## 122. Math Puzzle $7 / 75$

Peter rides his bike from school to the sports field. When he had covered 3/4 of the distance, he met his friend Hans, who was also riding at the same speed.

How fast are both of them riding if the teacher, traveling by moped ( $v=40 \mathrm{~km} / \mathrm{h}$ ), meets Peter and Hans exactly at their departure from the school and the sports field, respectively?

(Since the route passes through a sparsely trafficked and open area, the speeds can be considered constant.)
The distance from the school to the sports field is $s \mathrm{~km}$. Peter covers $\frac{3}{4} s \mathrm{~km}$ until the meeting point and takes $t_{P}=\frac{3}{4} \frac{s}{v}$ hours to do so. (Let $v$ be the speed of both Peter and Hans.) Hans covers the distance $\frac{1}{4} s \mathrm{~km}$ and starts $\frac{s}{40} \mathrm{~h}$ later. Since both meet at the same time, it follows that

$$
t_{P}=\frac{3 s}{4 v}=\frac{s}{4 v}+\frac{s}{40} \quad \text { or } \quad \frac{2}{4 v}=\frac{1}{40}
$$

$v=20 \mathrm{~km} / \mathrm{h}$. Peter and Hans ride at a speed of $20 \mathrm{~km} / \mathrm{h}$.