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