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Partial or Negative Work (Leak or Undo Work)

Introduction

Partial या Negative Work problems में ऐसा होता है कि कोई agent पहले किया हुआ काम undo कर देता है - जैसे tank में leak, या कोई व्यक्ति पहले पूरा किया गया काम हटाता है। यह pattern important है क्योंकि कई Time & Work questions में ऐसी conditions दी होती हैं जहाँ progress घटती है (leaks, theft, cancellation), और इन्हें negative rates की तरह treat करने से logic बहुत आसान हो जाता है।

Key idea: हर contributor को एक rate (work per unit time) की तरह model करें। Leak/undo वाले agent की rate negative रखें और सभी rates को जोड़कर net progress निकालें।

Pattern: Partial or Negative Work (Leak or Undo Work)

Pattern

Key concept: Undoing actions को negative rates मानें; net rate = positive rates का sum - negative rates का sum। Time = Total work ÷ Net rate.

Formula summary:
Worker rate = 1 / time_to_complete (positive for filling/doing)।
Leak/undo rate = -(1 / time_to_empty_or_undo) (negative)।
Net rate = Σ(positive rates) + Σ(negative rates).

Step-by-Step Example

Question

Pipe A 10 hours में tank भरता है, Pipe B 15 hours में भरता है। एक leak पूरा भरा tank 30 hours में खाली कर देता है। अगर A, B और leak तीनों साथ खुले हों, तो tank कितने समय में भरेगा?

Solution

  1. Step 1: Individual rates पहचानें (positive और negative):

    A = 1/10 (filling), B = 1/15 (filling), Leak = -1/30 (emptying)।

  2. Step 2: Net rate निकालें:

    Net rate = 1/10 + 1/15 - 1/30. LCM = 30 → (3 + 2 - 1)/30 = 4/30 = 2/15 tank/hour.

  3. Step 3: पूरा tank भरने का time निकालें:

    Time = 1 ÷ (2/15) = 15/2 = 7.5 hours.

  4. Final Answer:

    Tank 7 hours 30 minutes में भर जाएगा।

  5. Quick Check:

    7.5 hours में: A = 0.75, B = 0.5, leak = 0.25 remove करता है → net = 1.0 ✔️

Quick Variations

1. Multiple leaks या undoers: सभी negative rates जोड़कर positive rates से subtract करें।

2. Leak बाद में start या पहले stop हो - phases में work calculate करें।

3. कोई worker बीच में चला जाए और leak चलता रहे - phases में split करें और कुल work जोड़ें।

4. Leak का अकेला emptying time पूछें - leak rate invert करके निकालें (time = 1 / leak rate)।

Trick to Always Use

  • Step 1 → हर actor को rate (1/time) में बदलें; undoing actor को negative रखें।
  • Step 2 → सभी rates को algebraically जोड़कर net rate निकालें।
  • Step 3 → अगर events phases में हों, तो हर phase का work अलग निकालें (Work = rate × time)।
  • Step 4 → Final phase का time = Remaining work ÷ उस phase की net rate। अंत में quick-check करें कि total work = 1 आ रहा है।

Summary

Summary

Partial / Negative Work problems में:

  • सभी participants को positive या negative rates में model करें।
  • Net progress = algebraic sum of rates; time = 1 ÷ net rate।
  • Phased actions होने पर हर phase का work अलग निकालें और जोड़ें।
  • Quick check: positive contributions - undoing contributions = 1 होना चाहिए।

Practice

(1/5)
1. Pipe A can fill a tank in 12 hours, and a leak at the bottom can empty the full tank in 24 hours. How long will it take to fill the tank if the leak is also open?
easy
A. 18 hours
B. 16 hours
C. 24 hours
D. 20 hours

Solution

  1. Step 1: Identify rates:

    Pipe A filling rate = 1/12 tank/hour; Leak emptying rate = -1/24 tank/hour.

  2. Step 2: Compute net rate:

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

  3. Step 3: Find total time:

    Time = 1 ÷ (1/24) = 24 hours.

  4. Final Answer:

    24 hours → Option C.

