A country road between points $A$ and $B$ has a gas station, which needs to be relocated due to railway construction. Two new locations are proposed. According to the first proposal, the distance of the gas station from point $A$ would decrease by one-fifth, but at the same time, the distance from $B$ would increase by one-eighth of the current distance and by $120 \mathrm{~m}$. If the second proposal were to be implemented, the distance between the gas station and $B$ would decrease by one-fifth, but the distance from $A$ would increase by one-seventh of the current distance and by $180 \mathrm{~m}$. How far are $A$ and $B$ from the gas station currently?
I. Solution: The task is clearly to be understood such that every distance is measured along the route (so if the road is not straight, it is not measured as the crow flies), and based on the available data, it can only be solved in this case.

Let the distance of the gas station from point $A$ in km be $x$, and from point $B$ be $y$. By how much the distance from one point to the station decreases, the distance from the other point must increase by the same amount. Thus, according to the first relocation proposal,

$$
\frac{x}{5}=\frac{y}{8}+0.12
$$

and according to the second proposal,

$$
\frac{y}{5}=\frac{x}{7}+0.18
$$

From (1), expressing $x$ and substituting into (2):

$$
\frac{y}{5}=\frac{1}{7}\left(\frac{5 y}{8}+0.6\right)+0.18
$$

from which

$$
y=2.4 \text{ km}
$$

and thus

$$
x=2.1 \text{ km}
$$

This gives us the original distance of $A$ and $B$ from the gas station.

Rátkay Zsolt (Bp. VI., Kölcsey g. II. o. t.)

II. Solution: The task can also be solved without equations, just by reasoning.

The distance from $A$ to the gas station is $\frac{1}{5}$ of the distance from $B$ to the station plus $120 \text{ m}$, so the distance from $A$ to the station is $\frac{5}{8}$ of the distance from $B$ to the station plus $600 \text{ m}$.

Similarly, based on the second proposal, the distance from $B$ to the station is $\frac{5}{7}$ of the distance from $A$ to the station plus $900 \text{ m}$. Thus, the $\frac{5}{8}$ of the current distance from $B$ to the station is $\frac{5}{7} \cdot \frac{5}{8} = \frac{25}{56}$ of the distance from $A$ to the station plus $\frac{5}{8} \cdot 900 = \frac{1125}{2} \text{ m}$.

If we use this result from the first relocation proposal, it can be seen that $\frac{25}{56}$ of the distance from $A$ to the station plus $\left(\frac{1125}{2} + 600\right) \text{ m}$ gives the total distance from $A$ to the station, so $\frac{31}{56}$ of the distance from $A$ to the station is $\frac{1125}{2} + 600 = \frac{2325}{2} \text{ m}$. Thus, the distance from $A$ to the station is currently $\frac{2325}{2} \cdot \frac{56}{31} = 2100 \text{ m}$; based on the second proposal, the distance from $B$ is $\frac{5}{7} \cdot 2100 + 900 = 2400 \text{ m}$.

Magos András (Bp. II, Rákóczi g. II. o. t.)