Problem 3. While Peter was going to school, after he had walked $1 \mathrm{~km}$ and half of the remaining distance, he met his uncle Stojan, who asked Peter how many more kilometers he had to go to reach the school. Peter replied that he still had to go $1 \mathrm{~km}$ and a third of the total distance of the path. How long is the path that Peter travels to school?
Solution. First method. When Petre passed $1 \mathrm{~km}$ and half of the remaining part of the road, the second half of the remaining part of the road represented $\frac{1}{3}$ of the total length of the road plus $1 \mathrm{~km}$. This means that after traveling $1 \mathrm{~km}$ at the beginning, Petre had $\frac{2}{3}$ of the total length of the road plus $2 \mathrm{~km}$ left. The remaining part is $1 \mathrm{~km}$ less than the length of the entire road. Therefore, $\frac{1}{3}$ of the total length of the road is $2 \mathrm{~km} + 1 \mathrm{~km} = 3 \mathrm{~km}$, and the total length of the road is $9 \mathrm{~km}$.

Second method. $1+\frac{x-1}{2}+\frac{x}{3}+1=x ; \quad x=9 \mathrm{~km}$.