# 5. CONDITION

A tourist goes on a hike from $A$ to $B$ and back, and completes the entire journey in 3 hours and 41 minutes. The route from $A$ to $B$ first goes uphill, then on flat ground, and finally downhill. Over what distance does the road pass on flat ground, if the tourist's speed is 4 km/h when climbing uphill, 5 km/h on flat ground, and 6 km/h when descending, and the distance $AB$ is 9 km?
Solution. Let $x$ km of the path be on flat ground, then $9-x$ km of the path (uphill and downhill) the tourist travels twice, once (each of the ascent or descent) at a speed of 4 km/h, the other at a speed of 6 km/h, and spends $(9-x) / 4+(9-x) / 6$ hours on this part. Since the tourist walks $2 x / 5$ hours on flat ground, and the round trip takes 3 hours and 41 minutes, then $2 x / 5+(9-x) / 4+(9-x) / 6=221 / 60$, from which $x=4$ km.

Answer: 4 km.