Why does Galaxy S3 battery drain so fast?

[galaxyS3user1]

I recently was given a Galaxy S3 and I was shocked to see how fast the battery would drain. It went down about 2% every minute and would die suddenly at 25%-15%. I bought a brand new battery thinking the old one must be faulty, but the new one still drains very fast. I can turn it off at 100% and when I power it on it's at 80%. It also drops rapidly in sleep mode when it's not charging. I've tried power saving tips and restarted my phone, but it still drains fast. It also charges pretty quickly, so I'm thinking the phone says the battery is at 100% even if its not fully charged, which would explain why it dies so fast. But is there any way to fix this? Thanks so much!