\nWhen immersed in water, the apparent weight of an object is its true weight in air minus the buoyant force. The buoyant force is equal to the weight of the water displaced, which is (volume of the object) * (density of water) * g. The two pieces have equal apparent weight in water: W_air,gold – B_gold = W_air,silver – B_silver. Since both are solid and immersed, B_gold = V_gold * density_water * g and B_silver = V_silver * density_water * g. Given that the density of gold is greater than the density of silver, and they displace the same weight of water to have equal apparent weight difference from their air weight, the object with lower density (silver) must have a larger volume to displace the amount of water needed to satisfy the equality in apparent weight.
\nW_air = density * Volume * g.
\nW_air,gold – V_gold * density_water * g = W_air,silver – V_silver * density_water * g
\nV_gold * (density_gold – density_water) = V_silver * (density_silver – density_water)
\nSince density_gold > density_silver, it follows that (density_gold – density_water) > (density_silver – density_water). For the equation to hold, V_gold must be less than V_silver.
\nNow comparing weight in air: W_air,gold = density_gold * V_gold * g and W_air,silver = density_silver * V_silver * g.
\nWe know V_gold < V_silver. To determine which one weighs more in air, consider the relation derived from the apparent weight equation: W_air,gold - W_air,silver = (V_gold - V_silver) * density_water * g. Since V_gold < V_silver, (V_gold - V_silver) is negative. Therefore, W_air,gold - W_air,silver is negative, meaning W_air,gold < W_air,silver. The silver piece weighs more in air.\n<\/section>\n