For example, if I take out a personal loan from the bank for 1k for 1 year with 10% interest, I would pay $100 in interest. What would happen if I paid back $900 on the first day of the loan, and then made payments through the rest of the year on the remaining $100. Would I end up only paying 10% of that remaining $100, so $10, or 10% of the principal?
First the bank wouldn't allow you to pay off the loan, so quickly, without a severe penalty. And primarily loan is amortized over the whole time period of your payments. The sum you pay goes to a part of the principal and part as interest.
But there are also Interest only loans too.
An interest-only loan is a loan in which, for a set term, the borrower pays only the interest on the principal balance, with the principal balance unchanged. At the end of the interest-only term the borrower may enter an interest-only mortgage, pay the principal, or (with some lenders) convert the loan to a principal and interest payment (or amortized) loan at his/her option.
Primarily depends on what type of loan you have taken out. And banks use compounding rather than simple interest calculations.
The loan would have $100 interest only if it were a single $1100 payment after 12 months. If it were paid with level payments, the total interest would be just over $50. The answer to your question can only be known by looking at the terms of the loan. I agree with DC that the bank isn't likely to permit such an early payoff, but of course, if they got a fee for issuing the loan, they may not care. Most normal loans credit a payment first to accrued interest since last payment, then to principal. See the loan docs.
A few years ago, my daughter wanted to buy a car. They offered her a loan at 10%. Fortunately for her and unfortunately for the sales people I was there. The sales price / loan would have been £8,000 and over 5 years she was supposed to pay back a total of £12,000 at that 10% interest rate.
One of the regrets in my life is not reporting them to Trading Standards straight away.
To explain the math: £4,000 interest is exactly 50% of £8,000. So they expected her to pay 10% per year interest on the original principal. However, as you pay back a loan, you owe a lot less than that on average; the amount owed goes down from £8,000 to £0. The APR gives the correctly calculated interest rate, which with these payments would have been about 19.5%. So they tried to charge her about twice as much interest as they said.
In the UK, any loan has to state the APR, by law. If a company charged you a fixed percentage on the original principal for the duration of the loan on a normal repayment loan, the APR would be close to twice as high as the nominal rate. (An interest only loan where you pay interest only through the loan term and then repay the principal) would be different.