The percentage errors of measurement in a, b, c and d are 1%, 3%, 4% and 2%, respectively. What is the percentage error in the quantity P? If the value of P calculated using the above relation turns out to be 3.763, to what value should you round off the result?

Answer:

$$$P=\frac{{{a}^{3}}{{b}^{2}}}{\sqrt{c}d}$

$\frac{{{a}^{3}}{{b}^{2}}}{\sqrt{c}d}\frac{\Delta P}{P}=\frac{3\Delta a}{a}+\frac{2\Delta b}{b}+\frac{1}{2}\frac{\Delta c}{c}+\frac{\Delta d}{d}$

= 3 x 1 + 2 x 3 + 1/2​ x 4 + 2

= 3 + 6 + 2 + 2

= 13 %

$\left( \frac{\Delta P}{P}\times 100 \right)%=\left( \frac{3\Delta a}{a}\times 100+\frac{2\Delta b}{b}\times 100+\frac{1}{2}\frac{\Delta c}{c}\times 100+\frac{\Delta d}{d}\times 100 \right)%$

$\Delta P=\frac{13}{100}\times P$ $\Delta P=\frac{13}{100}\times 4.235=0.55$

The error here, lies in the first decimal point. So the value of p is 4.3.