You now carefully compress the sphere so its radius is r/2. what is the density of the compressed sphere

Density = mass/volume.

the volume of the uncompressed sphere is : 4/3.π. R³ and the density is:

D₁ = m / (4/3.π. R³)

the volume of the compressed sphere is 4/3.π. (R/2) ³ = π. R³/6 and its density is D₂ = m / (π. R³/6) (since the mass didn't change).

Density increase: D₂/D₁ = [m / (π. R³/6) ] / [m / (4/3.π. R³) ] = 8 times