A loan of 3,500 dollars is repaid with equal annual payments over 4 years at 9% interest compounded annually. What is the amount of each annual payment?

Introduction / Context:
This is a standard loan amortization problem. You are given the loan principal, term, and annual interest rate, and you must compute the fixed annual payment that exactly pays off the loan, including both principal and interest, over the specified time period.

Given Data / Assumptions:

• Loan principal PV = 3,500 dollars.
• Annual interest rate r = 9% per annum.
• Number of annual payments n = 4.
• Payments are made at the end of each year.
• Interest is compounded annually at the same rate.

Concept / Approach:
The fixed payment for a fully amortizing loan is given by the annuity formula:
Payment R = PV * r / (1 - (1 + r)^(-n)) Here PV is the present value of all payments, r is the periodic rate, and n is the total number of payments.

Step-by-Step Solution:
Step 1: Convert interest rate to decimal: r = 0.09. Step 2: Substitute PV = 3500, r = 0.09, n = 4 into the formula. Step 3: Compute the denominator 1 - (1.09)^(-4). Step 4: Compute 3500 * 0.09 = 315. Step 5: Divide 315 by the denominator to obtain R ≈ 1,080.34 dollars. So the borrower must pay approximately 1,080.34 dollars each year.

Verification / Alternative Check:
You can verify by constructing a repayment schedule: each year compute interest on the outstanding balance at 9%, subtract it from the payment to find the principal reduction, and update the balance. After the fourth payment, the balance should be near zero, confirming that the payment amount is correct.

Why Other Options Are Wrong:
Options A, B, C, and E either underestimate or overestimate the yearly payment needed to amortize the 3,500 dollar loan at 9% in 4 years. For instance, 890.60 dollars is too small and would leave a remaining balance after 4 years, while 1,089 dollars is slightly higher than necessary.

Common Pitfalls:
A common error is to divide the principal by the number of years and then add simple interest, which does not match the compound interest structure of an amortized loan. Another mistake is to forget to use the negative exponent when computing (1 + r)^(-n) in the denominator.

Final Answer:
Each annual payment required to repay the loan is approximately 1,080.34 dollars.