Julian trains on a $3 \mathrm{~km}$ track. He walks the first kilometer, runs the second, and cycles the third. If he had cycled the entire track, it would have taken him 10 minutes less. Julian runs at twice the speed he walks, and cycles at three times the speed he walks. How long does Julian take to run $1 \mathrm{~km}$?
Solution

Let $t$ be the time, in minutes, that Julian takes to cycle one kilometer. Since he cycles at three times the speed he walks, he walks one kilometer in $3 t$ minutes, and since he runs at twice the speed he walks, he runs one kilometer in $3 t / 2$ minutes. Thus, he took

$$
t+3 t+\frac{3 t}{2}=\frac{11 t}{2}
$$

minutes to cover the three kilometers of the track. If he had covered the three kilometers of the track by bicycle, he would have taken $3 t$ minutes. Thus, according to the problem statement, we have (in minutes)

$$
\frac{11 t}{2}-3 t=10
$$

and solving the above equation, we get $t=4$ minutes. Finally, Julian runs one kilometer in $3 t / 2=6$ minutes.