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Pipes and Cisterns (Inflow–Outflow)

Introduction

Pipes and Cisterns problems, Time and Work concepts का direct application हैं। यहाँ pipes (या taps) tank को या तो fill करते हैं (inflow) या empty करते हैं (outflow)। लक्ष्य यह पता लगाना होता है कि दिए गए conditions में tank को भरने या खाली करने में कितना time लगेगा।

यह pattern इसलिए important है क्योंकि इसमें काम की rate को दो opposite actions - inflow बढ़ाता है और outflow घटाता है - के साथ extend किया जाता है।

Pattern: Pipes and Cisterns (Inflow–Outflow)

Pattern

Key idea: Filling को positive work और emptying को negative work माना जाता है।

अगर कोई pipe x hours में tank भरता है, उसकी rate = 1/x (per hour)। कोई pipe y hours में tank खाली करता है, उसकी rate = -1/y। जब दोनों एक साथ open हों, तो combined rate = (1/x - 1/y).

Formula: Time to fill (or empty) = 1 ÷ (सभी pipes की net rate)

Step-by-Step Example

Question

Pipe A 12 hours में tank भरता है। Pipe B वही tank 15 hours में भरता है। Pipe C उसे 20 hours में खाली करता है। अगर तीनों pipes एक साथ खोले जाएं, तो tank कितने time में भरेगा?

Solution

  1. Step 1: Individual rates निकालें (fill / empty)

    A = 1/12 (filling), B = 1/15 (filling), C = -1/20 (emptying)。

  2. Step 2: Combined rate निकालें

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

  3. Step 3: Common denominator लेकर rates add करें

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

  4. Step 4: Time निकालें

    Time = 1 ÷ (1/10) = 10 hours.

  5. Final Answer:

    10 hours

  6. Quick Check:

    10 hours में A करता है 10×(1/12)=5/6, B करता है 10×(1/15)=2/3, C empty करता है 10×(1/20)=1/2 → net = 5/6 + 2/3 - 1/2 = 1 पूरा काम ✅

Quick Variations

1. दो या अधिक filling pipes + एक emptying pipe।

2. Outflow pipe कुछ delay से खुले (partial operation)।

3. Partial filling (जैसे आधा tank) के लिए time निकालना।

4. ऐसे cases जहाँ outflow > inflow हो, tank भरने के बजाय खाली होता है।

Trick to Always Use

  • Step 1: Inflow को (+) और outflow को (-) मानें।
  • Step 2: उनकी rates add/subtract करके net rate निकालें।
  • Step 3: Formula लगाएं: Time = 1 ÷ (Net rate).
  • Step 4: Partial work के लिए required fraction से time multiply करें।

Summary

Summary

Pipes and Cisterns (Inflow-Outflow) problems में:

  • Filling = positive rate; Emptying = negative rate।
  • Combined rate = inflow और outflow rates का sum।
  • Time = 1 ÷ (net rate)।
  • Half या partial tanks के लिए time proportionally adjust करें।

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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