If the radius of the earth be increased by a factor of `5`, by what factor its density be changed to keep the value of `g` of the same?

Explanation

Let `R, rho` be the radius and density of
Earth. Then mass of Earth, `M = (4)/(3) pi R^(3) rho`
`g = (GM)/(R^(2)) = (G)/(R^(2)) xx (4)/(3) pi R^(3) rho = (4)/(3) pi GR rho`
`:. rho = (3g)/(4pi GR)`
where `R' = 5 R`, then
`rho' = (3g)/(4 pi G(5R)) = (1)/(5) (3g)/(4pi G rho) = (rho)/(5)`
Thus the density of Earth would change by a factor `1//5`.