A metal block floats in mercury at $0°C$ with $k_1$ fraction of its total volume submerged. When temperature is raised to $60°C,$ a fraction $k_2$ of total volume is found submerged. If the coefficient of volumetric expansion of metal is ${\gamma }_1$ and coefficient of volumetric expansion of mercury is ${\gamma }_2,$ then ratio of $\frac{k_1}{k_2}$ is

For equilibrium in case $\left(1\right)$ at $0°C$ upthrust =weight of body $V_{in}{d}_{2}g=V{d}_{1}g$

$\frac{V_{in}}{V}=\frac{d_1}{d_2}=K_1\to \left(1\right)$

For equilibrium in case $\left(2\right)$ at $60°C,$ when the temperature is increased, the density will decreases $d_1^1=d_1/(1+{\gamma }_{1}60)d_2^1=d_2/(1+{\gamma }_{2}60)$

Again upthrust = weight of the body 

$V_{in}^{\text{'}}{d}_2^{\text{'}}g=V\text{'}{d}_1^{\text{'}}g$

$K_2=\frac{V_1^{1}n}{V^1}=\frac{d_1^1}{d_2^1}=\frac{d_1}{1+{\gamma }_{1}60}\times \frac{1+{\gamma }_{2}60}{d_2}=\frac{d_1}{d_2}\left(\frac{1+{\gamma }_{2}60}{1+{\gamma }_{1}60}\right)$

$\therefore \frac{K_1}{K_2}=\frac{d_1}{d_2}\times \frac{d_2}{d_1}\left(\frac{1+{\gamma }_{1}60}{1+{\gamma }_{2}60}\right)=\frac{1+{\gamma }_{1}60}{1+{\gamma }_{2}60}$