The road connecting two villages on a mountain consists only of uphill or downhill sections. A bus always travels at $15 \mathrm{~km} / \mathrm{h}$ on uphill sections and at $30 \mathrm{~km} / \mathrm{h}$ on downhill sections. Find the distance between the villages if the bus takes exactly 4 hours to complete the round trip.
Observe that the uphill sections on the outbound journey are exactly the downhill sections for the return journey and vice versa. Thus, in a round trip, the distance covered on the ascents is equal to the distance covered on the descents.

Let's call the distance between the two villages $\mathrm{d}$. Since the total distance traveled was equal to $2d$, the time spent climbing was $d / 15$ hours.

Suggestion: Show that the situation in item (a) is possible and the situation in item (b) is not.

![](https://cdn.mathpix.com/cropped/2024_05_01_9f6ce5ffa52bbf7500dfg-2.jpg?height=514&width=442&top_left_y=868&top_left_x=173)

Suggestion: Notice that to arrive within 2 hours and 40 minutes, each person must cover at least half of the journey by bicycle. The time spent descending was $\mathrm{d} / 30$ hours. Since the total time was 4 hours, we have

$$
\frac{d}{15}+\frac{d}{30}=4
$$

Solving the equation, we find $d=40$, that is, the distance between the villages is equal to $40 \mathrm{~km}$.