cleanly.
Common Mistakes: Wrong sign for emptying pipe.
5. Two pipes A and B can fill a tank in 12 hours and 18 hours respectively. A third pipe C can empty it in 36 hours. If all three are opened together, how long will it take to fill the tank?
medium
A. 8 hours
B. 10 hours
C. 12 hours
D. 9 hours

Solution

  1. Step 1: List individual rates

    A = 1/12, B = 1/18, C = -1/36.

  2. Step 2: Combine rates using LCM

    LCM = 36 → (3 + 2 - 1)/36 = 4/36 = 1/9 per hour.

  3. Step 3: Compute time

    Time = 1 ÷ (1/9) = 9 hours.

  4. Final Answer:

    9 hours → Option D

  5. Quick Check:

    9×(1/12 + 1/18 - 1/36) = 1 → complete tank ✅
Hint: Add inflow rates, subtract outflow rate, invert.
Common Mistakes: Incorrect LCM or sign error on outflow.

Mock Test

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