Answer
The fraction of the iron that is submerged is 0.57
Work Step by Step
Let $V_I$ be the total volume of the iron. Let $V_M$ be the volume of the iron that is submerged in the mercury. Note that $V_M$ is the volume of mercury that is displaced. The buoyant force is equal to the weight of the iron. Therefore;
$F_B = M_I~g$
$\rho_M~V_M~g = \rho_I~V_I~g$
$\frac{V_M}{V_I} = \frac{\rho_I}{\rho_M}$
$\frac{V_M}{V_I} = \frac{7.8\times 10^3~kg/m^3}{13.6\times 10^3~kg/m^3}$
$\frac{V_M}{V_I} = 0.57$
The fraction of the iron that is submerged is 0.57
