Two liquids of densities $$2\text{ g cm}^{-3}$$ and $$4\text{ g cm}^{-3}$$ of equal volumes form a homogenous mixture. A solid object made of materials of density $$1\text{ g cm}^{-3}$$ and $$5\text{ g cm}^{-3}$$ mixed homogenously is just found to just float in this mixture. What is ratio by volumes/masses of the materials of the solid object?

PLACEHOLDER ANSWER - the question has two answers. The liquid mixture has density $$\frac{2 + 4}{2} = 3\text{ g cm}^{-3}$$, and just floating means the solid also has density 3. If $$x$$ is the volume fraction of the lighter material, $$1 \times x + 5(1 - x) = 3$$, so $$x = \frac{1}{2}$$ and the volumes are in the ratio $$1 \colon 1$$. The masses are then in the ratio $$1 \times 1 \colon 5 \times 1 = 1 \colon 5$$.