Let $x$ and $y$ be the original numbers of adults and children respectively.
$\left\{ \begin{array}{ll}
x : y = 13 : 6 & \ldots \unicode{x2460} \\
x + 9 : y + 24 = 8 : 7 & \ldots \unicode{x2461}
\end{array}\right.$
From $\unicode{x2460}$, we have
$\begin{array}{rcl}
x : y & = & 13 : 6 \\
\dfrac{x}{y} & = & \dfrac{13}{6} \\
x & = & \dfrac{13y}{6} \ \ldots \unicode{x2462}
\end{array}$
Sub. $\unicode{x2462}$ into $\unicode{x2461}$, we have
$\begin{array}{rcl}
\dfrac{13y}{6} + 9 : y + 24 & = & 8 : 7 \\
\dfrac{13y + 54}{6} : y + 24 & = & 8 : 7 \\
\dfrac{13y + 54}{6(y + 24)} & = & \dfrac{8}{7} \\
7(13y + 54) & = & 48(y + 24) \\
91y + 378 & = & 48y + 1152 \\
43y & = & 774 \\
y & = & 18
\end{array}$
Sub. $y = 18$ into $\unicode{x2462}$, we have
$\begin{array}{rcl}
x & = & \dfrac{13(18)}{6} \\
& = & 39
\end{array}$
Therefore, there are $39$ adults originally in the playground.
