## Task 2 - 240832

To determine the durability of a motorcycle tire type, two tires were tested. It was found that the tire on the rear wheel was no longer within the required profile depth and thus worn out after 15000 kilometers, and the tire on the front wheel after 25000 kilometers.

a) It is now desired to achieve that two such tires are worn out simultaneously by swapping them after a certain number of kilometers driven. Determine this number of kilometers!

b) After how many kilometers will both tires be worn out under the conditions of part a)?

It is assumed that the wear on both the front and rear wheels is proportional to the distance driven.
If both the front and rear tires were replaced only after they were completely worn out, the front tire would need to be replaced after $25000 \mathrm{~km}, 50000 \mathrm{~km}, 75000 \mathrm{~km}, \ldots$ and so on, and the rear tire after $15000 \mathrm{~km}, 30000 \mathrm{~km}, 45000 \mathrm{~km}, 60000 \mathrm{~km}, 75000 \mathrm{~km}, \ldots$ and so on.

Therefore, with this approach, both the front and rear tires would first need to be replaced simultaneously after 75000 km, and by then, three front tires and five rear tires, a total of eight tires, would have been used.

If one can drive 75000 km with eight tires in total, then with two tires, one can drive $\frac{75000}{4} \mathrm{~km}=18750 \mathrm{~km}$ in total.

Each of the two tires must cover the same distance as both a front and a rear tire, so that both tires are subjected to the same wear conditions. This means the tires must be swapped after $\frac{18750}{2} \mathrm{~km}=9375 \mathrm{~km}$.

The kilometer values sought in a) and b) are therefore 9375 km and 18750 km, respectively.