1. Two cyclists started from places $A$ and $B$ towards each other. When they met, the first cyclist had traveled $\frac{4}{7}$ of the distance and an additional $\frac{24}{10} \mathrm{~km}$, while the second cyclist had traveled half the distance of the first. Find the distance from $A$ to $B$.
Solution. If we denote the distance from $A$ to $B$ by $x$, then from the conditions of the problem, we get the equation:

$$
\frac{4}{7} x+\frac{24}{10}+\frac{1}{2}\left(\frac{4}{7} x+\frac{24}{10}\right)=x
$$

From this, it follows that

$$
\left(1-\frac{4}{7}-\frac{2}{7}\right) x=\frac{24}{10}+\frac{12}{10}
$$

from which it is obtained that $x=25.2 \text{ km}$.