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I've been using my iPhone 5 for a while and a few months back it started to shut down when it still had 10% battery remaining.
I tried to send it to repair while it still had coverage but they told me that my phone was OK and that with the iOS 7 update it would get better. They also told me that it was a normal behavior, and this was supposed to happen if the remaining battery wasn't enough for the phone to operate (e.g. if there was an opened app needing a lot of power). So, since then I always close the apps that I'm not using but this hasn't solved anything.
Since I updated, the issue has just gotten worse. Now it's shutting down when it has 15%-20% battery.
Does anyone else has had the same issue?
Most likely faulty battery. If you have convenient Apple Store, make reservation and then go and ask for free quick battery diagnositic testing. Also refer them to your previous attempt to take care of this while still under Warranty.