A spherical shell of outer radius R and inner radius R/2 contains a solid sphere of radius R/2 (see figure). The density of the material of the solid sphere is ρ and that of the shell is ρ/2. What is the average mass density of the larger sphere thus formed ?

The total volume of the larger sphere of radius R is V = (4/3)πR³. The inner solid sphere has radius R/2, so its volume is (4/3)π(R/2)³ = (1/8)(4/3)πR³ = V/8, with mass ρ·V/8. The shell occupies the remaining volume V − V/8 = 7V/8 with density ρ/2, giving mass (ρ/2)(7V/8) = 7ρV/16. Total mass = ρV/8 + 7ρV/16 = 2ρV/16 + 7ρV/16 = 9ρV/16. Average density = total mass / total volume = (9ρV/16)/V = 9ρ/16.