Buoyancy without density of fluid

[Vanessa23]

Homework Statement

A ball of mass m_b and volume V is lowered on a string into a fluid of density rho_f. Assume that the object would sink to the bottom if it were not supported by the string.
What is the tension T in the string when the ball is fully submerged but not touching the bottom?
It claims that rho_f is not needed in the part representing buoyancy

Homework Equations

I think it should be:
T=Fg - B
where T is the force of tension and B is the buoyancy force.
Fg=rho(object)*V*g
B=rho_f*V*g
therefore, T=mg - rho_f*V*g

how can you not use rho_f in calculating the buoyancy force?

The Attempt at a Solution

 

[mgb_phys]

You need the density, unles you are going to quote the answer in terms of density.
Imagine the fluid is air and a metal ball, if you change the fluid to water - then the tension in the string is going to change.