## 195. Math Puzzle $8 / 81$

Frank is having a race with his younger brother Dirk. He gives him a 30-meter head start. While running, Frank covers four meters with every four steps, while his brother only covers three meters with every four steps (in the same time).

At what distance does Frank catch up to Dirk? How many steps does he need to take?
Until Frank catches up with Dirk, both boys run the same number of 8 steps. Since Frank covers $4 \mathrm{~m}$ with every four steps and Dirk only $3 \mathrm{~m}$, but has a $30 \mathrm{~m}$ head start, the following equation applies:

$$
4 \cdot 4 a=3 \cdot 4 a+30 \quad, \quad 4 a=30
$$

After 30 times 4 steps, Frank thus catches up with his brother. This is 120 steps and therefore also 120 meters for Frank.