Is the information sufficient 9?

Relevant wiki: Sine Rule (Law of Sines)

$\begin{aligned} Q & = \frac {{\color{#3D99F6}\overline{BE}}\cdot{\color{#D61F06}\overline{AD}}}{{\color{#3D99F6}\overline{AE}} \cdot {\color{#D61F06} \overline{DC}}} & \small \color{#3D99F6} \text{Using sine rule} \\ & = \frac {{\color{#3D99F6}\sin \angle BAE}\cdot{\color{#D61F06}\sin \angle C}}{{\color{#3D99F6}\sin \angle B}\cdot{\color{#D61F06}\sin \angle DAC}} \\ & = \frac {\sin 40^\circ\cdot\sin \angle C}{\sin \angle B\cdot \sin 40^\circ} \\ & = \frac {\sin \angle C}{\sin \angle B} & \small \color{#3D99F6} \text{Using sine rule again} \\ & = \frac {\overline{AB}} {\overline{AC}} = \frac {18}{24} = \frac 34 \end{aligned}$

$\implies \dfrac {2016}{m+n} = \dfrac {2016}{3+4} = \boxed{288}$

Consider the $\Delta ABD$ where AE is the bisector of $\angle BAD$ .So, applying bisector theorem

$\dfrac{AB}{AD}=\dfrac{BE}{ED}$ ..............................Equation(1)

Consider $\Delta AEC$ where AD is the bisector of $\angle EAC$ .Again applying the theorem

$\dfrac{AC}{AE}=\dfrac{CD}{ED}$ ..............................Equation(2)

Now dividing equation (1) by equation (2):

$\dfrac{AB}{AC}. \dfrac{AE}{AD}=\dfrac{BE}{CD}\\ \dfrac{BE.AD}{CD.AE}=\dfrac{AB}{AC}=\dfrac{18}{24}=\dfrac{3}{4}$

By Using the Bisector Theorem in $\Delta ABD$

$\implies \dfrac{18}{\overline{BE}} = \dfrac{\overline{AD}}{\overline{DE}} \\ \implies \overline{BE} = \dfrac{18\overline{DE}}{\overline{AD}} \rightarrow (1)$

By Using the Bisector Theorem in $\Delta AEC$

$\implies \dfrac{\overline{AE}}{\overline{ED}} = \dfrac{24}{\overline{DC}} \\ \implies \overline{DE} = \dfrac{\overline{AE} \cdot \overline{DC}}{24} \rightarrow (2)$

Combining $(1)$ and $(2)$ ,

$\implies \overline{BE} = \dfrac{18 \left( \dfrac{\overline{AE} \cdot \overline{DC}}{24} \right)}{\overline{AD}} \\ \therefore \dfrac{ \overline{BE} \cdot \overline{AD} }{\overline{AE} \cdot \overline{DC}} = \dfrac{18}{24} = \dfrac{3}{4} = \dfrac{m}{n} \\ \implies \boxed{\dfrac{2016}{m + n} = 288 }$