  5. Quick Check:

    24×(1/12 - 1/24)=24×(1/24)=1 (full tank) ✅
Hint: Subtract leak rate from filler rate to get net rate; invert net rate to get time.
Common Mistakes: Forgetting to subtract the leak's rate (treating it as positive).
2. Pipe A can fill a tank in 10 hours and Pipe B in 15 hours. A leak can empty the tank in 30 hours. All are open together. How long will it take to fill the tank?
easy
A. 8.5 hours
B. 9.25 hours
C. 7.25 hours
D. 7.5 hours

Solution

  1. Step 1: Identify rates:

    Pipe A fills 1/10 of the tank per hour, Pipe B fills 1/15 per hour, and the leak empties 1/30 per hour.

  2. Step 2: Compute net rate:

    Net rate = 1/10 + 1/15 - 1/30 = (3 + 2 - 1)/30 = 4/30 = 2/15 tank/hour.

  3. Step 3: Find total time to fill:

    Time = 1 ÷ (2/15) = 15/2 = 7.5 hours.

  4. Final Answer:

    The tank will be filled in 7.5 hours → Option D.

  5. Quick Check:

    7.5 × (1/10 + 1/15 - 1/30) = 7.5 × (2/15) = 1 ✅
Hint: Treat the leak as a negative rate, add all rates, and take the reciprocal for total time.
Common Mistakes: Using the leak rate as positive instead of negative or forgetting to take LCM correctly.
3. Pipe A can fill a tank in 6 hours. But due to a leak, it takes 8 hours to fill it. How long will the leak alone take to empty the full tank?
easy
A. 24 hours
B. 16 hours
C. 20 hours
D. 18 hours

Solution

  1. Step 1: Identify rates:

    Filling rate (A) = 1/6 tank/hour. Net (with leak) = 1/8 tank/hour.

  2. Step 2: Compute leak rate:

    Leak rate = A - Net = 1/6 - 1/8 = (4 - 3)/24 = 1/24 tank/hour (emptying).

  3. Step 3: Leak alone empties tank in:

    Time = 1 ÷ (1/24) = 24 hours.

  4. Final Answer:

    24 hours → Option A.

  5. Quick Check:

    1/6 - 1/24 = 1/8 (net), which matches the given 8-hour fill time ✅
Hint: Leak rate = filling rate - net rate; invert to get emptying time.
Common Mistakes: Subtracting in the wrong order (net - fill instead of fill - net).
4. Two pipes A and B can fill a tank in 12 and 18 hours respectively. A leak can empty the full tank in 36 hours. If all are opened together, how long will it take to fill the tank?
medium
A. 10 hours
B. 9 hours
C. 8 hours
D. 7.2 hours

Solution

  1. Step 1: Filling and leak rates:

    A = 1/12, B = 1/18, Leak = -1/36 (tank/hour).

  2. Step 2: Net rate:

    1/12 + 1/18 - 1/36 = (3 + 2 - 1)/36 = 4/36 = 1/9 tank/hour.

  3. Step 3: Total time:

    Time = 1 ÷ (1/9) = 9 hours.

  4. Final Answer:

    9 hours → Option B.

  5. Quick Check:

    9×(1/12 + 1/18 - 1/36) = 9×(1/9) = 1 ✅
Hint: Add filler rates, subtract leak rate, then invert to find time.
Common Mistakes: Using wrong LCM or forgetting the negative sign for the leak.
5. A pipe can fill a tank in 8 hours. Because of a leak, the tank is filled in 10 hours. How long will the leak take to empty the tank completely?
medium
A. 40 hours
B. 20 hours
C. 30 hours
D. 24 hours

Solution

  1. Step 1: Filling and net rates:

    Filling rate = 1/8 tank/hour; Net (with leak) = 1/10 tank/hour.

  2. Step 2: Compute leak rate:

    Leak = Filling - Net = 1/8 - 1/10 = (5 - 4)/40 = 1/40 tank/hour (emptying).

  3. Step 3: Leak empties tank in:

    Time = 1 ÷ (1/40) = 40 hours.

  4. Final Answer:

    40 hours → Option A.

  5. Quick Check:

    1/8 - 1/40 = 1/10 (net), which matches the given 10-hour fill time ✅
Hint: Leak time = 1 ÷ (filling rate - net rate).
Common Mistakes: Mixing denominators when subtracting fractional rates.

Mock Test

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