Question: A certain body weighs 22.42 g and has a measured volume of 4.7 cc. The possible error in the measurement of mass and volume are 0.01 g and 0.1 cc. Then maximum error in the density will be: [PMT/NEET-1991]

a. 22%
b. 2%
c. 0.2%
d. 0.02%

Answer: b. 2%

Solution:
Density

$$ \rho = \dfrac{m}{V} \Rightarrow \ln \rho = \ln m – \ln V $$

Differentiating gives

$$ \dfrac{\Delta \rho}{\rho} = \dfrac{\Delta m}{m} + \dfrac{\Delta V}{V} $$

(errors add in magnitude).

Using data:

$$ \dfrac{\Delta m}{m}=\dfrac{0.01}{22.42}\approx 0.00045 $$

$$ \dfrac{\Delta V}{V}=\dfrac{0.1}{4.7}\approx 0.0213 $$

Thus percentage error

$$ \approx (0.00045+0.0213)\times100\% \approx 2\% $$