Speaking as someone who just got their first horse 2 months ago at the age of 27, there is a LOT more to owning a horse than knowing how to ride. Things that come to mind that you'll need to know a fair bit about: nutrition, hoof care, first aid, recognizing symptoms of serious problems that require vet care, and some training. Even if your going to be under the watchful eye of your boss at the ranch, they have 30 odd horses to look after, they may not notice if your horse looks slightly off or is exhibiting early symptoms of disease or injury, so that's going to be up to you to recognize.
Hoof care, that will mostly be up to your farrier, but it IS important that you can recognize a good trim from a bad one to know whether your farrier is doing a good job or setting your horse up for lameness. Also, your $35 estimate for trimming every 9-10 weeks does not take into account the possibility of requiring shoeing, and some horses require more frequent trimming regardless.
Nutrition, even if feed is included and you allow your horse to follow the same nutritional plan as your ranch owners horses, you still need to know what is going into your horse, how to recognize if your ranch owners feed plan is a sound one, how and when to supplement.
First aid, horses are notorious for getting into things that they shouldn't and hurting themselves. You need to know when a scrape just needs a cleaning and to be left alone, and when its severe enough to call in a vet. Also, recognizing signs of lameness, illness etc.
I would NEVER, as a 16 year old, purchase a horse without my parents knowledge and approval. They are after all, the ones who you may have to rely on if you lose your job, if things don't work out at this ranch you work at, if your horse gets severely sick or injured and needs emergency vet care that you don't have enough saved for.
Anyways, those are just a few things that come to mind at the moment that you may want to consider.
ETA: your math doesn't add up if the $175/month includes feed. You've added it to your total twice.
Last edited by kenda; 04-12-2013 at 01:33 PM.