## Task 6 - 020826

Klaus is riding his moped at a constant speed along a road and passes a kilometer stone with a two-digit number before the decimal point at the beginning. After exactly $1 \frac{1}{2}$ hours, he passes another kilometer stone, on which the digits before the decimal point are the same but in reverse order.

After another $1 \frac{1}{2}$ hours, he reaches his destination and sees a kilometer stone with a three-digit number before the decimal point, which consists of the two digits of the first stone, with a zero in between. The digit after the decimal point was the same in all three cases.

a) What distance did Klaus cover?

b) At what average speed did he travel?
The sections between the first and second as well as the second and third milestones must be equal. Let the number on the first be $10 a+b$, on the second $10 b+a$, and on the third $100 a+b$ with $1 \leq a, b \leq 9$. Then it follows that $(10 b+a)-(10 a+b)=(100 a+b)-(10 b+a)$ or, rearranged, $108 a=18 b$ or $6 a=b$.

In the given range, this is only satisfied by $a=1, \quad b=6$. Therefore, the numbers on the milestones were 16, 61, and 106 - Klaus thus covered $90 \mathrm{~km}$. Since he took $3 \mathrm{~h}$, his average speed was $30 \frac{\mathrm{km}}{\mathrm{h}}$.