A solid cone is cut by two planes parallel to its base, one at one-third of the way down from the apex and the other at two-thirds of the way down. Find the ratio of the volumes of the three pieces, taken from the apex downwards.

Each cut leaves a small cone similar to the whole, and volumes of similar solids go as the cube of the linear ratio, so the cones down to the two cuts and the whole cone are as $1^3:2^3:3^3=1$ : 8 : 27. The three PIECES are the differences: 1, 8 − 1 = 7 and 27 − 8 = 19, giving 1 : 7 : 19. (1 : 8 : 27 lists the stacked cones, not the separate slices; 1 : 2 : 3 uses the heights, as though equal-height slices held equal amounts; 1 : 7 : 27 subtracts for the middle slice but forgets to for the last.)