A solid cone is separated into two pieces by a straight cut parallel to its base, made exactly halfway up its vertical height. What is the ratio of the volume of the small cone above the cut to the volume of the piece below it?

The small cone on top is a scaled copy of the whole cone, and every length in it is halved, so its volume is $(1/2{)}^3=1/8$ of the whole. The piece below is what is left, 7/8 of the whole, so the ratio is (1/8) : (7/8) = 1 : 7. The trap 1 : 8 compares the top piece with the ORIGINAL cone rather than with the piece below it; 1 : 1 assumes that cutting a cone halfway up its height halves the contents, but the wide bottom holds far more; 1 : 3 uses a linear or area ratio where volumes need the cube.