Angle Bisector Theorem

Suppose that for $\Delta{ABC}$, the angle bisector from $A$ meets $BC$ at $D$ and the angle bisector from $B$ meets $AC$ at $E$. Suppose that the area of $\Delta{ABC}$, denoted by $[ABC]$, equals $10$. Given that $[ADC]=7$ and $[ABE]=4$, the ratio of side lengths $BC$ to $AC$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are coprime positive integers. Find the value of $m+n$.