Consider two solid spheres of radii  $R_1=1m,R_2=2mand$  masses  $M_{1}and{M}_2$ , respectively. The  gravitational field due to sphere  $\left(1\right)and\left(2\right)$  are shown. The value of  $\frac{m_1}{m_2}$  is 

Gravitation field at the surface
 $$\begin{gathered} E=\frac{Gm}{r^2} \\ \therefore E_1=\frac{G{m}_1}{r_1^2}and{E}_2=\frac{G{m}_2}{r_2^2} \end{gathered}$$
 
From the diagram given in question,
 
 $\frac{E_1}{E_2}=\frac{2}{3}(r_1=1m,R_2=2mgiven)$   $\therefore \frac{E_1}{E_2}={\left(\frac{r_2}{r_1}\right)}^2\left(\frac{m_1}{m_2}\right)\Rightarrow \frac{2}{3}={\left(\frac{2}{1}\right)}^2\left(\frac{m_1}{m_2}\right)$

$\Rightarrow \left(\frac{m_1}{m_2}\right)=\frac{1}{6}$