Simple looking but difficult fraction problem

Question

If I use my phone continuously, its battery will last for 6 hours. If I do not use my phone, its battery will last for 60 hours. On a plane trip, I use my phone for only half of the trip. The phone is completely drained the exact moment the trip ended. how long was the plane trip?

Assume that the battery is at max capacity at the start of the trip. I thought this problem was simple, meaning that it could be solve as a linear equation where I had to solve for t. But, I found that as I began solving it, I just couldn't find a way to find a function where at t = 0 battery percent = 100 and at t = 6 or 60 battery percent = 0. It seemed like a quadratic but I wasn't sure how that could be possible. So as a result, I am stuck and I don't know where to begin since I don't know how to frame the problem. I am looking for a way to write this problem as a function of time, which yields the percentage remaining as I believe that is the key to solving this problem.

Answer 1

The battery consumption based on the $c$ hours of continuous used and $i$ hours of idle time is $$B(c,i)=100\%\cdot \frac{c}{6}+100\%\cdot \frac{i}{60}$$

Now you know that half the time $t$ you used you phone, the other half you didn't, and the total battery consumption was $100\%$, can you take it from here?

Answer 2

This is a rate problem.

While using the phone we burn $\frac{1}{6}$ battery/hour and when we are not using the phone we burn $\frac{1}{60}$ battery/hour where one battery is the complete charge on the phone.

Let $\tau$ be the length of the trip. Then $$\frac{1}{6}\frac{1}{2}\tau+\frac{1}{60}\frac{1}{2}\tau=1\implies\frac{11}{120}\tau=1\implies\tau=\frac{120}{11}\text{ hours.}$$