Battery dies at around 55%. Battery or Magsafe?

MacBook Pro 13" Early 2011. Unibody.

My Laptop holds charge. Gets charged to 100, but can die at any point from 40-67%.

Looking at my charge port ( Magsafe) one of the pin's looks burned out- black.

So I know I need to replace something. I feel it is the MagSafe. But, Can the MagSafe connector work partially? or it is one of those things that either work or dont work. Cause it does charge, but i feel it only works partially and could be the main problem.

Sorry I dont know how to articulate this problem well enough.