In triangle ABC, DE || BC where D is a point on AB and is a point on AC. DE divides the area of A ABC into two equal parts. Then DB : AB is equal to

DE || BC

DE divides the area of $$\triangle ABC$$ into two equal parts => D and E are midpoints of AB and AC.

$$ \triangle ADE and \triangle ABC are similar$$.

$$\frac{area of \triangle ABC}{area of  \triangle ADE} = \frac{AB^2}{AD^2}$$

=>$$\frac{AB^2}{AD^2} = 2$$

=>AB = $$\sqrt{2}AD$$

=>AB = $$\sqrt{2}(AB - BD)$$

=> $$(\sqrt{2} - 1)AB = \sqrt{2}BD$$

=> $$\frac{BD}{AB} = \frac{(\sqrt{2} - 1)}{\sqrt{2}}$$

So, the answer would be option d)$$ (\sqrt{2} - 1) : \sqrt{2} $$