In an experiment, four quantities $a$, $b$, $c$ and $d$ are measured with percentage error $1\%$, $2\%$, $3\%$

Question

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: $P=\frac{a^3b^2}{cd}$. Percentage error in $P$ is:

Accepted Answer

1. **Identify the Given Formula:** The quantity $P$ is given by $P = \frac{a^3b^2}{cd}$.
2. **Apply Error Propagation Rules:** For a quantity that is a product or quotient of other quantities, the maximum relative error is the sum of the relative errors of the individual quantities multiplied by their respective powers. Thus, $\frac{\Delta P}{P} = 3\frac{\Delta a}{a} + 2\frac{\Delta b}{b} + \frac{\Delta c}{c} + \frac{\Delta d}{d}$.
3. **Calculate Percentage Error:** Multiplying by $100$ converts this to percentage error:
$\% 	ext{ error in } P = 3(\% 	ext{ error in } a) + 2(\% 	ext{ error in } b) + (\% 	ext{ error in } c) + (\% 	ext{ error in } d)$.
4. **Substitute Given Values:**
$\% 	ext{ error in } P = 3(1\%) + 2(2\%) + 1(3\%) + 1(4\%)$
$\% 	ext{ error in } P = 3\% + 4\% + 3\% + 4\% = 14\%$.

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