A solid cone is cut by a plane parallel to its base so that the small cone removed has one-third the height of the original cone. Find the ratio of the curved surface area of the small cone to the curved surface area of the frustum left behind.

The small cone is similar to the whole cone with linear ratio 1 : 3, so its curved surface is $(1/3{)}^2=1/9$ of the whole cone's curved surface. Taking the whole as 9 parts, the small cone is 1 part and the sloping surface of the frustum is 9 − 1 = 8 parts. Ratio = 1 : 8. (The volumes would instead be in the ratio 1 : 26.)