Two solid pieces, one of gold and the other of silver when immersed completely in water have equal weighs. When weighed in air

Suppose,

Weight of gold in air = xg,

Weight of silver in air = yg

Density of gold = ${\mathrm{d}}_1$

Density of gold = ${\mathrm{d}}_2 (\mathrm{where} {\mathrm{d}}_1 >{\mathrm{d}}_2)$

Volume of gold = $\frac{\mathrm{x}}{{\mathrm{d}}_1}, \mathrm{volume} \mathrm{of} \mathrm{silver} = \frac{\mathrm{y}}{{\mathrm{d}}_2}$

Weight of gold in water =Weigh of silver in water

or $(\mathrm{x}-\frac{\mathrm{x}}{{\mathrm{d}}_1}\times 1)\mathrm{g} = (\mathrm{y}-\frac{\mathrm{y}}{{\mathrm{d}}_2}\times 1)\mathrm{g}$

or $\mathrm{x}(1-\frac{1}{{\mathrm{d}}_1}) = \mathrm{y}(1-\frac{1}{{\mathrm{d}}_2})$

or $$\begin{gathered} \frac{\mathrm{x}}{\mathrm{y}} = \frac{(1-{\displaystyle \frac{1}{\mathrm{d}2}})}{(1-{\displaystyle \frac{1}{{\mathrm{d}}_1}})}< 1 as d_1>d2 \\ \mathrm{x} < \mathrm{y} \mathrm{or} \mathrm{y} > \mathrm{x} \end{gathered}$$

i.e., weight of silver in air is more than that of gold in air.