Let \(R = (8,6)\). The lines whose equations are \(8y = 15x\) and \(10y = 3x\) contain points \(P\) and \(Q\), respectively, such that \(R\) is the midpoint of \(\overline{PQ}\). The length of \(PQ\) equals \(\frac {m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\).

(第十九届AIME2 2001 第4题)