Problem 4. A phone charges fully in 1 hour 20 minutes on fast charging, and in 4 hours on regular charging. Fedya first plugged in a completely discharged phone to regular charging, and then, when he found the necessary adapter, switched it to fast charging until it was fully charged. Find the total charging time of the phone, given that it was on fast charging for one third of the total charging time. Assume that the phone charges at a constant rate both on fast and regular charging. [6 points] (based on problem 7.3 of the Math Festival 2007)
Answer: 144 minutes = 2 hours 24 minutes.

Solution. Since the fast charging lasts 1 hour 20 minutes, which is 80 minutes, the phone charges by $1 / 80$ of the full charge per minute. For the regular charging, which lasts 4 hours, or 240 minutes, the phone charges by $1 / 240$ of the full charge per minute.

Let's denote the total charging time of the phone in minutes by $t$. Then $t / 3$ is the time for fast charging, and $2 t / 3$ is the time for regular charging. During the fast charging, the phone charges to $\frac{t}{3} \cdot \frac{1}{80}=\frac{t}{240}$ of the full charge. During the regular charging, it charges to $\frac{2 t}{3} \cdot \frac{1}{240}=\frac{t}{360}$ of the full charge. Since the phone is fully charged at the end of the charging process, we can set up the equation $\frac{t}{240}+\frac{t}{360}=1$, solving which we find

$$
t=\frac{1}{\frac{1}{240}+\frac{1}{360}}=\frac{1}{\frac{3}{720}+\frac{2}{720}}=\frac{720}{5}=144
$$