Numerical answer
A container holds 80 litres of pure milk. A certain quantity x litres is removed and replaced with water. This operation is performed once more with the same x. The container now has milk and water in the ratio 16 : 9. Find x (in litres).
Show answer and solution
Answer: 16
After n replacements of x litres from a V-litre container, milk left = V(1 − x/V)ⁿ.
Final milk fraction = 16/(16 + 9) = 16/25.
So (1 − x/80)² = 16/25 → 1 − x/80 = 4/5 → x/80 = 1/5 → x = 16.
Check: after first step milk = 64 L; second step removes 16 L of mixture containing 64/80 × 16 = 12.8 L milk, leaving 51.2 L milk, i.e. 51.2 : 28.8 = 16 : 9. ✓
Final milk fraction = 16/(16 + 9) = 16/25.
So (1 − x/80)² = 16/25 → 1 − x/80 = 4/5 → x/80 = 1/5 → x = 16.
Check: after first step milk = 64 L; second step removes 16 L of mixture containing 64/80 × 16 = 12.8 L milk, leaving 51.2 L milk, i.e. 51.2 : 28.8 = 16 : 9. ✓
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