There seem to be a lot of comments hung up on the length of the ropes. The whole point is to discard the physical dimensions and consider only the TIME. Even if a rope were constructed so that the first 80% of it’s length burnt in 30 min, and the last 20% in the remaining 30 minutes, by lighting both ends at once the entire rope will still be consumed in exactly 30 minutes (although the flames will meet far from the center). With this principle in mind, the second step for the 15 minute measurement should follow without much difficulty.

All the answers that suggest to cut the secon rope into half, or to make an 8 shape and start burn in the intersections are wrong … think about it …
(for the 8 shape, what if the upper o burns only 5 minutes, and the lower o burns 25minutes.. you could not to measure 15 minutes)

Tie the ends of the first rope together to make it into an O. That rope will burn in half an hour (because it burns in 2 directions).

Tie the ends of the second rope together, but then fold it over to make an 8 shape. If you start burning at the intersection, it will burn in 15 minutes (because it burns in 4 directions – thus 4 times faster).

Light one right after the other, and you got your 45 minutes.

@Sigil
You are mistaken about the solution being faulty. The second rope has been burning for 30 minutes, it does not mention ‘half a rope’ that is simply you misinterpreting the answer.

As it has been burning for 30 minutes means 30 minutes of rope is left, lighting the other side will reduce the time to 15 minutes regardless of the physical length that remains.

Your answer also borders on spoiler material, you should have flagged it as such.

Light both ends of rope A, and one end of rope B. Start the timer. When rope A is completely burnt out, 30 minutes have elapsed, and half the burn time of rope B has been expended. Light the other end of rope B. When rope B completely burns out, 15 minutes further will have elapsed. There’s your 45 minutes.

You could also do it this way: Label the two ropes A and B. Leave A as a normal length of rope. Fold B in half. Burn rope A at both ends to that it is consumed in 30 minutes. As soon as this half hour is done, Burn rope B, which is folded in half, at both ends, so that it is consumed in 15 minutes. Total elapsed time, 45 minutes.

it is true because you independently know that the rope has been burning for 1/2 hour because you know the first rope (which you lit at the same time) has been burning for 1/2 hour

The solution for this puzzle is incorrect. It assumes that the second rope burns at a constant rate. “Once the one rope (which is burning from both ends) finally burns out (and you know a 1/2 hour has elapsed), you also know that the other rope (which is burning from only one end) has exactly 1/2 hour left to burn.” This is not true. The puzzle itself states, “burning half of the rope is not necessarily 1/2 hour”. Thus the solution is incorrect.

the way i thought it… you burn the first rope at both ends, and cut the second rope in half… doesn’t really matter where… after the first rope has burned up you light the second rope that has been cut at the now four ends. This will burn for 15 minutes and total exactly 45 minutes

I thought of this a different way. I sliced each rope (length ways and not easy) into 4 equal narrow strips. Ending up with 8 strips. Tie any of the 3 strips together to make one long strip – light one end at it should burn for 45 minutes. I’d say this is more elegant since you have rope left over. Especially if you are on a desert island….