How much time should be spent playing versus studying? I have been learning fairly rapidly lately (will probably go about 400 points in about a year), although I am sure that pace will slow with time.

However, my learning is actually slowed down because I like to play too much. That sounds silly because the whole point is to play and have fun. But what happens is I start playing a few games to try to apply what I have learned and then I just keep playing until I get tired and forget about whatever study I was doing. 

As much as I like playing it's not as satisfying as learning. I know some like to win, some like to play, but I like to understand. 

So, knowing that my goal is to understand as much as possible, how do I get the right balance between studying and trying to apply my studies with games?
I think you are already on the right track.  Obviously the more you play the more you learn, so even though you may think you are not learning when playing, you are in fact learning something new with every game.  Finding the right balance is subjective, but what I would do is I would learn small things and try to apply them in as many games as possible, basically lather-rinse-repeat.  For example, if you study checkmating pattern(s), you can try to apply some of the patterns you learned to some games.  I would not try to study too much and then apply everything you learned.  In a nutshell, learn a little bit, try to understand it, if you have trouble, just ask here for more clarification and play some games and try to apply what you have learned.  I did this with the Scotch Game, basically applying what I had studied over and over again until I basically became one with the Scotch and comfortable with it.  I was later able to come back and answer a question on here pertaining to the Scotch due to playing, learning, and understanding the opening that I would not have been able to answer before.