Problem 4. On the segment $A B$, a point $C$ is chosen. The segment $A B$ is 4 times longer than the segment $A C$. Determine the lengths of the segments $A B$ and $A C$, if the length of the segment $C B$ is $24 \mathrm{~cm}$.
Solution. First method. If we divide the segment $A B$ into 4 equal parts, since it is 4 times longer than the segment $A C$, this means that the segment $A C$ is one of the four equal parts of the segment $A B$. This also means that the segment $C B$ contains the remaining three parts of the segment $A B$. Since the length of the segment $C B$ is $24 \mathrm{~cm}$, it means that one of the four parts of the segment $A B$ is $24: 3=8 \mathrm{~cm}$. Therefore, the segment $A C$ is $8 \mathrm{~cm}$ long, and the segment $A B$ is $8 \cdot 4=32 \mathrm{~cm}$ long.

Second method. If we denote the length of the segment $A C$ by $x$, then for the length of the segment $A B$ we have:

\[
\begin{aligned}
& \overline{A C}+\overline{C B}=\overline{A B} \\
& x+24=4 x \\
& 4 x-x=24 \\
& 3 x=24 \\
& x=8 \mathrm{~cm}
\end{aligned}
\]

The segment $A B$ is $8 \cdot 4=32 \mathrm{~cm}$ long.