= Solution
At identical applied load and fixed geometric constraints, minimum <mass> is inversely proportional to the appropriate <material performance index>. Therefore
$$
\boxed{\text{Design A reduction}=1-\frac{(40/550)}{(2000/7800)}=71.6\%},
$$
$$
\boxed{\text{Design B reduction}=1-\frac{(\sqrt{2000}/7800)}{(\sqrt{40}/550)}=50.1\%}.
$$
The same percentages apply to weight under the same gravitational acceleration. It would be incorrect to compare densities alone, since the required beam dimensions also change.
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