Example 7. In $\triangle A B C$, $D, E$ are points on $B C$, $C A$ respectively, and $B D: D C=m: 1, C E: E A$ $=n: 1, A D$ intersects $B E$ at $F$. Then the area of $\triangle A B F$ is $\qquad$ times the area of $\triangle A B C$.
Let $\angle E B C$
$$
=\alpha, \angle A B E=P \text {. }
$$

By the Angle Bisector Theorem,
$$
n=\frac{C E}{E A}=\frac{B C \cdot \sin \alpha}{A B \cdot \sin \beta} .
$$

Therefore, $\sin \alpha / \sin \beta=n A B / B C$.
Thus, $\frac{D F}{F A}=\frac{B D \cdot \sin \alpha}{A B \cdot \sin \beta}=\frac{B D}{A B} \cdot \frac{n A B}{B C}$
$$
\begin{array}{l}
\text { Let } S_{\triangle A B F}=\frac{m+1}{m n+m+1} \cdot \frac{m}{m+1} \cdot S \triangle A R C \\
=\frac{m}{m n+m+1} \cdot S \triangle \triangle B C . \\
\end{array}
$$