1.1. From the village to the station along one road, a dachnik A set off on foot and a motorcyclist with a passenger - dachnik B - at the same time. Not reaching the station, the motorcyclist dropped off the passenger and immediately headed back to the village, while dachnik B continued on foot to the station. Meeting dachnik A, the motorcyclist picked him up and brought him to the station. As a result, both dachniks arrived at the station simultaneously. What part of the journey from the village to the station did dachnik A travel by motorcycle, if the dachniks walked at the same speed, 9 times slower than the speed of the motorcycle?
Answer: $\frac{5}{6}$.

Solution. Let the distance from the village to the station be 1. Since both cottagers spent the same amount of time traveling to the station and used the same motorcycle, the distance they walked is the same (this can also be seen on the graphs of their movement). Let this distance be $x$. Then the distance the motorcyclist traveled to meet cottager $\mathrm{A}$ is $(1-x)+(1-2 x)=2-3 x$. Since the speed of the motorcyclist is 9 times the speed of the cottager, the equation $2-3 x=9 x$ holds, from which $x=1 / 6$.