In triangle ABC, D lies on AB and E lies on AC so that AD : DB = 1 : 2 and AE : EC = 1 : 2. What is the ratio of the area of triangle ADE to the area of the quadrilateral DBCE?

Since AD/AB = AE/AC = 1/3, DE is parallel to BC and triangle ADE is similar to triangle ABC with ratio 1 : 3. So area ADE : area ABC = 1 : 9. The quadrilateral DBCE is the rest of the triangle, i.e. 9 − 1 = 8 parts, giving 1 : 8.