## Task 2 - 010722

The railway line Leipzig - Halle - Köthen - Magdeburg is 123.6 km long. A passenger train departs from Leipzig at 12:32. It has an average speed of $32.7 \frac{\mathrm{km}}{\mathrm{h}}$. An express train (D-Zug) departs from Leipzig at 13:11. Its average speed is $75.2 \frac{\mathrm{km}}{\mathrm{h}}$.

a) At what time does the express train catch up with the passenger train?

b) How many kilometers have both trains traveled by then?
The distance to be covered is the product of the average speed and the time required for each train until the faster one catches up. The travel times of the two trains differ by $39 \mathrm{~min}=\frac{13}{20} \mathrm{~h}$ (which is the difference in their departure times).

The equation is thus, if $t$ represents the travel time of the D-train:

$$
t \cdot 75.2 \frac{\mathrm{km}}{\mathrm{h}}=\left(t+\frac{13}{20} \mathrm{~h}\right) \cdot 32.7 \frac{\mathrm{km}}{\mathrm{h}}=t \cdot 32.7 \frac{\mathrm{km}}{\mathrm{h}}+21.255 \mathrm{~km}
$$

a) From this, $t=0.5 \mathrm{~h}=30 \mathrm{~min}$ is calculated. The D-train catches up with the passenger train half an hour after 13:11, which is at 13:41.

b) At this time, both trains have covered $37.6 \mathrm{~km}$.