A body of volume V and density `P_S`, floats with volume v inside a liquid of density `P_L` . Show that
`v/V=P_S/P_L`

Explanation

Volume of the solid body = V
Density of the solid body = `P_S`
Mass of the solid body = `VP_S`
Weight of the solid body = `VP_Sg`
Volume of solid inside liquid = v
So, the volume of displaced liquid = v
Density of the liquid = `P_L`
Mass of the displaced liquid = `vP_L`
Weight of the displaced liquid = `vP_Lg`
According to the law of floatation,
Weight of solid body = weight of displaced liquid
`VP_Sg=vP_Lg`
Or, `Ve_S=vP_L`
`therefore v/V=e_S/P_L`