jmc

Embed the tetrahedron in 4-space to make calculations easier. Its vertices are $(1,0,0,0)$, $(0,1,0,0)$, $(0,0,1,0)$, $(0,0,0,1)$. To get the center of any face, we take the average of the three coordinates of that face. The vertices of the center of the faces are: $(\frac{1}{3},\frac{1}{3},\frac{1}{3},0)$,$(\frac{1}{3},\frac{1}{3},0,\frac{1}{3})$,$(\frac{1}{3},0,\frac{1}{3},\frac{1}{3})$,$(0,\frac{1}{3},\frac{1}{3},\frac{1}{3})$. The side length of the large tetrahedron is $\sqrt{2}$ by the distance formula. The side length of the smaller tetrahedron is $\frac{\sqrt{2}}{3}$ by the distance formula. Their ratio is $1:3$, so the ratio of their volumes is ${\left(\frac{1}{3}\right)}^3=\frac{1}{27}$. $m+n=1+27=\boxed{28}$.