## 9. By car and by bicycle

Jean and Jules set off simultaneously from city $A$ to city $B$, one by car and the other by bicycle.

After some time, it turned out that if Jean had traveled three times more, he would have had to travel twice less than he currently has left, and that if Jules had traveled half as much, he would have had to travel three times more than he currently has left.

What is the name of the cyclist?
9. Let Jean have traveled a distance $x$, and let $x^{\prime}$ be the distance he has left to travel; similarly, let Jules have traveled a distance $y$, and let $y^{\prime}$ be the distance he has left to travel. In this case,

$$
x+x^{\prime}=3 x+x^{\prime} / 2, \quad y+y^{\prime}=y / 2+3 y^{\prime}
$$

Therefore,

$$
x=x^{\prime} / 4 \quad \text { and } \quad y=4 y^{\prime} .
$$

Thus, Jean has traveled a quarter of the distance he still has to cover, while Jules has traveled a distance four times greater than the distance he has left to cover. The latter, therefore, was traveling much faster; it is obvious that he was using a car. The name of the cyclist is Jean.