Q.62 A container originally contains 10 litres of pure spirit. From this container 1 litre of spirit is replaced with 1 litre of water. Subsequently, 1 litre of the mixture is again replaced with 1 litre of water and this process is repeated one more time. How much spirit is now left in the container?
Option (D) 7.29 Litres
After three 1L replacements from 10L pure spirit:
Final Spirit = 10 × (9/10)³ = 10 × 0.729 = 7.29L
Master Formula: Successive Dilution
Final Amount = Initial × (1 – Fraction Removed)n
Here: 10 × (9/10)3 = 10 × 0.729 = 7.29L
Here: 10 × (9/10)3 = 10 × 0.729 = 7.29L
Each replacement removes 10% of current mixture, compounding geometrically.
Detailed Step-by-Step Calculation
0
Initial: 10L pure spirit
1
Step 1: Remove 1L → 9L spirit + 1L water = 90%
2
Step 2: Remove 0.9L spirit → 8.1L spirit = 81%
3
Step 3: Remove 0.81L spirit → 7.29L
| Option | Amount | Calculation Error | ✅ Correct? |
|---|---|---|---|
| (A) | 7.58L | Overestimates remaining | ❌ Wrong |
| (B) | 7.84L | 10 × (0.9)² only (2 steps) | ❌ Wrong |
| (C) | 7L | Linear: 10 – 1×3 (no compounding) | ❌ Wrong |
| (D) | 7.29L | 10 × 0.9³ = 7.29 ✓ | ✅ CORRECT |
Exam Shortcuts & Pro Tips
- 🔥 Pattern Recognition: Same volume replaced = geometric series
- ⚡ Instant Formula: V₀ × (1 – r/V)ⁿ
- ✅ Quick Check: 3 steps × 10% removal ≈ 73% left (7.3L range)
- 🚫 Avoid Traps: Linear math (7L), partial compounding (7.84L)
- 💡 Pro Tip: Calculate (1 – fraction removed)ⁿ first → 0.9³ = 0.729!
🚨 Common Mistakes Explained
- 7.84L (B): Forgot 3rd replacement, did only (0.9)²
- 7L (C): Simple subtraction ignores mixture concentration
- 7.58L (A): Wrong formula or calculator error
