The density $$\rho$$ of a uniform cylinder is determined by measuring its mass $$m$$, length $$l$$ and diameter $$d$$. The measured values of $$m$$, $$l$$ and $$d$$ are 97.42 $$\pm$$ 0.02 g, 8.35 $$\pm$$ 0.05 mm and 20.20 $$\pm$$ 0.02 mm, respectively. Calculated percentage fractional error in $$\rho$$ is _______.

$$\rho = \frac{m}{V} = \frac{m}{\pi \left(\frac{d}{2}\right)^2 l} = \frac{4m}{\pi d^2 l}$$

$$\frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + 2\frac{\Delta d}{d} + \frac{\Delta l}{l}$$

$$\frac{\Delta \rho}{\rho} = \frac{0.02}{97.42} + 2\left(\frac{0.02}{20.20}\right) + \frac{0.05}{8.35}$$

$$\frac{\Delta \rho}{\rho} \approx 0.000205 + 2(0.00099) + 0.005988$$

$$\frac{\Delta \rho}{\rho} \approx 0.000205 + 0.00198 + 0.005988 \approx 0.008173$$

$$\% \text{ error} = \frac{\Delta \rho}{\rho} \times 100 \approx 0.008173 \times 100 \approx 0.82\%$$