A material has normal density $\mathrm{\rho }$ and bulk modulus K. The increase in the density of the material when it is subjected to an external pressure P from all sides is

$\frac{{\mathrm{\rho }}^{\mathrm{\prime }}}{\mathrm{\rho }}=\frac{\frac{\mathrm{M}}{\mathrm{V}-\mathrm{\Delta V}}}{\mathrm{M}/\mathrm{V}}=\frac{\mathrm{V}}{\mathrm{V}-\mathrm{\Delta V}}$

$\text{ or }\frac{{\mathrm{\rho }}^{\mathrm{\prime }}}{\mathrm{\rho }}=\mathrm{V}(\mathrm{V}-\mathrm{\Delta V}{)}^{-1}\text{ or }\frac{{\mathrm{\rho }}^{\mathrm{\prime }}}{\mathrm{\rho }}=1+\frac{\mathrm{\Delta V}}{\mathrm{V}}$

(using binomial theorem)

$\text{ or }\frac{\mathrm{\Delta V}}{\mathrm{V}}=\frac{{\mathrm{\rho }}^{\mathrm{\prime }}}{\mathrm{\rho }}-1\text{ or }\frac{\mathrm{\Delta V}}{\mathrm{V}}=\frac{{\mathrm{\rho }}^{\mathrm{\prime }}-\mathrm{\rho }}{\mathrm{\rho }}$

$\text{ Again, }\mathrm{k}=\frac{\mathrm{\Delta P}}{{\displaystyle \frac{\mathrm{\Delta V}}{V}}}=\frac{\mathrm{\rho \Delta P}}{{\mathrm{\rho }}^{\mathrm{\prime }}-\mathrm{\rho }}\text{ or }{\mathrm{\rho }}^{\mathrm{\prime }}-\mathrm{\rho }=\frac{\mathrm{\rho \Delta }P}{\mathrm{K}}$

$\text{ But, }\mathrm{\Delta P}=\mathrm{P}\text{ (given) }$

$\therefore {\mathrm{\rho }}^{\mathrm{\prime }}-\mathrm{\rho }=\frac{\mathrm{\rho P}}{\mathrm{K}}$