In this question, we are given that a solid floats in two different liquids, A and B.

In liquid A, the solid floats with half of its volume immersed. This means that the buoyant force on the solid is equal to the weight of the liquid displaced by the immersed portion of the solid.

Similarly, in liquid B, the solid floats with 2/3 of its volume immersed. Again, the buoyant force on the solid is equal to the weight of the liquid displaced by the immersed portion of the solid.

Now, let`s compare the buoyant forces in the two liquids. Since the solid is the same in both cases, the weight of the liquid displaced by the immersed portion of the solid is also the same.

Therefore, the buoyant force in both liquids is the same, which means that the densities of the liquids A and B must be in the same ratio.

Looking at the options, we can see that option 4 (4:3) satisfies this condition. Thus, the correct answer is option 4.

Therefore, the densities of the liquids A and B are in the ratio of 4:3.