A physical quantity P is related to four observations a, b, c and d as follows:

P= a³b²/ c√d

The percentage errors of measurement in a, b, c and d are 1%, 3%, 2%, and 4% respectively. The percentage error in the quantity P is

To find the percentage error in the physical quantity $P$, we start with the given relationship:
$P=\frac{a^3{b}^2}{c\sqrt{d}}$

This expression can also be written in terms of exponents as:
$P=a^3{b}^2{c}^{-1}{d}^{-1/2}$

When calculating errors, the relative error (or fractional error) in a quantity that is a product or quotient of other quantities is the sum of the relative errors of the individual quantities, each multiplied by the absolute value of its exponent. Therefore, the maximum relative error in $P$, denoted as $\frac{\Delta P}{P}$, is given by:
$\frac{\Delta P}{P}=3\left(\frac{\Delta a}{a}\right)+2\left(\frac{\Delta b}{b}\right)+1\left(\frac{\Delta c}{c}\right)+\frac{1}{2}\left(\frac{\Delta d}{d}\right)$

To express this as a percentage error, we multiply both sides by 100:
$(\frac{\Delta P}{P}\times 100)\%=\left[3\right(\frac{\Delta a}{a}\times 100)+2(\frac{\Delta b}{b}\times 100)+1(\frac{\Delta c}{c}\times 100)+\frac{1}{2}(\frac{\Delta d}{d}\times 100\left)\right]\%$

We are given the individual percentage errors for each measurement:
• Percentage error in $a=1\%$
• Percentage error in $b=3\%$
• Percentage error in $c=2\%$
• Percentage error in $d=4\%$

Substituting these values into our error formula:
$(\frac{\Delta P}{P}\times 100)\%=\left[3\right(1)+2(3)+1(2)+\frac{1}{2}(4\left)\right]\%$

Now, we compute the sum:
$(\frac{\Delta P}{P}\times 100)\%=[3+6+2+2]\%$

Combining the terms yields:
$(\frac{\Delta P}{P}\times 100)\%=13\%$

Thus, the percentage error in the quantity $P$ is 13%.