A solid sphere of volume V and density floats at the interface of...

$V=$ Volume of solid sphere. Let $V_1=$ Volume of the part of the sphere immersed in a liquid of density $\rho_1 and V_2=$ Volume of the part of the sphere immersed in liquid of density $\rho_2$. According to law of floatation, Hence from eqns (i) and (ii), $$\begin{aligned} & V_1 \rho g+V_2 \rho g=V_1 \rho_1 g+V_2 \rho_2 g \\ & \text{ or } V_1\left(\rho-\rho_1\right) g=V_2\left(\rho_2-\rho\right) g \\ & \text{ or } \frac{V_1}{V_2}=\frac{\rho_2-\rho}{\rho-\rho_1}\end{aligned}$$