## Task 4 - 090624

Jürgen and his younger sister Elke have the same route to school. Elke takes exactly 30 minutes to get from the family home to the school gate, while Jürgen takes exactly 20 minutes. One day, Elke left the house exactly 5 minutes before Jürgen.

After how many minutes did Jürgen catch up with his sister?

(It is assumed that both walked at a constant speed.)
According to the task, Elke covers exactly $\frac{1}{30}$ of the school route every minute, and Jürgen covers exactly $\frac{1}{20}$ of the same route every minute. Since Elke started 5 minutes before Jürgen, she has a head start of $\frac{5}{30}=\frac{10}{60}$ of the route.

Because $\frac{1}{20}-\frac{1}{30}=\frac{1}{60}$, this head start decreases by $\frac{1}{60}$ of the total route each minute.

The number of minutes required for Jürgen to catch up can therefore only be the number by which $\frac{1}{60}$ must be multiplied to obtain $\frac{10}{60}$. This number is 10.

Jürgen catches up with his sister in exactly 10 minutes under the given conditions.

Solutions of the II. Round 1969 taken from [5]

### 3.11 X. Olympiad 1970

### 3.11.1 I. Round 1970, Class 6