In an experiment to find the Young's modulus of a wire using Y=(4FL)/(π d^2 l), the following readings are taken: load F=10.0 N measured with least…

For a product or quotient of powers, the maximum fractional errors add with weights equal to the powers: $\frac{\Delta Y}{Y}=\frac{\Delta F}{F}+\frac{\Delta L}{L}+2\frac{\Delta d}{d}+\frac{\Delta l}{l}$. Substituting: $\frac{0.1}{10.0}=1\%$; $\frac{1\ \mathrm{mm}}{1000\ \mathrm{mm}}=0.1\%$; $2\times\frac{0.01}{0.50}=2\times2\%=4\%$; $\frac{0.01}{0.40}=2.5\%$. The total is $1+0.1+4+2.5=7.6\%$. The diameter enters squared, so its error is doubled, and the small extension $l$ together with the thin diameter dominate the uncertainty. Hence the maximum percentage error in $Y$ is $7.60$. Since the question asks for the maximum error, the individual errors are added and never combined in quadrature, and the least count of each instrument is the error assumed in its reading. Therefore the answer is $7.60\%$.