A loan of $4,800 is made at an annual percentage rate (APR) of 12%, with repayments made monthly for 24 months. What is the total finance charge (total interest paid) over the life of the loan?

Introduction:
This problem involves an installment loan repaid in equal monthly payments. A loan of 4,800 dollars is taken at an APR of 12%, and the loan is to be repaid in 24 monthly installments. We are asked for the finance charge, that is, the total interest paid over the entire loan term.

Given Data / Assumptions:

• Loan principal, P = 4,800 dollars
• APR (nominal annual interest rate) = 12% per annum
• Compounding and payment frequency: monthly, 12 periods per year
• Number of payments, n = 24 months
• Payments are equal and made at the end of each month

Concept / Approach:
For a loan repaid by equal monthly installments, the fixed payment R is given by the standard annuity formula: R = P * i * (1 + i)^n / ((1 + i)^n - 1) where:

• i is the monthly interest rate, i = APR / 12
• n is the total number of payments

Once R is found, the total amount repaid is: Total repaid = R * n The finance charge (total interest paid) is: Finance charge = Total repaid - P

Step-by-Step Solution:
Step 1: Compute the monthly interest rate. APR = 12% per annum, so i = 12% / 12 = 1% per month = 0.01 Step 2: Use the loan payment formula. R = 4800 * 0.01 * (1.01)^24 / ((1.01)^24 - 1) Step 3: Compute (1.01)^24. (1.01)^24 ≈ 1.26824 Step 4: Substitute into the formula. R ≈ 4800 * 0.01 * 1.26824 / (1.26824 - 1) R ≈ 48 * 1.26824 / 0.26824 R ≈ 60.876 / 0.26824 ≈ 226.0 (approximately 225.95 with more precision) Step 5: Compute total amount repaid. Total repaid ≈ 225.95 * 24 ≈ 5422.80 Step 6: Compute the finance charge. Finance charge ≈ 5422.80 - 4800 = 622.80

Verification / Alternative check:
We can quickly check reasonableness: a 12% annual rate on 4,800 dollars over 2 years might be expected to produce total interest a little over 10% of the principal, since the balance is declining as it is repaid. An interest total around 600 to 650 dollars is reasonable, and 622.80 fits this expectation well.

Why Other Options Are Wrong:
522.80 and 322.80: These underestimate the finance charge and would correspond to either a lower APR or a shorter term. 632.80: This slightly overestimates the amount of interest and does not match the precise calculated value. 422.80: This is too low for a 12% APR on this size of loan over 24 months. 622.80: This matches the finance charge obtained from the correct amortization calculation.

Common Pitfalls:
Some learners mistakenly apply simple interest P * APR * (term in years) and ignore the fact that the loan balance is declining, which leads to an overestimate of interest. Others try to divide the principal by 24 and then add a flat interest amount each month, which does not correctly reflect amortization. Miscomputing the monthly rate (for example, using 12% instead of 1%) is another frequent error that can completely distort the results.

Final Answer:
The total finance charge (total interest paid) on the 4,800 dollar loan repaid over 24 months at 12% APR is approximately 622.80 dollars.