13.348. A passenger train departed from station $A$ towards station $B$ (Fig. 13.17). After $a$ hours, a train "Strela" departed from station $B$ towards station $A$. The trains met at station $C$.

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-1215.jpg?height=149&width=667&top_left_y=1619&top_left_x=367)

Fig. 13.17

After the meeting, the passenger train traveled for $b$ hours, and the train "Strela" traveled for $c$ hours. How much time did each of these trains need for the entire journey between stations $A$ and $B$? It is assumed that the speed of the trains is constant throughout the journey.
## Solution.

Let $V_{1}, V_{2}$ be the speeds of the train and "Sarrow" respectively. According to the condition $\frac{c V_{2}-a V_{1}}{V_{1}}=b \frac{V_{1}}{V_{2}}$, from which $\frac{V_{2}}{V_{1}}=\frac{a+\sqrt{a^{2}+4 b c}}{2 c}$. Then the train traveled from station $A$ to $B a+\left(c \frac{V_{2}}{V_{1}}-a\right)+b=\frac{a}{2}+b+\frac{\sqrt{a^{2}+4 b c}}{2}$ (hours), "Sarrow" traveled from station $B \kappa A-c+b \frac{V_{1}}{V_{2}}=c-\frac{a}{2}+\frac{\sqrt{a^{2}+4 b c}}{2}$ (hours).

$$
\text { Answer: } \frac{a+2 b+\sqrt{a^{2}+4 b c}}{2} \text { and } \frac{2 c-a+\sqrt{a^{2}+4 b c}}{2} \text { hours. }
$$