What is the total amount of monthly payments for a \( \$ 6,100 \), two-year loan with an APR of \( 5 \% \) \( \$ 406.94 \) \( \$ 322.88 \) \( \$ 6,422.88 \) \( \$ 267.62 \)

To find the total amount of monthly payments for a loan, we can use the formula for the monthly payment on an installment loan, which is given by: \[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \] Where: - \( M \) is the monthly payment - \( P \) is the loan principal (amount borrowed) - \( r \) is the monthly interest rate (annual interest rate divided by 12) - \( n \) is the number of payments (loan term in months) ### Step 1: Extract known conditions - Loan amount \( P = 6100 \) - Annual interest rate \( APR = 5\% \) - Loan term \( = 2 \) years \( = 24 \) months ### Step 2: Calculate the monthly interest rate The monthly interest rate \( r \) is calculated as follows: \[ r = \frac{APR}{12} = \frac{5\%}{12} = \frac{0.05}{12} \] ### Step 3: Calculate the number of payments The total number of payments \( n \) is: \[ n = 2 \times 12 = 24 \] ### Step 4: Calculate the monthly payment \( M \) Now we can substitute the values into the formula to find \( M \): \[ M = 6100 \cdot \frac{\frac{0.05}{12}(1 + \frac{0.05}{12})^{24}}{(1 + \frac{0.05}{12})^{24} - 1} \] Let's calculate \( M \) using the above formula. Calculate the value by following steps: - step0: Calculate: \(\frac{6100\left(\frac{0.05}{12}\left(1+\frac{0.05}{12}\right)^{24}\right)}{\left(\left(1+\frac{0.05}{12}\right)^{24}-1\right)}\) - step1: Remove the parentheses: \(\frac{6100\times \frac{0.05}{12}\left(1+\frac{0.05}{12}\right)^{24}}{\left(1+\frac{0.05}{12}\right)^{24}-1}\) - step2: Divide the terms: \(\frac{6100\times \frac{0.05}{12}\left(1+\frac{1}{240}\right)^{24}}{\left(1+\frac{0.05}{12}\right)^{24}-1}\) - step3: Add the numbers: \(\frac{6100\times \frac{0.05}{12}\left(\frac{241}{240}\right)^{24}}{\left(1+\frac{0.05}{12}\right)^{24}-1}\) - step4: Divide the terms: \(\frac{6100\times \frac{0.05}{12}\left(\frac{241}{240}\right)^{24}}{\left(1+\frac{1}{240}\right)^{24}-1}\) - step5: Add the numbers: \(\frac{6100\times \frac{0.05}{12}\left(\frac{241}{240}\right)^{24}}{\left(\frac{241}{240}\right)^{24}-1}\) - step6: Divide the terms: \(\frac{6100\times \frac{1}{240}\left(\frac{241}{240}\right)^{24}}{\left(\frac{241}{240}\right)^{24}-1}\) - step7: Multiply: \(\frac{\frac{61\times 241^{24}}{12\times 5^{23}\times 48^{24}}}{\left(\frac{241}{240}\right)^{24}-1}\) - step8: Subtract the numbers: \(\frac{\frac{61\times 241^{24}}{12\times 5^{23}\times 48^{24}}}{\frac{241^{24}-240^{24}}{240^{24}}}\) - step9: Multiply by the reciprocal: \(\frac{61\times 241^{24}}{12\times 5^{23}\times 48^{24}}\times \frac{240^{24}}{241^{24}-240^{24}}\) - step10: Rewrite the expression: \(\frac{61\times 241^{24}}{12\times 5^{23}\times 48^{24}}\times \frac{12^{24}\times 20^{24}}{241^{24}-240^{24}}\) - step11: Reduce the numbers: \(\frac{61\times 241^{24}}{5^{23}\times 48^{24}}\times \frac{12^{23}\times 20^{24}}{241^{24}-240^{24}}\) - step12: Rewrite the expression: \(\frac{61\times 241^{24}}{5^{23}\times 12^{24}\times 4^{24}}\times \frac{12^{23}\times 20^{24}}{241^{24}-240^{24}}\) - step13: Reduce the numbers: \(\frac{61\times 241^{24}}{5^{23}\times 12\times 4^{24}}\times \frac{20^{24}}{241^{24}-240^{24}}\) - step14: Rewrite the expression: \(\frac{61\times 241^{24}}{5^{23}\times 12\times 4^{24}}\times \frac{5^{24}\times 4^{24}}{241^{24}-240^{24}}\) - step15: Reduce the numbers: \(\frac{61\times 241^{24}}{12\times 4^{24}}\times \frac{5\times 4^{24}}{241^{24}-240^{24}}\) - step16: Rewrite the expression: \(\frac{61\times 241^{24}}{4\times 3\times 4^{24}}\times \frac{5\times 4^{24}}{241^{24}-240^{24}}\) - step17: Reduce the numbers: \(\frac{61\times 241^{24}}{3\times 4}\times \frac{5}{241^{24}-240^{24}}\) - step18: Multiply the fractions: \(\frac{61\times 241^{24}\times 5}{12\left(241^{24}-240^{24}\right)}\) - step19: Multiply: \(\frac{305\times 241^{24}}{12\times 241^{24}-12\times 240^{24}}\) The calculated monthly payment \( M \) is approximately \( 267.62 \). ### Step 5: Calculate the total amount of monthly payments To find the total amount of monthly payments over the loan term, we multiply the monthly payment by the number of payments: \[ \text{Total Payments} = M \times n = 267.62 \times 24 \] Now, let's calculate the total amount of monthly payments. Calculate the value by following steps: - step0: Calculate: \(267.62\times 24\) - step1: Multiply the numbers: \(6422.88\) The total amount of monthly payments over the two-year loan is \( 6422.88 \). ### Summary of Results - Monthly Payment: \( 267.62 \) - Total Amount of Monthly Payments: \( 6422.88 \) Thus, the correct answer is \( \$ 6,422.88 \).