## 60. Math Puzzle 5/70

A tractor with a trailer, loaded with prefabricated parts, drives from the company to a construction site. An average speed of $15 \frac{\mathrm{km}}{\mathrm{h}}$ is achieved. After unloading, an average speed of $30 \frac{\mathrm{km}}{\mathrm{h}}$ is achieved on the return trip to the company (the same distance).

What is the average speed for the entire journey (to and from)?
They mean: $s$ driving distance from operation to construction site, $t_{1}$ driving time to the construction site, $t_{2}$ driving time to the operation, $v_{1}=15 \frac{\text{km}}{\text{h}}, v_{2}=30 \frac{\text{km}}{\text{h}}$ and $v_{M}$ average speed. Then it holds:

$$
t_{1}+t_{2}=\frac{s}{v_{1}}+\frac{s}{v_{2}}=\frac{s\left(v_{1}+v_{2}\right)}{v_{1} v_{2}} \rightarrow v_{M}=\frac{2 s}{t_{1}+t_{2}}=\frac{2 v_{1} v_{2}}{v_{1}+v_{2}}
$$

With the given values, $v_{M}=20$, i.e., the average speed is $20 \frac{\text{km}}{\text{h}}$.