A conical glass standing point downwards is filled with juice to exactly half its depth. What fraction of the glass's full capacity is the juice?

The juice forms a small cone similar to the whole glass with half its height, so every length in it is halved and its volume is $(1/2{)}^3=1/8$ of the full capacity. Half depth is far from half full — the glass is widest at the top, where the missing juice would sit. (1/2 assumes volume follows depth directly; 1/4 uses the square, which belongs to areas; 1/6 has no basis in the similar-solids rule.)