The key to this solution lies in what happens when you light the ropes from both ends. If the ropes takes one hour to burn completely, then when you light a rope from both ends it will take half the time because the consumption rate of the fire is doubled.
To time forty-five minutes, you must do this: Light the first rope from both ends and the second rope from just one end. Once the first rope is completely burnt, you will know that thirty minutes have gone by and you will know that the second rope has thirty minutes of time left before it burns completely. As soon as the first rope is completely burnt you light the other end of the second rope, which will double the rate of consumption and cut the remaining time in the rope in half, meaning that is will only take fifteen more minutes for the second rope to burn out completely.
The thirty minutes from the first rope and the fifteen minutes from the second rope gives you forty-five minutes.
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