1. The front tire of a motorcycle wears out after $25000 \mathrm{~km}$, and the rear tire after $15000 \mathrm{~km}$. After how many kilometers should the tires be swapped so that both tires wear out simultaneously? After how many kilometers should the motorcyclist buy new tires?
Solution. To ensure the tires wear out evenly, the same number of kilometers must be driven on the front and rear tires. Let's assume that each tire travels $x \mathrm{~km}$ on the front and $x \mathrm{~km}$ on the rear.

Then for $x$ we have $\frac{1}{25000}+\frac{1}{15000}=\frac{1}{x}$, from which we get $x=9375 \mathrm{~km}$. Therefore, the motorcyclist will switch the positions of the tires after $9375 \mathrm{~km}$, and after $18750 \mathrm{~km}$, he will need to buy new tires.