37. Will the cyclist make it to the train?

- Wasn't the previous problem sad? the Turtle Quasi asked. - Just think! The poor frog spent so many days in a dark well! And to get out of there, she had to undertake a climb like a real mountaineer!
- Nonsense! - the Griffin interrupted him. - The saddest part of the whole story is that I kept hearing your voice and couldn't sleep for a minute! Tell us another problem!
- Alright, - the Turtle Quasi agreed. - A man set out on a bicycle for twelve miles to the railway station to catch a train. He reasoned as follows:
- I have one and a half hours to catch the train. Four miles of the road go uphill, and I will have to walk them. I think I can manage it in an hour. Then four miles of the road go downhill. On this section, I can reach a speed of twelve miles per hour. The last four miles of the road are on flat ground. On this section, I can reach a speed of eight miles per hour. On average, this is eight miles per hour, so I will get to the station on time.

Was the cyclist's reasoning correct?

![](https://cdn.mathpix.com/cropped/2024_05_21_5afa4e43618f5800e92dg-056.jpg?height=366&width=925&top_left_y=1453&top_left_x=608)
37. Will the cyclist make it to the train? The cyclist reasoned incorrectly: he averaged distances, not time. If he had traveled at 4 miles per hour, 8 miles per hour, and 12 miles per hour for the same amount of time, his average speed would indeed have been 8 miles per hour. However, he spent more time climbing the hill (at 4 miles per hour) and less time descending the hill (at 12 miles per hour).

It is not difficult to calculate how long he was on the road. Climbing the hill took him 1 hour, he spent half an hour (or 30 minutes) traveling on the flat section of the road, and a third of an hour (or 20 minutes) descending the hill. In total, he was on the road for 1 hour and 50 minutes, missing the train by 20 minutes.