## 10. Math Puzzle 3/66

Two hikers have planned a distance of $40 \mathrm{~km}$. Since one of them has a bicycle, he suggests to the other: "We will take turns using the bicycle, and you will ride for half an hour, then leave the bicycle by the road and continue on foot. In the meantime, I will walk to the bicycle, ride after you until I catch up, then you will ride for another half an hour, etc."

How much time will the two need if they walk at $5 \frac{\mathrm{km}}{\mathrm{h}}$ and cycle at $20 \frac{\mathrm{km}}{\mathrm{h}}$?
The first hiker covers $10 \mathrm{~km}$ in 30 minutes. The other one walks $2.5 \mathrm{~km}$ in 30 minutes, so he needs another $1.5 \mathrm{~h}$ to reach the 10 km mark.

By then, the first hiker has already reached km 17.5, and thus needs only 30 more minutes to reach km 20. In this half hour, the second hiker, on a bicycle, has also reached km 20. This is halfway through the route, and each has taken $2.5 \mathrm{~h}$ for the $20 \mathrm{~km}$. Since the process repeats, the hikers will need $5 \mathrm{~h}$ for the $40 \mathrm{~km}$.