Leaks, Drains & Alternate Openings

🎯 Learning Objective

Solve leak-delay problems, two inlets plus a drain, and alternate-hour opening questions.

💡 Concept

• Leak delay: (rate without leak) − (rate with leak) = leak's own rate → invert for its emptying time
• Two inlets + drain: net = r₁ + r₂ − r₃ per hour, then capacity ÷ net
• Alternate opening: compute units filled per CYCLE, repeat cycles till just below full
• The last partial slot is filled only by the pipe whose turn it is

🧮 Key Formulas

Leak rate = 1/t − 1/t′

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Cycle units = A's units + B's units

✏️ Easy Example

Q. A pipe can fill a tank in 6 hours, but due to a leak it takes 9 hours. In how long can the leak alone empty a full tank?

1. Capacity = LCM(6, 9) = 18 units
2. Leak = 3 − 2 = 1 unit/h
3. 18 ÷ 1 = 18

Answer: 18 hours

🇮🇳 Real-Life Example

Your overhead tank taking longer to fill each summer? A worn washer or hairline leak is silently draining it — plumbers estimate leak size from exactly this delay.

📝 Exam-Level Example

Q. Pipes A and B can fill a tank in 12 and 15 minutes; pipe C can empty it in 20 minutes. All three open — time to fill?

1. Capacity = LCM = 60 units
2. Net = 5 + 4 − 3 = 6/min
3. 60 ÷ 6 = 10

Answer: 10 minutes

📝 Exam-Level Example

Q. A fills a tank in 4 hours, B in 6 hours. They are opened alternately for 1 hour each, A first. When is the tank full?

1. Capacity = 12 → A = 3/h, B = 2/h
2. Each 2-hour cycle = 5 units → after 4 h: 10 units
3. 2 units left, A's turn at 3/h → 2/3 h = 40 min

Answer: 4 hours 40 minutes

🪄 Memory Trick

In alternate problems, run full cycles till JUST below capacity — never overshoot — then hand the tail to whichever pipe's turn it is.

⚠️ Common Mistakes

• ❌ Overshooting the capacity with one extra full cycle
• ❌ Subtracting leak TIME instead of leak RATE
• ❌ Giving the final partial hour to the wrong pipe

🏆 Exam Tips

• ✅ Track filled units on a number line for alternate questions
• ✅ With a leak, filling time always increases — if your answer got smaller, a sign flipped

📌 Summary

• Leak rate = normal rate − delayed rate
• Inlets + drain → signed sum
• Alternate: cycle units × cycles + careful tail
• Never overshoot the capacity