Milk and water are mixed in a vessel A in the ratio 3:2 and in vessel B in the ratio 3:1. What quantities should be taken from the vessels to have a new mixture consisting of 5 kg of milk and 2 kg of water?

Let x be the amount from vessel A (3/5 milk, 2/5 water) and y be the amount from vessel B (3/4 milk, 1/4 water). We need 3/5x + 3/4y = 5 and 2/5x + 1/4y = 2. Solving this system: 0.6x + 0.75y = 5 and 0.4x + 0.25y = 2. Multiplying the second by 3 gives 1.2x + 0.75y = 6. Subtracting the first from this gives 0.6x = 1, so x = 1.666 kg. Then y = 5.334 kg.