In rectangle $abcd$, $p$ is a point on $bc$ so that $\angle apd=90^{\circ}$. $ts$ is perpendicular to $bc$ with $bp=pt$, as shown. $pd$ intersects $ts$ at $q$. point $r$ is on $cd$ such that $ra$ passes through $q$. in $\triangle pqa$, $pa=20$, $aq=25$ and $qp=15$. find $sd$. (express your answer as a common fraction.)