Answer: the average density of the combined world is 256/81 times the average density of the original two worlds.
Step-by-step explanation:
The total mass of the two original worlds is 2M, and their average density is given by:
ρ0 = 2M/(4/3 πR^3) = 3M/(2πR^3)
The final radius of the combined world is 3/4R, so its volume is:
V = 4/3 π(3/4R)^3 = 27/64 πR^3
The mass of the combined world is still 2M, so its density is:
ρ = 2M/V = 128M/(27πR^3)
The ratio of the average density of the combined world to that of the original worlds is:
ρ/ρ0 = (128M/(27πR^3)) / (3M/(2πR^3)) = 256/81