The side AB of triangle ABC is marked so that AD = 3k and DB = 5k for some positive number k, and the line DE through D is parallel to BC, meeting AC at E. If AC = 32 cm, by how many centimetres does EC exceed AE?

By BPT, AE : EC = AD : DB = 3k : 5k = 3 : 5, and the unknown k cancels out entirely. So AC = 32 cm splits into 3 + 5 = 8 parts of 4 cm, giving AE = 12 cm and EC = 20 cm, and EC − AE = 8 cm. The answers 12 and 20 stop at AE or EC without taking the difference the question asks for; 6.4 comes from dividing AC by 5 (one of the ratio terms) instead of by the total 8.