Consider the figure
Gravitational force inside a solid sphere =GMmR3×r.....(1)
Where r is distance from centre.
Gravitational force due to solid sphere outside=GMmr2....(2) where r is distance from centre.
Gravitational force due to remaining body = gravitational force due to 9 solid sphere due to cavity)
Mass of material inside cavity
=M1=M×43π(R2)343πR3=M8
∴Fbody=GMmR3×(R2)−GMm8(R1)2
GMm2R2−GMm8R2GMR2.m(12−18)mg.(4−18)=3mg8=Fbody