Pipes — Filling & Emptying Rates

🎯 Learning Objective

Treat inlets as positive rates and outlets as negative, then combine them with the LCM method.

💡 Concept

• A pipe that fills in n hours works at +1/n tank per hour
• An outlet or leak that empties in m hours works at −1/m per hour
• Net rate = sum of all rates WITH signs; Time = capacity ÷ net rate
• LCM method: capacity = LCM of the hours → every pipe gets whole units/hour
• The tank fills only if the net rate is positive

🧮 Key Formulas

Inlet: +1/n per hour

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Outlet: −1/m per hour

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Time = Capacity ÷ Net rate

✏️ Easy Example

Q. Pipe A fills a tank in 12 hours and pipe B in 6 hours. How long do they take together?

1. Capacity = LCM(12, 6) = 12 units
2. A = 1/h, B = 2/h → net 3/h
3. 12 ÷ 3 = 4

Answer: 4 hours

🇮🇳 Real-Life Example

Your rooftop tank fills from the motor while the kitchen tap keeps drawing — it fills only because inflow beats outflow. Every housing society runs this equation daily.

📝 Exam-Level Example

Q. Pipe A fills a tank in 10 hours while leak B empties it in 15 hours. Both open — in how long is the tank full?

1. Capacity = LCM(10, 15) = 30 units
2. Net = 3 − 2 = 1 unit/h
3. 30 ÷ 1 = 30

Answer: 30 hours

📝 Exam-Level Example

Q. A fills in 6 h, B fills in 8 h and C empties in 12 h. All three open — time to fill?

1. Capacity = LCM = 24 units
2. Net = 4 + 3 − 2 = 5/h
3. 24 ÷ 5 = 4.8 h

Answer: 4.8 hours (4 h 48 min)

🪄 Memory Trick

Same LCM trick as Time & Work — only the sign changes: fill = +, empty = −.

⚠️ Common Mistakes

• ❌ Adding an outlet's rate with a plus sign
• ❌ Averaging the hours ((12 + 6)/2 = 9 ✗) instead of adding rates

🏆 Exam Tips

• ✅ Two fillers together must beat the faster pipe's solo time — sanity check
• ✅ Convert decimal hours to minutes: 0.8 h = 48 min

📌 Summary

• Inlet +1/n, outlet −1/m
• Net rate = signed sum
• Capacity = LCM of times
• Positive net rate, warna tank kabhi nahi bharega