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

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

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