Repeated Replacement Formula

🎯 Learning Objective

Find how much pure liquid remains after some is repeatedly drawn out and replaced with water.

💡 Concept

• When x litres are removed from V litres and replaced by water, done n times, the pure liquid left follows a fixed formula.
• Final pure quantity = Initial × (1 − x/V)^n
• Each operation keeps the same FRACTION (1 − x/V) of the pure liquid that was present.
• Water added does not change the total volume V, so the fraction stays the same every round.
• After finding pure liquid left, water left = V − (pure left).

🧮 Key Formulas

Pure left = Initial × (1 − x/V)^n

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Water left = V − Pure left

✏️ Easy Example

Q. A vessel has 20 L of pure milk. 4 L is drawn out and replaced with water. How much milk is left?

1. Here V = 20, x = 4, n = 1
2. Milk left = 20 × (1 − 4/20) = 20 × (16/20)
3. = 20 × 0.8 = 16 L

Answer: 16 L of milk

🇮🇳 Real-Life Example

A milk dairy tank of 40 L gets 4 L skimmed off and topped up with water every round for quality checks — the replacement formula predicts the exact milk left after each round.

📝 Exam-Level Example

Q. A container has 40 L of milk. 4 L is drawn and replaced with water. This is repeated 3 times. Find the milk remaining.

1. V = 40, x = 4, n = 3
2. Milk left = 40 × (1 − 4/40)³ = 40 × (36/40)³
3. = 40 × (9/10)³ = 40 × 0.729
4. = 29.16 L

Answer: 29.16 L of milk

📝 Exam-Level Example

Q. After two identical replacement operations, the ratio of milk to water in a vessel is 16 : 9. What fraction was drawn out each time?

1. Milk fraction left = 16 / (16 + 9) = 16/25
2. So (1 − x/V)² = 16/25
3. Take square root: (1 − x/V) = 4/5
4. x/V = 1 − 4/5 = 1/5

Answer: 1/5 of the vessel was drawn out each time

🪄 Memory Trick

Keep only the fraction (1 − x/V) and raise it to the power n — never track the water separately.

⚠️ Common Mistakes

• ❌ Using the replaced amount x as the whole answer instead of the fraction
• ❌ Forgetting the power n (using ×1 when it should be repeated)
• ❌ Subtracting from the wrong total when finding water left

🏆 Exam Tips

• ✅ Turn (36/40) into (9/10) before cubing — cleaner arithmetic
• ✅ If asked for water, compute milk first, then subtract from V

📌 Summary

• Pure left = Initial × (1 − x/V)^n
• Same fraction survives every round
• Reverse questions → take the n-th root
• Water left = V − pure left