Two cylinders have equal curved surface areas and their radii are in the ratio 2 : 3. Find the ratio of their volumes.

Equal curved surfaces mean 2π$r$₁h₁ = 2π$r$₂h₂, so $r$₁h₁ = $r$₂h₂ and the heights are in the ratio 3 : 2. Volumes are then $r$₁²h₁ : $r$₂²h₂ = $(4\times 3):(9\times 2)=12$ : 18 = 2 : 3 — the same ratio as the radii. (3 : 2 is the ratio of the heights; 4 : 9 squares the radii but forgets that the heights differ; 9 : 4 turns that upside down.)