What does archimedes' principle say about how the buoyant force relates to the fluid involved?

The buoyant force is equal to the weight of the volume of fluid displaced by the solid immersed into it, and it is equal to:

where
is the density of the fluid
is the volume of the part of the solid immersed in the fluid
g is the gravitational acceleration

We see that the greater the density of the fluid, the greater the buoyant force that pushes the object upward. When the object floats in the fluid, the buoyant force is equal to the weight of the object, mg:
(1)
and since the mass of the solid is equal to the product between its density and its volume: 
we can rewrite (1) as

where  is the density of the object and  is its total volume. So the fraction of the object immersed in the fluid is