3.3.1 Gavriil got on the train with a fully charged smartphone, and by the end of the trip, his smartphone was completely drained. For half of the time, he played Tetris, and for the other half, he watched cartoons. It is known that the smartphone fully discharges in 3 hours of video watching or in 5 hours of playing Tetris. What distance did Gavriil travel if the train moved half of the distance at an average speed of 80 km/h and the other half at an average speed of 60 km/h? Provide the answer in kilometers, rounding to the nearest whole number if necessary.
Answer: 257 km.

Solution. Let's take the "capacity" of the smartphone battery as 1 unit (u.e.). Then the discharge rate of the smartphone when watching videos is $\frac{1}{3}$ u.e./hour, and the discharge rate when playing games is $\frac{1}{5}$ u.e./hour.

If the total travel time is denoted as $t$ hours, we get the equation $\frac{1}{3} \cdot \frac{t}{2}+\frac{1}{5} \cdot \frac{t}{2}$, from which $\frac{(5+3) t}{2 \cdot 3 \cdot 5}=1$, that is, $t=\frac{15}{4}$ hours.

Then we get (where $\mathrm{S}$ is the distance): $\frac{S}{2 \cdot 80}+\frac{S}{2 \cdot 60}$, that is, $\frac{S}{40}+\frac{S}{30}=15$,

$S=15 \cdot \frac{40 \cdot 3}{4+3}=\frac{1800}{7} \approx 257 \mathrm{km}$.