Calculate AB

Since D lies on AB, the total length of segment $AB$ is the sum of $BD$ and $AD$.

$AB = BD + AD = 8 + 7 = 15 \text{ cm}$

Establish Similarity

Because $DE \parallel AC$, $\triangle BDE$ is similar to $\triangle BAC$ by the AA criterion.

$\triangle BDE \sim$\triangle BA$C$

Apply Area Ratio Theorem

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

\frac{\text{Area}(\triangle BDE)}{\text{Area}(\triangle BAC)} = \left(\frac{BD}{BA}\right)^2 = \left(\frac{8}{15}\right)^2 = \frac{64}{225}

Find Trapezium Area

The area of the trapezium $ADEC$ is the difference between the total area of $\triangle BAC$ and the area of $\triangle BDE$.

$\text{Area}(ADEC) = \text{Area}($\triangle BAC$) - \text{Area}($\triangle BDE$) = 225 - 64 = 161$

Calculate Final Ratio

The ratio of Area($\triangle BDE$) to Area(Trapezium $ADEC$) is $64: 161$.

$\text{Ratio} = \frac{64}{161}$