IPhone three keeps shutting off, was in perfect condition?

My HTC Inspire stopped charging all together but my contract isn't up yet so my mom got an iPhone three. It was in PERFECT condition, not even a single tiny scratch on it. The first day I was using it it shut off five times out of random and the only way to turn it back on was to plug it into my computer. Now it is the second day I have had it and it keeps doing the same thing. Every time it shuts off it says it has a low battery even if it's fully charged. It over heats really easily and now i have found about 6 cracks down the back of the phone and it's raised (I'm guessing where the battery is) I can't take this anymore! I don't even like iPhones in the first place!

Added (1). My mom got the phone from a friend that had used it before with no problems, the only reason she doesn't use it is because she had an upgrade. However it's not under warranty anymore and I can't afford to pay to fix anything.

Toss it! Apple will give you a new one

Hi thanks for answering my question.

but I think you should try bringing it back to Apple, I brought my iPhone 4 back because of the same issues and they replaced my phone, or you can try updating (if there's a software update) you can also try restoring your phone that could work too? Anyway I know apple has way to many problems I don't really like iPhones either… But best of luck!

I'm assuming "iPhone three", is iPhone 3G, or iPhone 3GS?
Well, that means that the battery is (nearly) dead, and should either be replaced, or should be rendered unusable and you should throw it away.

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