1. The front tire of a motorcycle wears out after $25000 \mathrm{~km}$, while the rear tire wears out after $15000 \mathrm{~km}$. After how many kilometers should the tires be swapped so that they wear out simultaneously? After how many kilometers must the motorcyclist put on new tires?
Solution. To ensure the tires are used equally, they must travel the same number of kilometers on the front and rear wheels. Let's assume that each tire travels $x \mathrm{~km}$ on the front and $\mathrm{xkm}$ on the rear until it is completely worn out. Then the unknown $x$ satisfies the condition $\frac{1}{25000}+\frac{1}{15000}=\frac{1}{x}$, from which we get $x=9375 \mathrm{~km}$.

The motorcyclist will swap the positions of the tires after $9375 \mathrm{~km}$ $\mathrm{km}$, and after $2.9375 \mathrm{~km} 18750 \mathrm{~km}$ he will need to put on new tires.