63. One bicycle for two. Two brothers had to set off on a journey and arrive at their destination simultaneously. They had only one bicycle, on which they took turns riding, and when the rider's time was up, he would dismount and leave the bicycle by the fence, then walk forward without waiting for his brother, while the one walking behind would pick up the bicycle upon reaching that spot and ride for his turn, and so on. Where should they switch the bicycle? If the walking and cycling speeds are the same, the problem is extremely easy to solve. They should simply divide the route into an even number of equal segments and switch the bicycle at the end of each such segment, which can be determined, for example, by a distance counter. In this case, each brother would walk half the distance and ride the other half.

But here is a similar problem that is not so easy to solve. Anderson and Brown have to cover a distance of 20 km and arrive at their destination simultaneously. They have one bicycle between them. Anderson walks at only 4 km/h, while Brown walks at 5 km/h. However, on the bicycle, Anderson rides at 10 km/h, and Brown at 8 km/h. Where should they switch the bicycle? Each of them either rides or walks, without taking any breaks along the way.
63. Let Anderson travel \(11 \frac{1}{9}\) km, leave the bicycle, and walk the remaining part of the journey. Brown will walk until he picks up the bicycle and then ride the remaining part of the way. In this case, he will arrive at the destination simultaneously with Anderson, and the entire journey will take them 3 hours and 20 minutes. The 20 km can also be divided into 9 segments of \(2 \frac{2}{9}\) km each, with Anderson traveling first. In this case, Anderson will ride each of his 5 segments in \(\frac{2}{9}\) hours and walk each of the remaining 4 segments in \(\frac{5}{9}\) hours, spending a total of \(3 \frac{1}{3}\) hours on the journey. Brown will ride each of his 4 segments in \(\frac{5}{18}\) hours and walk each of the remaining 5 segments in \(\frac{4}{9}\) hours, also spending a total of \(3 \frac{1}{3}\) hours on the journey. The distances traveled by Anderson and Brown on the bicycle are in the ratio of 5 to 4, and the distances they walk are in the ratio of 4 to 5.