In triangle LMN the points P and Q lie on LM in the order L, P, Q, M. A line through P parallel to MN meets LN at R, and a line through Q parallel to MN meets LN at S. Given LP = 3 cm, PQ = 5 cm, QM = 4 cm and LN = 24 cm, find RS (in cm).

Both PR and QS are parallel to MN, so each divides the two sides in the same ratio and the whole of LN is cut in the same proportions as LM. Since LM = 3 + 5 + 4 = 12 cm, the parts LR : RS : SN are 3 : 5 : 4. Applying these to LN = 24 cm gives one part = 2 cm, so RS = 5 parts = 10 cm. The value 6 is LR and 8 is SN, the other two parts; 5 copies PQ, wrongly assuming parallel cuts transfer lengths rather than ratios.