Center of mass and distance ratio

[kthouz]

how to show that the ratio of the distance between two particles from their centers of mass is the inverse of their masses? I tried but i found a way which can be possible if only we assume that they are constituted by a same number of small particles.

 

[lpfr]

Start with the definition of CM:
$$x_{cm}={m_1\ell_1+ m_2\ell_2 \over m_1+m_2}$$
Then ask that the coordinate of the CM be zero. This is just a change in the position of the origin of x.
Then compute de ratio of distances to the CM. Don't forget that now one of the distances in negative. See why?