iPhone Dies with 80% Battery Remaining
While the iPhone 6s has a service plan - reduced cost replacement1 for having an issue where the battery reaches an end-of-life state after around a year of use (causing the battery to die at ~30%), older iPhones normally reach this state when their batteries grow old and tired (lithium-ion batteries can only be recharged so many times before they become unable to hold most of the original charge). Your iPhone 5 has a 3-3.5-year-old battery2, and thus it's the battery's fault that the phone is dying so quickly.

The only fix is to replace your iPhone's battery, something you can safely attempt yourself following this video or bring your phone to the Apple Store to have them do it (cost is usually 80-100 dollars without AppleCare).

1 The battery malfunctions are blamed on a manufacturer leaving the batteries exposed to ambient air too long; note that it is not a safety issue, merely a convenience issue. Recall information here.

2 Based on a release date of September 2014.

tl;dr: It's the old battery's fault, bring it to an Apple Store and get them to replace the battery to fix the issue.