Burning Ropes for 45 Minutes

Lighting a rope at both ends halves its remaining time, whatever the burn rate.

Use the first rope as a 30-minute timer.

1. 1. Start

   Light rope 1 at both ends and rope 2 at one end. Rope 1 finishes after $30$ minutes.

2. 2. Second stage

   Rope 2 has $30$ minutes left; light its other end so it finishes in $15$ minutes. Total: $45$ minutes.

3. 3. Key idea

   Irregular burning does not matter: only total burn time is preserved.

Answer: Light rope 1 at both ends and rope 2 at one end; when rope 1 finishes (30 min) light rope 2's other end; it finishes 15 minutes later