Question

if the radius of the earth were increased by a factor of 3 . By what factor could its density hare to be changed to keep 'g' the same?

Answer 1

if you see the formula of gravitational acceleration on earth.
g=GM/R^2
now writing M as density*volume
M= 4π/3 R^3*density
so G being the gravitational constant
g(acceleration due to gravity) will be directly proportional to density*R
so if radius increased by a factor of 3
so the density needs to be decreased by a factor of 3 to keep g constant.

Answer 2

the formula for calculating the density of the earth is given as

ρ = M /V = [gR^2 / G] / [(4/3)πR^3]

as g = GM/R^2 and V = (4/3)πR^3

so,

ρ = (3/4).(g / πRG)

now, if the radius is increased, the new density will be given as

ρ' = (3/4).(g / πR'G)

now as

R' = 3R

so,

ρ' = (3/4).(g / π(3R)G)

so,

ρ' = (1/3).(3/4).(g / πRG)

thus,

ρ' = (1/3)ρ

so, the new density would decrease to one-third its initial value.