When a block of iron floats in mercury at $0^{0}C,$  fraction $k_1$  of its volume is submerged, while at the temperature $60{}^{0}C$ , a fraction  $k_2$ is seen to be submerged. If the coefficient of volume expansion of iron is ${\gamma }_{Fe}$ and that of mercury is ${\gamma }_{Hg}$ , then the ratio $k_1/k_2$  can be expressed as 

 For equilibrium  when temperature  $0^{0}C$
Upthrust = Wt. of body
 $\therefore K_{1}V{d}_{2}g=V{d}_{1}g$
  $\Rightarrow K_1=\frac{d_1}{d_2}\mathrm{...............}\left(i\right)$

 For equilibrium when temperature increases to  ${60}^0$
When the temperature is increased the density will decrease.
 $\therefore d_1\text{'}=d_1/\left(1+{\gamma }_{Fe}\times 60\right)$
And  $d_2\text{'}=d_2/\left(1+{\gamma }_{Hg}\times 60\right)$
 $\therefore K_2\left[\frac{d_2}{1+{\gamma }_{Hg}\times 60}\right]=\frac{d_1}{1+{\gamma }_{Fe}\times 60}$
 $\therefore K_2\left[\frac{1+{\gamma }_{Fe}\times 60}{1+{\gamma }_{Hg}\times 60}\right]=\frac{d_1}{d_2}\Rightarrow \frac{K_1}{K_2}=\frac{1+{\gamma }_{Fe}\times 60}{1+{\gamma }_{Hg}\times 60}$