A vessel contains a mixture of milk and water in the ratio of 4 : 1. When 15 litres of the mixture is taken out and 15 litres of water is poured into the vessel, the ratio becomes 2 : 3. How many litres of milk contain in the vessel finally?

Let initial mixture be 4x milk + x water = 5x total. After removing 15 litres, remaining: 4x - 12 milk (since 15 × 4/5 = 12) and x - 3 water. Adding 15 water gives: milk = 4x - 12, water = x - 3 + 15 = x + 12. New ratio 2:3 means (4x - 12)/(x + 12) = 2/3. Solving: 12x - 36 = 2x + 24, so 10x = 60, x = 6. Final milk = 4(6) - 12 = 12 litres.