The relative density of a material is found by weighing the body first in air and then in water. If the weight in air is (10.0 $\pm$ 0.1) gf and the weight in water is (5.0 $\pm$ 0.1) gf, then the maximum permissible percentage error in relative density is

Relative density $=\frac{{\mathrm{W}}_{\mathrm{a}}}{{\mathrm{W}}_{\mathrm{a}}-{\mathrm{W}}_{\mathrm{w}}},\mathrm{\rho }=\frac{{\mathrm{W}}_{\mathrm{a}}}{\mathrm{w}}$

where $\mathrm{\rho }$ is relative density, W_a is weight in air, and w is loss in weight.

$\frac{\mathrm{\Delta \rho }}{\mathrm{\rho }}=\frac{{\mathrm{\Delta W}}_{\mathrm{a}}}{{\mathrm{W}}_{\mathrm{a}}}-\frac{\mathrm{\Delta w}}{\mathrm{w}}$

For maximum error $\frac{\mathrm{\Delta \rho }}{\mathrm{\rho }}=\frac{{\mathrm{\Delta W}}_{\mathrm{a}}}{{\mathrm{W}}_{\mathrm{a}}}+\frac{\mathrm{\Delta w}}{\mathrm{w}}$

For maximum percentage error, 

$\frac{\mathrm{\Delta \rho }}{\mathrm{\rho }}\times 100=\frac{{\mathrm{\Delta W}}_{\mathrm{a}}}{{\mathrm{W}}_{\mathrm{a}}}\times 100+\frac{\mathrm{\Delta w}}{\mathrm{w}}\times 100$

$\text{ Given }{\mathrm{\Delta W}}_{\mathrm{a}}=0.1\mathrm{gf}\text{ and }{\mathrm{W}}_{\mathrm{a}}=10.0\mathrm{gf}$

$\begin{array}{l}\mathrm{w}=10.0-5.0=5.0\mathrm{gf}\\ \mathrm{\Delta w}={\mathrm{\Delta W}}_{\mathrm{a}}+{\mathrm{\Delta W}}_{\mathrm{w}}=0.1+0.1=0.2\mathrm{gf}\\ \frac{\mathrm{\Delta \rho }}{\mathrm{\rho }}\times 100=\left(\frac{0.1}{10.0}\right)\times 100+\left(\frac{0.2}{5.0}\right)\times 100=1+4=5\end{array}$