13.400 On the highway, points D, A, C, and B are arranged sequentially. A motorcyclist and a cyclist set off from A and B simultaneously, heading to C and D, respectively. Meeting at E, they exchanged vehicles and each continued their journey. As a result, the first one spent 6 hours on the trip from A to C, while the second one spent 12 hours on the trip from B to D. Determine the length of the route AB, given that each person traveling on a motorcycle develops a speed of 60 km/h, and on a bicycle - 25 km/h, and furthermore, the average speed of the first person on the route AC is equal to the average speed of the second person on the route BD.
Solution. Let $x=DA, y=AE, z=EC, u=CB$. Then $\frac{y}{60}+\frac{z}{25}$ is the total travel time of the first, $\frac{z+u}{25}+\frac{x+y}{60}$ is the total travel time of the second, $\frac{y}{60}=\frac{z+u}{25}$ is the time of their travel until they meet. In the end, we have:

$\frac{y}{60}+\frac{z}{25}=6$

$\left\{\begin{array}{l}\frac{z+u}{25}+\frac{x+y}{60}=12, \\ \frac{y}{60}=\frac{z+u}{25}, \\ \frac{y+z}{6}=\frac{x+y+z+u}{12}\end{array} \Rightarrow\left\{\begin{array}{l}\frac{z+u}{25}+\frac{z}{25}=6 ; \\ \frac{y}{60}+\frac{x+y}{60}=12 \\ y+z=x+u\end{array},\left\{\begin{array}{l}2 z+u=150, \\ 2 y+x=720, \\ y+z=x+u\end{array} \Rightarrow y+z=290\right.\right.\right.$.

Considering $\frac{y}{60}+\frac{z}{25}=6, y=240, z=50, u=50 \Rightarrow y+z+u=340$.

Answer: 340 km.