Question 12
Rhombus $PQRS$ is inscribed in rectangle $ABCD$ so that vertices $P,$ $Q,$ $R,$ and $S$ are interior points on sides $\overline{AB},$ $\overline{BC},$ $\overline{CD},$ and $\overline{DA},$ respectively. It is given that $PB=15,$ $BQ=20,$ $PR=30,$ and $QS=40.$ Let $\frac{m}{n},$ in lowest terms, denote the perimeter of $ABCD.$ Find $m+n.$