## 3. Math Puzzle $8 / 65$

A crane is being transported to a construction site. To measure the length of the crane, Peter walks from the tip of the crane to the end and takes 15 steps (step length 80 $\mathrm{cm}$).

To walk back from the end to the tip of the crane while the train is moving at a constant speed, he needs 75 steps at the same walking speed.

How long is the crane?
With $v=$ speed, $s=$ distance, $t_{1}=$ time for 15 steps, $t_{2}=$ time for 75 steps, and $l=$ crane length, we have $l-v t_{1}=12 \mathrm{~m}, l+v t_{2}=60 \mathrm{~m}$.

It holds that $\frac{t_{1}}{t_{2}}=\frac{1}{5} \rightarrow t_{1}=\frac{1}{5} t_{2}$.

Thus, $l-v \cdot \frac{1}{5} t_{2}=12 \mathrm{~m}$ and $l+v t_{2}=60 \mathrm{~m}$. Multiplying the first equation by 5 and adding both equations yields $6 l=120 \mathrm{~m}, l=20 \mathrm{~m}$. The length of the crane is $20 \mathrm{~m}$.