constant rate of drain + constant rate of fill

Question

I am starting with 155 stamina in a video game. Every 55 seconds I lose 1 stamina. Every 5 minutes I gain 1 stamina. I'm trying to find out how many minutes until I have 0 stamina. I set t = 1 second, so -55t + 300t = 245t but 155 isn't in units of t, I'm stuck.

Answer 1

I assume the increases and decreases happen in exactly the intervals given, so at $55$ seconds you drop to $154,$ at $110$ seconds you drop to $153$, by $5\cdot 55=275$ you are down to $150$ then at $300$ you go back up to $151$.

The pattern will repeat after $\operatorname {LCM}(55,300)=3300$ seconds. In that time you will have taken $60$ decreases and $11$ increases, so you will lose $49$ net. At the last of the $3300$ seconds you will have both an increase and a decrease for no net change. The two counters are then back to zero just like the start. After three cycles of this, $9900$ seconds, you will have lost $147$ so be at $8$. You need $9$ losses to take care of these $8$ plus the $1$ you pick up at $10200$ seconds, so the time is $9900+9\cdot 55=10395$ seconds.

Answer 2

If this is too complex then start with simpler problems, starting with 3,4 or 5 points, loose 1 point every 2 seconds gain one point every 3 seconds. Then try more difficult problems. Make a table that shows what happens every second:

t           s
0           3
1           3
2   -1      2
3       +1  3
4   -1      2
5           2
6   -1  +1  2
7           2
8   -1      1
...

Then try to calculate this without using a table.