We are installing new tires on both wheels of a motorcycle. A tire is considered completely worn out if it has run $15000 \mathrm{~km}$ on the rear wheel, or $25000 \mathrm{~km}$ on the front wheel. What is the maximum distance the motorcycle can travel before the tires are completely worn out, if we timely switch the front tire with the rear tire?
Solution. If both the front and rear wheels wear out at the same time, they must travel the same distance. Let this distance be $\frac{s}{2}$.

If the wear of the front wheel is $\frac{1}{25000}$ per $1 \mathrm{~km}$, then over $\frac{s}{2} \mathrm{~km}$, it is $\frac{s}{2} \cdot \frac{1}{25000}$. The wear of the rear wheel is $\frac{1}{15000}$ per $1 \mathrm{~km}$, so over $\frac{s}{2} \mathrm{~km}$, it is $\frac{s}{2} \cdot \frac{1}{15000}$.

The equation for the tire wear can be written as:

$$
\frac{s}{2} \cdot \frac{1}{25000} + \frac{s}{2} \cdot \frac{1}{15000} = 1
$$

From this:

$$
3s + 5s = 150000, \quad s = 18750
$$

Therefore, the motorcycle can travel a maximum of $18750 \mathrm{~km}$.