Final answer:
The ratio of water in the tetra pack to milk in the vessel after the described transfers is 4:9.
Step-by-step explanation:
Calculation of Ratios in Mixtures
When 0.5 liters of milk is transferred from the tetra pack to the vessel, the vessel contains 0.5 liters of milk and 2 liters of water, totaling 2.5 liters of mixture. After mixing, when 0.5 liters of this mixture is transferred back to the tetra pack, it will contain the same fraction of milk and water as the mixture in the vessel. Since the mixture in the vessel after transfer is (0.5/2.5) milk and (2/2.5) water, the returned mixture will similarly be 1/5 milk and 4/5 water.
Thus, the tetra pack will have (0.5 liters × 1/5) = 0.1 liters of milk added to the 0.5 liters it already had, totaling 0.6 liters of milk. It will also contain (0.5 liters × 4/5) = 0.4 liters of water, which is the quantity 'x'. The vessel, originally containing 2.5 liters of mixture, after sending back 0.5 liters to the tetra pack, is left with 2 liters of mixture. Since 0.4 liters of water was removed, it has (2 liters - 0.4 liters) = 1.6 liters of water and (2.5 liters - 1.6 liters) = 0.9 liters of milk remaining, which is the quantity 'y'.
Therefore, the ratio of water in the tetra pack (x) to milk in the vessel (y) is 0.4:0.9, which simplifies to 4:9 when the common factor is removed.