Vector

Let the position vectors of $A$ , $B$ and $C$ be $a$ , $b$ and $c$ respectively.

$\begin{aligned} c & =ha+kb & \\ \left( \begin{matrix} 15 \\ 9 \end{matrix} \right) & =h\left( \begin{matrix} 1 \\ 2 \end{matrix} \right) +k\left( \begin{matrix} 7 \\ 1 \end{matrix} \right) \, =\, \left( \begin{matrix} h+7k \\ 2h+k \end{matrix} \right) & \\ h\, +\, 7k\, & =\, 15 \quad \quad \longrightarrow (1) & \\ 2h\, +\, k\, & =\, 9\quad \quad \, \, \, \longrightarrow (2) & \\ 2h\, +\, 14k\, & =\, 30\quad \quad \longrightarrow (3) & (1) \, \times \, 2 \\ 13k\, & =\, 21 & (3)\,-\,(2) \\ k\, & =\, \frac { 21 }{ 13 } & \\ h\, & =15-7\left( \frac { 21 }{ 13 } \right) \, =\, \frac { 48 }{ 13 } & \text{Substituting the value of k into (1)} \\ \therefore \quad h\, +\, k\, & =\, \frac { 48+21 }{ 13 } \, \, =\, \frac { 69 }{ 13 } & \\ \Longrightarrow \quad \frac { m }{ n } \, & =\, \frac { 69 }{ 13 } & \\ \therefore \quad \quad \quad \quad \quad m+n\, & =\,69+13\\ & =\,\boxed{82} \end{aligned}$