jmc

Let the radius of the large sphere be $R$, and of the inner sphere $r$. Label the vertices of the tetrahedron $ABCD$, and let $O$ be the center. Then pyramid $[OABC]+[OABD]+[OACD]+[OBCD]=[ABCD]$, where $[\dots ]$ denotes volume; thus $[OABC]=\frac{[ABCD]}{4}$. Since $OABC$ and $ABCD$ are both pyramids that share a common face $ABC$, the ratio of their volumes is the ratio of their altitudes to face $ABC$, so $r=\frac{h_{ABCD}}{4}$. However, $h_{ABCD}=r+R$, so it follows that $r=\frac{R}{3}$. Then the radius of an external sphere is $\frac{R-r}{2}=\frac{R}{3}=r$. Since the five described spheres are non-intersecting, it follows that the ratio of the volumes of the spheres is $5\cdot {\left(\frac{1}{3}\right)}^3=\frac{5}{27}\approx \boxed{.2}$.