In triangle ABC the point D is the midpoint of BC and the point E is the midpoint of AD. What is the ratio of the area of triangle BEC to the area of triangle ABC?

Triangles BEC and ABC stand on the same base BC. E is the midpoint of AD and D lies on BC, so the perpendicular height from E down to BC is exactly half the height from A. Same base, half the height, so the areas are in the ratio 1 : 2. The trap 1 : 4 halves twice — once for the median and once for the midpoint — but the median AD does not shorten the base here, since triangle BEC still uses the whole of BC.