Why does my iPhone die at 40%?

I've had my iPhone 5 for about two and a half months now. Recently, it keeps shutting off at around 40%! It's really annoying as I have to charge my phone twice as much which can't be good for it! I've heard others having the same problem. Maybe from an update? I was told to let the battery completely die and then charge it but it never does completely die. After it dies at 40% I can come back 20 min later and it will turn back on to 20-30% and then shut off in another 5min. It's seriously like clock work though. I wake up, phone has 60% battery, I go for a 40min run, the phone somehow drops to 40 something % and as I enter my house to turn off my music if does. Then as soon as I plug it in and it boots up it's right at 43%?! It did it yesterday at 51%. I'm in PA and it's getting colder out so i don't know if that has something to do with it but cold weather has been around forever so I doubt they didn't account for that. Anyways sorry I wrote a book but any help is greatly appreciated!

That can't be right, take it back and they will replace the battery as I'm guessing you just got a bad one. I have 5S and it won't shut off even if it goes down to 3%