You burn the ropes at the wooden door of a jewler's shop. The door burns down and you nick a wristwatch.
Finally - one right for me!
Good idea - lateral thinking - but wrong.
By burning the ropes, how do you measure exactly 45 minutes worth of time?
You just use one rope. (Put the other in the dustbin.)
Then you light the other rope.
After quarter of an hour, you put the fire out.
The remainder of the rope that's left, when re-lit, will only burn for 45 minutes, so when it expires, you'll know 45 minutes has passed.
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Wrong. You have no means of measuring time other than to burn the rope. You do have a lighter to light the rope though.