A piece of metal floats on mercury. The coefficients of volume expansion of the metal and mercury are  ${\gamma }_1$ and  ${\gamma }_2$  respectively. If the temperature of both mercury and metal are increase by an amount  $\Delta T$  the fraction the volume of the metal submerged in mercury changes by the factor.

Let density of the metal and mercury be d₁ and d₂ respectively.

The fraction of volume of metal submerged initially, f=d₁/d₂

Density of metal after increase of temperature by Δt, $d_1\text{'}=\frac{d_1}{1+{\gamma }_1\Delta T}$

Density of mercury after increase of temperature by Δt, $d_2\text{'}=\frac{d_2}{1+{\gamma }_2\Delta T}$

So, fraction of volume of metal submerged finally, $f\text{'}=\frac{d_1\text{'}}{d_2\text{'}}=\frac{d_1(1+{\gamma }_2\Delta T)}{d_2(1+{\gamma }_1\Delta T)}=\frac{d_1({\gamma }_2-{\gamma }_1)\Delta T}{d_2}$

$f\text{'}=f({\gamma }_2-{\gamma }_1)\Delta T$

$\frac{f\text{'}}{f}=({\gamma }_2-{\gamma }_1)\Delta T$