A piece of metal weighs x newton in air , y newton when completely immersed in water and z newton when completely immeresed in liquid . The relative density of liquid is

Given that

A piece of metal weighs

${\text{W}}_{\text{air}}=x$

${\text{W}}_{\text{water}}=y$

${\text{W}}_{\text{liquid}}=z$

We know when a body is immersed in a fluid

Weight lost = weight of fluid displaced

Upthrust $={\text{V}}_{\text{B}}{\rho }_l\text{g}$

Applying this to the given data

Weight lost in water $=V_B{\rho }_{w}g$

$x-y={\text{V}}_{\text{B}}{\rho }_{\text{w}}\text{g}$

${\rho }_{\text{w}}=\frac{x-y}{{\text{V}}_{\text{B}}\text{g}}$

Similarly

Weight lost in liquid $={\text{V}}_{\text{B}}{\rho }_l\text{g}$

$x-y={\text{V}}_{\text{B}}{\rho }_l\text{g}$

$\therefore {\rho }_l=\frac{x-z}{{\text{V}}_{\text{B}}\text{g}}$

Now we get relative density of liquid $=\frac{{\rho }_l}{{\rho }_w}=\frac{x-z}{x-y}$