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

$P=\frac{a^{\mathrm{³}}}{c\sqrt{d}}$

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 given relationship is

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

To find the percentage (relative) error in P, we use the rule that for a product or quotient the relative errors add, each multiplied by the absolute value of its exponent.

Hence

$\frac{\Delta P}{P}=3·\frac{\mathrm{\Delta a}}{a}+2·\frac{\mathrm{\Delta b}}{b}+\frac{\mathrm{\Delta c}}{c}+\frac{}{}·\frac{\mathrm{\Delta d}}{d}$

Insert the given percentage errors (Δa/a = 1 %, Δb/b = 3 %, Δc/c = 2 %, Δd/d = 4 %):

$\frac{\Delta P}{P}=3\times 1\%+2\times 3\%+2\%+0.5\times 4\%$

Calculate each term:

• 3 × 1 % = 3 %
• 2 × 3 % = 6 %
• 2 % (from c)
• 0.5 × 4 % = 2 %

Adding them together:

3 % + 6 % + 2 % + 2 % = 13 %

Therefore, the percentage error in the quantity P is

13 %.