What follows about $DE$ and $BC$ if $D$ lies on $AB$ and $E$ lies on $AC$ of triangle $ABC$, and $\frac{AD}{AB}=\frac{AE}{AC}=\frac{3}{7}$?

Equal ratios on the two sides mean, by the converse of BPT, that DE is parallel to BC. Triangle ADE is then similar to triangle ABC, so every pair of corresponding sides is in the ratio $\frac{3}{7}$, giving DE = $\frac{3}{7}$ BC. The option $\frac{3}{4}$ uses AD : DB (= 3 : 4) instead of AD : AB; $\frac{9}{49}$ squares the ratio, which belongs to areas, not lengths; and the data are quite enough to settle the comparison.