3.47 Two athletes start running simultaneously - the first from $A$ to $B$, the second from $B$ to $A$. They run at different but constant speeds and meet at a distance of 300 m from $A$. After running the entire path $A B$, each immediately turns around and meets the other at a distance of 400 m from $B$. Find the length of $\boldsymbol{A}$.
3.47 In Fig. 3.10, the graphs of the runs of two athletes are shown.

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-111.jpg?height=566&width=879&top_left_y=690&top_left_x=199)

Fig. 3.10

Let $A B=x$ m, $C$ and $D$ be the points of the first and second meetings. If $v_{1}$ and $v_{2}$ are the speeds of the first and second athletes, then

$$
\frac{300}{v_{1}}=\frac{x-300}{v_{2}}
$$

During the time between the meetings ( $K L$ ), the athletes ran: the first $(x-300)+400=x+100$ m; the second $300+(x-400)=x-100$ m. Therefore,

$\frac{x+100}{v_{1}}=\frac{x-100}{v_{2}}$

From (1) and (2) we get $\frac{x+100}{300}=\frac{x-100}{x-300}$,

from which $x=500($ m).

Answer: $\quad 500 \mathrm{m}$.