A loan of $3,500 is to be repaid with equal annual payments over 4 years at an interest rate of 9% per annum, compounded annually. What is the amount of each annual payment (to the nearest cent)?

Introduction:
This problem is an example of loan amortization. A loan of 3,500 dollars is repaid in equal annual installments over 4 years at an interest rate of 9% per annum. We must determine the size of each annual payment that will fully pay off the loan plus interest by the end of the 4th year.

Given Data / Assumptions:

• Principal, P = 3,500 dollars
• Annual interest rate, i = 9% = 0.09
• Number of payments (years), n = 4
• Payments are made at the end of each year (ordinary annuity)
• The loan is fully amortized by the end of the 4th payment

Concept / Approach:
The present value of an annuity of equal payments R, at interest rate i for n periods, is: P = R * (1 - (1 + i)^(-n)) / i Here P is the loan amount. We are given P, i, and n, and must solve for the payment R: R = P * i / (1 - (1 + i)^(-n))

Step-by-Step Solution:
Step 1: Substitute P, i, and n into the formula. P = 3500, i = 0.09, n = 4 R = 3500 * 0.09 / (1 - (1.09)^(-4)) Step 2: Compute (1.09)^4. (1.09)^2 ≈ 1.1881 (1.09)^4 ≈ 1.1881 * 1.1881 ≈ 1.4116 Step 3: Find (1.09)^(-4). (1.09)^(-4) = 1 / 1.4116 ≈ 0.7085 Step 4: Compute the denominator. 1 - (1.09)^(-4) ≈ 1 - 0.7085 = 0.2915 Step 5: Compute the numerator. 3500 * 0.09 = 315 Step 6: Divide numerator by denominator. R ≈ 315 / 0.2915 ≈ 1080.34 So each annual payment is about 1080.34 dollars.

Verification / Alternative check:
We can verify by computing the present value of four payments of 1080.34 dollars: P ≈ 1080.34 * (1 - (1.09)^(-4)) / 0.09 Using the previously computed factor: P ≈ 1080.34 * 0.2915 / 0.09 ≈ 1080.34 * 3.238 ≈ 3500 Minor differences arise from rounding, but the result is sufficiently close to 3,500 dollars, confirming that 1080.34 dollars is the correct payment amount.

Why Other Options Are Wrong:
890.60: This is too small and would not generate enough present value to cover the 3,500 dollar loan at 9% interest. 1089.00 and 1070.00: These are close but do not match the exact amortization formula, and would either slightly overpay or underpay over the 4 years. 980.34: This is significantly lower than the correct payment and would leave a balance at the end of 4 years. 1080.34: This matches the payment obtained using the proper loan amortization formula.

Common Pitfalls:
Some learners mistakenly divide the loan amount by 4 and then add yearly interest separately, which ignores the time value of money and the amortization structure. Others use the simple interest formula P(1 + i * n) instead of the annuity present value formula. It is also easy to miscalculate the negative exponent or to forget to convert the interest rate from percent to decimal form.

Final Answer:
Each annual payment required to amortize a 3,500 dollar loan over 4 years at 9% interest is approximately 1080.34 dollars.