Question

Consider the following two loans for $P_{0}=\$ 3,000$.
- Loan A: 4 year loan, monthly repayments, annual interest rate of $4 \%$.
- Loan B: 2 year loan, monthly repayments, annual interest rate of $8 \%$.

On which loan will you pay the least interest?

Select the correct answer below:
Loan B
Loan A

Step 1 ：Calculate the monthly payment for Loan A using the formula \(P = \frac{{r*P0*(1 + r)^n}}{{(1 + r)^n - 1}}\) where \(r = 0.003333\), \(P0 = 3000\), and \(n = 48\).

Step 2 ：The monthly payment for Loan A is \(P_A = \frac{{0.003333*3000*(1 + 0.003333)^{48}}}{{(1 + 0.003333)^{48} - 1}} = $68.41\) (rounded to the nearest cent).

Step 3 ：Calculate the total amount paid for Loan A by multiplying the monthly payment by the number of payments: \(TotalPaid_A = P_A * n = $68.41 * 48 = $3283.68\).

Step 4 ：Calculate the total interest paid for Loan A by subtracting the principal amount from the total amount paid: \(Interest_A = TotalPaid_A - P0 = $3283.68 - $3000 = $283.68\).

Step 5 ：Calculate the monthly payment for Loan B using the formula \(P = \frac{{r*P0*(1 + r)^n}}{{(1 + r)^n - 1}}\) where \(r = 0.006667\), \(P0 = 3000\), and \(n = 24\).

Step 6 ：The monthly payment for Loan B is \(P_B = \frac{{0.006667*3000*(1 + 0.006667)^{24}}}{{(1 + 0.006667)^{24} - 1}} = $130.24\) (rounded to the nearest cent).

Step 7 ：Calculate the total amount paid for Loan B by multiplying the monthly payment by the number of payments: \(TotalPaid_B = P_B * n = $130.24 * 24 = $3125.76\).

Step 8 ：Calculate the total interest paid for Loan B by subtracting the principal amount from the total amount paid: \(Interest_B = TotalPaid_B - P0 = $3125.76 - $3000 = $125.76\).

Step 9 ：Compare the total interest paid on each loan: \(Interest_A = $283.68\) and \(Interest_B = $125.76\).

Step 10 ：\(\boxed{\text{The correct answer is Loan B because you will pay less interest on Loan B.}}\)