I just got accepted to a university where my tuition is completely covered by grants, but I still need to pay for housing and a car. I am taking 16 units and I feel like it would be too much to add a part-time job on top of my classes. I would have to reduce my units, which would add on another year of schooling (and I would get less grants for less units.) I was offered a low federal subsidized and unsubsidized amount of $5,500 total, which I accepted, but it's nowhere near enough to pay for rent and a car. My plan is to take out $60k in private student loans to buy a decent car that will last me 10 years and cover rent/food/other necessities during my time at university. I am majoring in finance so ideally, I plan on getting a job immediately out of college making 70k+ annually. Does this sound like a good approach?
I would stay away from private student loans personally.
I would start off with 12 and a part time job.
Nothing is set in stone.
Try different ratios different semesters.
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Don't get yourself in deep water before you even know how to swim! Don't saddle yourself with a $60K student loan before you even know whether  you'll like and do well in finance classes and  you'll get a good job in finance.
 Most students change their majors. Out of all my friends, only a handful graduated with a degree in the major they declared their freshman year (molecular cell biology/pre-med major found she didn't like the college-level sciences so changed to english and went to law school, etc...). Of those, most aren't working in their field of study (e.g. material science, physics, architecture majors went to law school; chem e went to business school; econ major went to med school; lawyer became a doctor; PhD became a lawyer; PhD became a social worker, etc...).
 It's not fair, but unless you have connections, the competitive finance jobs tend to be directed towards the top performers in top tier schools. I suggest you see if you can get an intership (preferably paid - you can make very good money during the summer with the right intership) your freshman year and do well in that internship before you start thinking that you're going to land a job in finance after you graduate. Of course, if you have an inside track, you can ignore what I wrote.
Have you considered applying for merit scholarships? A few thousand here and there add up. Are there still work-study?
OP are you in the states? If so, do you qualify for work study? What about a job with your school's library? You can definitely kill two birds with one stone if you can get a campus related job. Typically you can do your school work in between your work duties.