## Task 1 - 050611

Two cyclists start simultaneously from Leipzig and Dresden (distance $119 \mathrm{~km}$). The cyclist from Leipzig is heading to Dresden, and the one from Dresden is heading to Leipzig. One of them travels at $15 \mathrm{~km}$ per hour, the other at $20 \mathrm{~km}$ per hour.

a) What is the distance between the two cyclists after $2 \frac{1}{2}$ hours?

b) How far are they from both cities when they meet?
a) After $2 \frac{1}{2}$ hours, one cyclist has covered a total of $2 \frac{1}{2} \cdot 20 = 50 \mathrm{~km}$, and the other has covered $2 \frac{1}{2} \cdot 15 = 37.5 \mathrm{~km}$ in the same time.

After $2 \frac{1}{2}$ hours, their distance from each other (in kilometers) is therefore $119 - (50 + 37.5) = 31.5$.

b) In $\frac{1}{5}$ of an hour, which is 12 minutes, one cyclist covers $3 \mathrm{~km}$, and the other covers $4 \mathrm{~km}$. Together, they cover $7 \mathrm{~km}$ in 12 minutes. Therefore, they will meet each other after $\frac{119}{7} \cdot 12 = 17 \cdot 12 \mathrm{~min}$.

In this time, one cyclist has covered $17 \cdot 3 \mathrm{~km}$, and the other has covered $17 \cdot 4 \mathrm{~km}$. The meeting point is therefore $51 \mathrm{~km}$ from one city and $68 \mathrm{~km}$ from the other city.