## 13. The Scout and the Drummer

On the occasion of the village festival, a procession was organized that stretched for 250 m; the scouts led the procession, and the musicians brought up the rear. Soon after the march began, the youngest scout remembered that he had not tied his neckerchief, which was left with his friend, the drummer, who was in the last row of the musicians. So the scout ran to get the neckerchief and returned to his place in 3 minutes and 18 seconds.

What was the speed of the procession, assuming the scout ran at a speed of 10 km/h?
13. Let $x$ be the unknown speed of the procession. When the scout ran to the end of the procession, he moved relative to the procession (considered stationary!) at a speed of $10+x$; when he returned, he moved relative to the procession at a speed of $10-x$. Since the length of the procession is 250 m, the total time spent by the scout is

$$
\frac{250}{1000} \cdot \frac{1}{10+x}+\frac{250}{1000} \cdot \frac{1}{10-x}=\frac{3}{60}+\frac{18}{3600}
$$

(here we measure distance in kilometers, time in hours, and speeds in km/h).

Transform our equation:

$$
(10-x)+(10+x)=4(10+x)(10-x)(3 / 60+18 / 3600)
$$

or

$$
20=4\left(100-x^{2}\right) \cdot(198 / 3600)
$$

From this, we find

$$
x^{2}=100 / 11, \quad x=\sqrt{100 / 11} \approx 3
$$

The speed of the procession is approximately 3 km/h.