How do you combine percent uncertainties for multiplication?

Multiply or divide the final value by the combined relative uncertainty, which you get by adding the relative uncertainties of each quantity; report the result with significant figures matching that combined uncertainty.

Think of relative uncertainty as the percent wiggle in each number; to find the wiggle in the answer when you multiply or divide, turn each wiggle into a fraction of the number, add those fractions together, and that sum is the wiggle fraction for the result. For example, if one length has a ten percent uncertainty and another has a five percent uncertainty and you multiply them, the product has a fifteen percent uncertainty. Convert that combined percent back into the absolute uncertainty by multiplying the product by the fifteen percent, and then report the product with digits that reflect that size of uncertainty.

A common mistake is adding absolute uncertainties instead of relative ones, which gives the wrong size of uncertainty; fix this by always converting absolute uncertainties to relative form before combining. Another mistake is keeping too many digits in the final answer even though the uncertainty is large; round the result to the same precision implied by the combined uncertainty. Some people forget to convert percentages to decimal form when calculating absolute uncertainty from the combined relative uncertainty, so remember to multiply by the decimal version of the percent.

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Combine Percent Uncertainties for Multiplication

1. Use relative uncertainty (fractional)
2. Add relative uncertainties together
3. Convert percent to fraction when needed
4. Multiply product by combined percent
5. Report with matching significant figures

Common Mistakes in Combining Percent Uncertainties

1. Adding absolute instead of relative uncertainties
2. Keeping too many digits despite large uncertainty
3. Forgetting to convert percent to decimal

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