Question

The repayment schedule is for a $1,300 loan at 7.5% for 6 months. What is the principal after the first payment?

Answer 1

First, you will need to calculate the monthly payment.
The loan payment formula is attached. 
(I'll leave it up to you to show the math).
Anyway, using a loan calculator, the monthly payment is 221.43
the monthly loan rate of 7.5% = rate / 1,200
.00625
Let's calculate the interest paid for the first payment

Interest Paid = Previous Balance * Monthly Interest Rate
Interest Paid = 1,300 * .00625
Interest Paid = $8.13

Now to calculate equity.
Equity = Monthly Payment -Interest Paid
Equity = 221.43 -8.13
Equity = 213.30

The equity (or what you own), is 213.30

The New Loan Balance (or the principal) is calculated by
New Balance = Previous Balance -Equity
New Balance (or principal) = 1,300 -213.30
= 1086.70

You might find this page VERY helpful:
http://www.1728.org/loanmath.htm