Volume of body = V
The volume of body submerged in liquid = v
Density of body = ρs
Density of liquid = ρL
Let weight of the body be W. W = volume of the body x density of the body x g = V ρs g
The weight of the liquid displaced by the body will be equal to the upthrust. Let it be FB.
FB = volume of the liquid displaced × density of the liquid × g = v ρL g
From the principle of floatation,
W = FB
⇒ V ρs g = v ρL g
⇒ {v\over V} = {ρ_s\over ρ_L}
Hence proved.