The two egg timers

I have two egg timers. One takes seven minutes for the sand in the top half to run through to the bottom half, and the other takes eleven minutes. How can I use the two to obtain hardboiled eggs that have been boiled for exactly a quarter of an hour? One way of doing this is as follows. Start both the eggtimers together. When the seven-minute timer runs out, put the eggs into boiling water. When the eleven-minute time runs out (four minutes later), start the eleven-minute timer again. When it runs out again, the eggs will have boiled for four minutes plus eleven minutes, a quarter of an hour in total. Unfortunately, I have to spend 22 minutes to get these 15-minute eggs. Can you see how I could spend less time in boiling the eggs for 15 minutes?
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Start both the timers together and pop the eggs into the boiling water. After seven minutes, when the smaller timer runs out, restart it. After eleven minutes, when the larger timer runs out, and the smaller timer has been running for four minutes since it was restarted, invert the smaller timer. It will then run for four more minutes, running out after 15 minutes. All told, then, I only have to spend 15 minutes to get 15 minute eggs.

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