# Vladimir and his motorbike

Vladimir wants to take his two daughters to their Dacha (weekend house) which is located 66 kilometers away from their home. Vladimir's motorbike has a maximum speed of 50 kilometers per hour; with one passenger (that is, driver plus one extra person), the maximum speed drops down to 40 kilometers per hour; the motorbike cannot carry three persons. Vladimir's walking speed is 9 kilometers per hour, and each daughter walks at a speed of 10 kilometers per hour.

Is there a schedule that makes all three of them (and the motorbike) reach the Dacha within three hours?

• Does each person weigh the same amount? Can the daughters drive the motorbike? – A E Mar 21 '15 at 14:46
• Can the daughters carry Vladimir and if so does it slow their walking pace? – A E Mar 21 '15 at 14:52
• If the daughters carry Vladimir, their walking speed drops down to 1 kilometer per hour. If Vladimir carries one of the daughters, his walking speed drops down to 1 kilometer per hour. – Gamow Mar 21 '15 at 14:54
• And no other form of transportation is available? – A E Mar 21 '15 at 14:56
• No other form of transportation is available. – Gamow Mar 21 '15 at 14:58

## 2 Answers

Vladimir starts with daughter 1 on the motorbike. After 48 km (that's one hour and 12 min at 40 kph), he stops and daughter 1 continues on foot. She reaches the dacha after exactly three hours.

Vladimir drives back to meet daughter 2, who has already covered 12 km. They meet 36 min later (one hour and 48 min after the start), 18 km from the starting point (he drove 30 km at 50 kph and she walked 6 extra km at 10 kph). They still have 48 km to cover, which is one hour and 12 min at 40 kph (as we saw before, not coincidentally). They will therefore reach the dacha exactly three hours after the start.

• Is there a formal way to solve this, without guessing solutions and simulating them? – Alexander Mar 23 '15 at 19:55

This picture shows the best strategy to get to the Dacha.