## 5. The Story of the Vagabonds

At the very moment when Pierrot left the "Commercial" bar, heading for the "Theatrical" bar, Jeanno was leaving the "Theatrical" bar, making his way to the "Commercial" bar. They were walking at a constant (but different) speed. When the vagabonds met, Pierrot proudly noted that he had walked 200 meters more than Jeanno. Already quite drunk, Jeanno got offended and hit Pierrot. Of course, Pierrot didn't hesitate to respond... When the fight was over, they hugged and cried, and each continued on their way, but at half the speed they had before, as the beatings were taking their toll. Therefore, Pierrot reached the "Theatrical" bar in 8 minutes, and Jeanno reached the "Commercial" bar in 18 minutes. What is the distance between the bars?
5. Let $x$ (m) - be the distance between the bars we are looking for; $d$ - the distance traveled by Pierrot by the time of the meeting; $V$ (m/s) - Pierrot's speed, and $v$ - Jeanno's speed before the fight.

The sum of the distances traveled by the vagrants is

$$
d+(d-200)=x, \quad \text { from which } \quad x=2 d-200 .
$$

Before the meeting, each traveled their path in the same amount of time; therefore,

$$
d / V=(d-200) / v
$$

After the meeting, Pierrot walked for 8 minutes:

$$
\frac{d-200}{V / 2}=8, \quad \text { or } \quad V=\frac{d-200}{4}
$$

and Jeanno - for 18 minutes:

$$
d /(v / 2)=18, \quad \text { or } \quad v=d / 9 .
$$

Substituting the values of $V$ and $v$, expressed in terms of $d$ from (3) and (4), into equation (2), we get

$\frac{4 d}{d-200}=\frac{9(d-200)}{d}$, or $5 d^{2}-3600 d+360000=0$.

The discriminant of this quadratic equation in $d \Delta^{\prime}=(1800)^{2}-5 \cdot 360000=(1200)^{2}$. Therefore,

$$
d=\frac{1800 \pm 1200}{5} \text {, i.e., } d=600 \text { or } 120 \text { (m). }
$$

Since $x=2 d-200$ [see (1)], we get two values for $x: 1000$ or 40.

However, since, according to the condition, the distance $d$ is only a part of the sought distance $x$, the second solution should be discarded. The first solution remains. Thus, the bars are 1 km apart.