How Do I Compare Their Densities?

Let the total volume of the submerged body be $V$ and let it's density be $\sigma$ . Let the densities of the three liquids be ${\rho}_{A}$ , ${\rho}_{B}$ and ${\rho}_{C}$ respectively. Using Archimedes' principle and the equation of buoyancy, we get:

$\sigma V g = \frac {3}{5} V {\rho}_{A} g$

and, $\sigma V g = \frac {2}{9} V {\rho}_{B} g$

and also, $\sigma V g = \frac {8}{11} V {\rho}_{C} g$

Now, solving for ${\rho}_{A}$ , ${\rho}_{B}$ and ${\rho}_{C}$ , we get:

${\rho}_{A} = \frac {5}{3} \sigma$

${\rho}_{B} = \frac {9}{2} \sigma$

${\rho}_{C} = \frac {11}{8} \sigma$

which shows ${\rho}_{B} > {\rho}_{A} > {\rho}_{C}$