Archimedes' Principle and Buoyant Force Steps:

• Apparent weight equals true weight minus buoyant force: in water, P = W - ρ_water V g, so ρ_water V g = W - P.
• In oil, Q = W - ρ_oil V g, so ρ_oil V g = W - Q.
• The volume V and g are the same for both fluids.
• Ratio of densities ρ_oil / ρ_water = (W - Q) / (W - P).

Why C is correct:

• It matches the ratio of buoyant forces, which equals the density ratio by Archimedes' principle (buoyant force = ρ_fluid V g).

Why the others are wrong:

• A: Numerator is water's buoyant force, but denominator Q - P lacks physical meaning as a force.
• B: Inverts the buoyant force ratio and uses unrelated Q - P term.
• D: Gives the reciprocal ρ_water / ρ_oil, not the asked ratio.

Final answer: C