There are two vessels - one is in the shape of a cylinder and the other in the shape of a right circular cone. Both the vessels have the same height and the same base radius. The cylindrical vessel and the conical vessel are filled with milk and water respectively and are both filled to half of their maximum heights. The cone is standing on its vertex. The contents of the conical vessel are emptied into the cylindrical vessel. What is the ratio of water to milk in the cylindrical vessel now

Let the common radius be r and the common maximum height be H. Because the cone is standing on its vertex and is filled to half its height, the empty part forms a smaller, similar cone at the top with half the height. The radius of the surface of the water is r/2, so the volume of water is 1/3 * pi * (r/2)^2 * (H/2) = pi * r^2 * H / 24. The volume of milk in the cylinder filled to half its height is pi * r^2 * (H/2) = pi * r^2 * H / 2. When the water is poured into the cylinder, the ratio of water to milk is (pi * r^2 * H / 24) to (pi * r^2 * H / 2), which simplifies to 1 : 12.

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