In a regular tetrahedron, the centres of the four faces are the vertices of a smaller tetrahedron. If the ratio of the volume of the larger tetrahedron to the volume of the smaller tetrahedron is $\frac{m}{n}$, find $m+n$.
My attempt:
I assumed the origin to be a vertex of the larger tetrahedron $A$. I then drew three non - colinear, non - coplanar vectors $(\vec a,\vec b, \vec c)$ from the origin. Thus volume of A : $$\frac{1}{3}.\frac {1}{2}.[\vec a \vec b\vec c]$$ Position vector of the three centroids will be : $$(\vec a + \vec b) \over 3$$$$(\vec a + \vec c) \over 3$$$$(\vec c + \vec b) \over 3$$
Now when I calculate the volume of the small tetrahedron by the previous formula : $$V = \frac{1}{6}.\frac{1}{27}.[(\vec a + \vec b) (\vec a +\vec c)(\vec b + \vec c)] = \frac{1}{6}.\frac{1}{27}.2.[\vec a \vec b \vec c]$$
But apparently the answer given is $28$. While I get $29$. What is the error I made?
You can do this with classical geometry as well. Let $A,B,C,D$ be the vertices of the large tetrahedron, and let $M$ be the midpoint of $CD$. Then we have an isosceles triangle $\triangle ABM$, where $AB$ is one of the edges of the tetrahedron, while $AM$ and $BM$ are the altitudes of two of the faces.
Note that the small tetrahedron has a vertex on each of $AM$ and $BM$. Say $E$ is the vertex on $AM$ and $F$ is the one on $BM$. We then have that $\triangle EFM$ is isosceles and similar to $\triangle ABM$, and $EF$ is one of the edges of the small tetrahedron.
So, the ratio between the sides of these two similar triangles is the same as the ratio between the side lengths of the two tetrahedra. The ratio of the two volumes is thus the cube of the side length ratio. What is the ratio?
An altitude on a face is also a median (because the faces are equilateral). $E$ is the intersection of all the medians on $\triangle ACD$, and it is known that the medians divide one another in $2:1$ ratio. Therefore, $\frac{AE}{EM} = 2$. This implies that $\frac{EM}{AM} = \frac13$, which means that $\frac{EF}{AB} = \frac13$. Therefore, the ratio between the volumes is $\left(\frac13\right)^3 = \frac1{27}$.