The physical quantity $ P $ is defined in terms of four observations $ a, b, c, $ and $ d $ by the formula:

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

To find the percentage error in $ P $, we use the formula for maximum percentage error. The general rule for percentage errors when a quantity is a product or a quotient is to add the percentage errors, while for powers, multiply the error by the power.

The percentage error for $ P $ is calculated using:

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

Given the percentage errors:

$ a $ has a percentage error of $ 1\% $

$ b $ has a percentage error of $ 3\% $

$ c $ has a percentage error of $ 2\% $

$ d $ has a percentage error of $ 4\% $

Substituting these into the formula:

$ = 3 \times (1) + 2 \times (3) + (2) + \frac{1}{2} \times (4) $

$ = 3 + 6 + 2 + 2 $

$ = 13\% $

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