<p>Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths <img alt="$\sqrt{21}$" height="18" src="https://latex.artofproblemsolving.com/5/4/b/54b7a75cd4076dc5e9aaed1c570289a4e27802d4.png" width="33" /> and <img alt="$\sqrt{31}$" height="18" src="https://latex.artofproblemsolving.com/9/8/f/98f84dc10eae5d7fe9624ef936a66f08882bcfab.png" width="33" />. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is <img alt="$\frac{m}{n}$" height="33" src="https://latex.artofproblemsolving.com/2/7/f/27f583697295fc6f723862c13228ee55107c33ca.png" width="18" />, where <img alt="$m$" height="8" src="https://latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png" width="15" /> and <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> are relatively prime positive integers. Find <img alt="$m + n$" height="12" src="https://latex.artofproblemsolving.com/5/3/b/53bf1afcb274a3bb84338109fffec8a1f1c02008.png" width="48" />. A parallelepiped is a solid with six parallelogram faces such as the one shown below.</p> <p><img alt="[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]" height="138" src="https://latex.artofproblemsolving.com/b/c/c/bcc8a67600e8a557236d37e2d01ae1737ef052b5.png" width="188" /></p>

Question No: 343

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 $\frac{m}{n}$ , 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.

$[asy] unitsize(2cm); pair o = (0, 0), u = (1, 0), v = 0.8*dir(40), w = dir(70); draw(o--u--(u+v)); draw(o--v--(u+v), dotted); draw(shift(w)*(o--u--(u+v)--v--cycle)); draw(o--w); draw(u--(u+w)); draw(v--(v+w), dotted); draw((u+v)--(u+v+w)); [/asy]$

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