In triangle ABC, D lies on AB and E lies on AC with DE parallel to BC. The line DE splits the triangle so that area(ADE) : area(trapezium DBCE) = 4 : 21.
Find AD : AB.

The trapezium is the rest of the triangle, so area(ADE) : area(ABC) = 4 : $(4+21)=4$ : 25. Then AD : AB = $\sqrt{\frac{4}{25}}=\frac{2}{5}$, i.e. 2 : 5. The option 4 : 25 is that area ratio left unrooted, 4 : 21 treats the triangle-to-trapezium comparison as though it were a length ratio, and 5 : 2 inverts the answer.