A right circular cone is sliced into four pieces by planes parallel to its base. All of these pieces have the same height. What is the ratio of the volume of the second-largest piece to the volume of the largest piece? Express your answer as a common fraction.
Let ABC be the original cone, and let DEF be the second-largest piece. Let the radius of the base of the cone be r and the height be h. The volume of the cone is 1/3πr^2h. The volume of DEF is 1/3*π(r/2)^2h=1/12πr^2h. The volume of ABC is1/3πr^2h, so the ratio of the volume of DEF to the volume of ABC is 1/12.