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

The correct answer is 13%.

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

We can rewrite the square root of $d$ as $d^{1/2}$:
$P=a^3{b}^2{c}^{-1}{d}^{-1/2}$

The maximum fractional error in $P$, denoted as $\frac{\Delta P}{P}$, is calculated by summing the individual fractional errors multiplied by their respective absolute powers:
$\frac{\Delta P}{P}=3\left(\frac{\Delta a}{a}\right)+2\left(\frac{\Delta b}{b}\right)+1\left(\frac{\Delta c}{c}\right)+\frac{1}{2}\left(\frac{\Delta d}{d}\right)$

To find the percentage error, we multiply each term by 100:
$\frac{\Delta P}{P}\times 100=3(\frac{\Delta a}{a}\times 100)+2(\frac{\Delta b}{b}\times 100)+1(\frac{\Delta c}{c}\times 100)+\frac{1}{2}(\frac{\Delta d}{d}\times 100)$

Given the percentage errors of measurement for the individual quantities:
Percentage error in $a$ = 1%
Percentage error in $b$ = 3%
Percentage error in $c$ = 2%
Percentage error in $d$ = 4%

Substitute these values into the percentage error formula:
$\text{Percentage error in}P=3(1\%)+2(3\%)+1(2\%)+\frac{1}{2}(4\%)$

Calculate each component:
$\text{Percentage error in}P=3\%+6\%+2\%+2\%$

Summing the percentage errors:
$\text{Percentage error in}P=13\%$