## 25. Math Puzzle $6 / 67$

A motorcyclist sets off at $x$ o'clock and notes the time as 9:00 AM after completing the first quarter of the journey. At the halfway point, he turns onto the main road and continues at double the speed. A clock at the end of the second third of the journey shows 9:30 AM.

When did the motorcyclist set off and when did he reach his destination?
The task is simply to solve using a diagram, plotting the distance over time. The abscissa should be divided into 9 parts. On the ordinate, only the points mentioned in the problem statement as distance indicators need to be marked.

For the first half of the distance, the driver needs 6 parts of the time scale, and for the second half at double the speed, only 3 parts. For the segment from $\frac{1}{4}$ to $\frac{2}{3}$ of the total distance, he needs only 4 parts of the time scale $=30$ min.

![](https://cdn.mathpix.com/cropped/2024_06_06_69be9c4355ed0dc2557cg-11.jpg?height=477&width=485&top_left_y=2080&top_left_x=794)

One part of the time scale is therefore 7.5 min. 3 parts of the time scale before 9:00 AM, he started, so at 8 h $37 \mathrm{~min} 30 \mathrm{~s}$. Two parts of the time scale after 9:30 AM, he arrives at the destination, so at 9 h $45 \mathrm{~min}$.