## 3. Before the Competition

A group of cyclists decided to train on a small road before the competition. All of them are riding at a speed of 35 km/h. Then one of the cyclists suddenly breaks away from the group and, moving at a speed of 45 km/h, covers 10 km, after which he turns back and, without slowing down, rejoins the others (who, ignoring him, continue to move at the same speed).

How much time has passed from the moment the cyclist broke away from the group until he rejoined them?
3. Let $x$ be the time that has passed since the cyclist broke away from the group until his return. Clearly, this cyclist has traveled a distance of $45 x$ km in this time, while the rest of the group has only traveled $35 x$ km. However, the sum of these two distances consists of the distance the cyclist traveled alone before the turn (10 km), the distance he traveled after the turn, and the distance the rest of the group of cyclists traveled in the corresponding time, which is to say, it is equal to $y$ twice the distance traveled by the cyclist before the turn. Thus,

$$
45 x + 35 x = 2 \cdot 10 = 20, \quad \text{i.e. } x = 1 / 4.
$$

Therefore, a quarter of an hour has passed from the moment the cyclist broke away from the group until his return.