In triangle ABC, the point P lies on AB with AP : PB = 1 : 3, and the point Q lies on AC with AQ : QC = 2 : 1. What is the ratio of the area of triangle APQ to the area of triangle ABC?

Triangles APQ and ABC share the angle at A, so their areas are in the ratio (AP × AQ) : (AB × AC). Here AP/AB = 1/4 and AQ/AC = 2/3, so the ratio of areas is (1/4) × (2/3) = 1/6, that is 1 : 6. Note PQ is not parallel to BC here, so the square-of-the-ratio rule for similar triangles does not apply.