A block is made with equal volumes of two materials whose densities are in the ratio $$1 \colon 2$$. This block floats with $$\frac{3}{4}$$ of its volume submerged in a certain liquid. Find the ratio of the volumes of the materials (lighter to denser) for $$\frac{2}{3}$$ of the volume of the block to be just submerged.

With densities $$\rho$$ and $$2\rho$$ in equal volumes, the block's density is $$1.5\rho$$, and floating $$\frac{3}{4}$$ submerged gives the liquid density $$\frac{1.5\rho}{3/4} = 2\rho$$. For $$\frac{2}{3}$$ submerged, the block's density must be $$\frac{4\rho}{3}$$. If $$f$$ is the volume fraction of the lighter material, $$\rho f + 2\rho(1 - f) = \frac{4\rho}{3}$$, so $$f = \frac{2}{3}$$ and the volumes are in the ratio $$2 \colon 1$$.