\section*{Task 2 - 111022}

Two cars started simultaneously and drove on the same road from A to B. The first car needed 4 hours for this distance, the second 3 hours. Both drove at a constant speed throughout the entire time.
a) At what time after the start was the first car exactly twice as far from B as the second?

b) What distance, expressed as fractions of the total distance from A to B, did each car cover by the time sought in a)?
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a) The distance from A to B is \(s \mathrm{~km}\). Then the first car drove at a speed of \(\frac{s}{4} \frac{\mathrm{km}}{\mathrm{h}}\) and the second at \(\frac{s}{3} \frac{\mathrm{km}}{\mathrm{h}}\).

The first car is exactly twice as far from B as the second car after \(t\) hours if

\[
s - \frac{s}{4} t = 2 \left( s - \frac{s}{3} t \right)
\]

holds. Since \(s \neq 0\), this is equivalent to \(1 - \frac{t}{4} = 2 \left( 1 - \frac{t}{3} \right)\), which simplifies to \(t = \frac{12}{5} = 2.4\).

Therefore, after exactly 2.4 hours, the first car was twice as far from B as the second car.

b) By this time, the first car had traveled \(\frac{s}{4} \cdot \frac{12}{5} = \frac{3}{5} s\), exactly \(\frac{3}{5} s\) of the distance, and the second car had traveled \(\frac{s}{3} \cdot \frac{12}{5} = \frac{4}{5} s\), exactly \(\frac{4}{5}\) of the distance.