MathDB

Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths

\sqrt{21}

and

\sqrt{31}

. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is

\frac mn

, where

m

and

n

are relatively prime positive integers. Find

m+n

. A parallelepiped is a solid with six parallelogram faces such as the one shown below. [asy] unitsize(2cm); pair o = (0, 0), u = (1, 0), v = 0.8*dir(40), w = dir(70); draw(o--u--(u+v)); draw(o--v--(u+v), dotted); draw(shift(w)*(o--u--(u+v)--v--cycle)); draw(o--w); draw(u--(u+w)); draw(v--(v+w), dotted); draw((u+v)--(u+v+w)); [/asy]

Problem Statement