Final answer:
To calculate the final payment for Fat Tires Ltd's loan, one must account for the principal reduction after each payment and calculate the interest accrued over each period using the simple interest formula. Calculations must be done for the period from loan inception until the payoff date, considering all payments made.
Step-by-step explanation:
The question involves determining the final payment made by Fat Tires Ltd on a loan with an initial principal of $10,000 at an annual simple interest rate of 7.5%. Payments of $2,800 on April 7 and $3,100 on June 5 were made towards the loan, which initially was taken on March 10 and was fully repaid on October 13.
To calculate the final payment, we need to account for the reduction in principal after each payment and subsequently calculate the interest accrued over the different periods. The loan accrues interest daily because the interest rate is annual, and the time periods involved are part of a year.
Assuming each month has 30 days for simplicity, we calculate the interest after the first payment on April 7 (28 days from March 10), after the second payment on June 5 (59 days from April 7), and the final interest from the last payment date to the loan payoff date on October 13 (130 days from June 5).
Each interest calculation will involve the formula I = P x r x t, where I is the interest, P is the principal, r is the daily interest rate (annual rate/365), and t is the time in days. After adding the interest to the remaining principal for each period, we can find out what the final payment needs to be to fully settle the loan.
It's important to note that there may be a small variance due to the actual number of days between dates and the assumption of a 30-day month. Nevertheless, the calculation will give us a close approximation of the final payment required.