A cone is cut by a plane parallel to its base through the mid-point of its axis. Find the ratio of the curved surface area of the small cone removed to the curved surface area of the frustum left.

The small cone has linear scale $\frac{1}{2}$, so AREAS scale by ${\left(\frac{1}{2}\right)}^2=\frac{1}{4}$. Its curved surface is $\frac{1}{4}$ of the whole cone’s, leaving $\frac{3}{4}$ for the frustum, so small cone : frustum = 1 : 3. 1 : 4 compares the small cone with the WHOLE cone, not with the frustum; 3 : 1 gives the two the wrong way round; 1 : 7 is the VOLUME ratio for this cut, not the surface-area ratio.