Without using the value of G or the actual value of any masses, estimate the ratio of the mean density of the Earth to that of the Sun. The only facts you may use are those that a nonscientist might be aware of, the fact that the Moon's radius is three and one half times smaller than Earth's radius and the scaling implicit to Kepler's 3rd Law.
The full breakdown of the solution is attached as a pdf file, after using Kepler's 3rd ...
The answer provides a ratio of the average density of the Earth to the Sun. By using Kepler's Law, and common knowledge that the Moon's radius is 3.5 times smaller than the Earth's radius, the ratio is thus obtained.