In an experiment, four quantities $a,b,c$ and $d$ are measured with percentage error $1\%$, $2\%$, $3\%$ and $4\%$ respectively. Quantity $P$ is calculated as follows: $P=\frac{a^3{b}^2}{cd}$. The percentage error in $P$ is:

To find the percentage error in $P$, we use the formula for propagation of errors in a quotient and product:
$$P=\frac{a^3{b}^2}{cd}$$
The relative error in $P$ is given by:
$$\frac{\Delta P}{P}=3\left(\frac{\Delta a}{a}\right)+2\left(\frac{\Delta b}{b}\right)+\frac{\Delta c}{c}+\frac{\Delta d}{d}$$
Multiplying by $100$ to get percentage errors:
$$\%\text{ error in }P=3(\%\text{ error in }a)+2(\%\text{ error in }b)+(\%\text{ error in }c)+(\%\text{ error in }d)$$
Substituting the given values:
$$\%\text{ error in }P=3(1\%)+2(2\%)+3\%+4\%$$
$$\%\text{ error in }P=3\%+4\%+3\%+4\%=14\%$$