Two metal spheres, one of radius R and the other of radius 2R respectively have the same surface charge density $\mathrm{\sigma }$. They are brought in contact and separated. What will be the new surface charge densities on them?

when spheres are kept in contact, charge redistribution takes place to get same potential. 

Total combined charge = $20{\mathrm{\pi R}}^2\mathrm{\sigma }$. 

Potential of both the spheres become equal, hence Q = $\frac{Q^I}{2}$ where Q is the charge on sphere of radius R and $Q^I$ is the charge on sphere of radius 2R.

Thus charge on sphere of radius R becomes $\frac{20{\mathrm{\pi R}}^2\mathrm{\sigma }}{3}$ , while charge on sphere of radius 2R becomes $\frac{40{\mathrm{\pi R}}^2\mathrm{\sigma }}{3}$

Hence new charge densities will be

${\mathrm{\sigma }}_1=\frac{5}{3}\mathrm{\sigma }, {\mathrm{\sigma }}_2=\frac{5}{6}\mathrm{\sigma }$