A physical quantity P is related to four observables a, b, c, and d by the equation P = a^3 b^2 / (sqrt(c) d). The percentage errors in a, b, c, and dare 1%, 3%, 4%, and 2% respectively. What is the maximum percentage error in P?

B. 13% — The maximum percentage error is calculated using the propagation of errors formula for products and powers.

JEE Main · PhysicsMedium

$A physical quantity P is related to four observables a, b, c, and d by the equation P = \frac{a ^{3} b ^{2}}{c d} . The percentage errors in a, b, c, and d are 1%, 3%, 4%, and 2% respectively. What is the maximum percentage error in P ?$

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Correct answer: B — $13%$

$The relative error is given by \frac{Δ P}{P} = 3 \frac{Δ a}{a} + 2 \frac{Δ b}{b} + \frac{1}{2} \frac{Δ c}{c} + \frac{Δ d}{d} .$
$Substituting the given percentage errors: \frac{Δ P}{P} \times 100 = 3 (1%) + 2 (3%) + 0.5 (4%) + 1 (2%) .$
$Calculating the sum: 3% + 6% + 2% + 2% = 13% .$
$Hence the answer is (B).$