The ratio of the weights of a body on the Earth's surface to that on the surface of a planet is $9:4$. The mass of the planet is $\frac{1}{9}th$ of that of the Earth. If 'R' is the radius of the Earth, what is the radius of the planet ?
(Take the planets to have the same mass density)

Since mass of the object remains same 
$\therefore$ Weight of object will be proportional to ${}^{\prime }{g}^{\prime }$ (acceleration due to gravity)
Given
$\frac{W_{earth}}{W_{planar}}=\frac{9}{4}\frac{g_{earth}}{g_{planet}}$
Also, $g_{surface}=\frac{GM}{R^2}$ ($M$ is mass planet, $G$ is universal gravitational constant, $R$ is radius of planet)

$\therefore \frac{9}{4}=\frac{G{M}_{earth}{R}_{planet}^2}{G{M}_{planet}{R}_{earth}^2}=\frac{M_{earth}}{M_{planet}}\times \frac{R_{planet}^2}{R_{earth}^2}=9\frac{R_{planet}^2}{R_{earth}^2}$

$\therefore R_{planet}=\frac{R_{earth}}{2}=\frac{R}{2}$