Propagating error question

Question

9.2(±0.4)× ( [5.4(±0.3)×10^−3]+[5.6(±0.1)×10^−3] )

For this question I need to solve the equation and then I need to figure out the absolute uncertainty and percent relative uncertainty. I understand how to do each one individually but cant seem to figure out how to get to the right answer in problems that have both addition and subtraction as well as multiplication and division. Also, I seem to keep getting the problem wrong mathematically as I come out with an answer of 0.101 or 0.105 and I'm not certain what I'm doing wrong.

Answer 1

Definitions (assumed)

absolute uncertainty - maximum variance in value given worst case errors in individual elements of equation divided by 2. Expressed as total interval or expected value with + - variance

relative uncertainty - ratio of maximum variance to expected value of equation (can be expressed as percentage)

expected value - value of equation without error/variance

Given: 9.2(±0.4)× ( [5.4(±0.3)×10^−3]+[5.6(±0.1)×10^−3] ) as equation with error limits.

Expected value = 9.2 x (5.4 x 10 ^-3 + 5.6 x 10 ^-3) = 9.2 x 1.1 x 10^-2 = 0.1012

minima = 8.8 x (5.1 x 10 ^-3 + 5.5 x 10 ^-3) = 8.8 x 1.06 x 10^-2 = 0.09328

maxima= 9.6 x (5.7 x 10 ^-3 + 5.7 x 10 ^-3) = 9.6 x 1.14 x 10^-2 = 0.10944

absolute uncertainty = (0.10944 , 0.09328) or (0.10944 - 0.09328)/2 {± 0.00808}

relative uncertainty = ± .00808 / .1012 or ± 7.99 ‰

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Calculating using uncertainty laws we can get the same answer with less work.

laws: I When adding values in an equation we add their absolute uncertainties

II When multiplying values in an equation we add their relative uncertainties

Uncertainty of values left of multiplication sign = ±.4 relative uncertainty = .4/9.2 or .0435

Uncertainty of values right of multiplication sign = ± (.3 + .1) x10^-3 or ± 0.4 x 10^-3

relative uncertainty = (.4 x 10^-3) / (11 x 10^-3) or .0364

The total relative uncertainty is .0435 + .0364 or .0799 (7.99%)

the total absolute uncertainty = relative uncertainty times expected value = .0799 x .1012 or ± .00808

Same answer, less work.