Answer: a 50 min
Step-by-step explanation:
Here is my reasoning:
We can easily do 30 minutes by burning one rope at both ends.
We can do 25 minutes by first burning one rope at both ends, and once in the middle. This leaves you with 2 ropes, each burning at the end. If either of these ropes ever burns out, light the other in the middle. This will guarantee that the rope is always burning at 4 places, so the rope will burn out in 15 minutes. After all of the parts burn out, light the other rope at both ends, and twice more in the middle. Then you have 3 ropes burning at both ends. If any of them burn out, light one of the other two somewhere in the middle, and continue doing this. Once all ropes burn out, 25 minutes will have passed (since the second rope will take 10 minutes to completely burn out).
We can do 35 minutes by first burning one rope at both ends, then starting to burn the other rope at both ends, plus an additional 5 times in the middle. This will create 6 ropes, each burning at both ends. If any of these ropes burns out completely, light one of the other 5 ropes in the middle. You can continue this process until all of the ropes burn out, and 35 minutes have passed.
Though I haven't thought too hard about it, I do not believe it is possible to make 50 minutes without being able to keep an odd number of flames on the rope (which I believe you aren't guaranteed to be able to do without being able to cut the rope or extinguish a flame).
