John borrowed money from a bank to invest in antiques. He took out a personal, amortized loan for \( \$ 27,000 \), at an interest rate of \( 7.95 \% \), with monthly payments for a term of 1 ye For each part, do not round any intermediate computations and round your final answers to the nearest cent. If necessary, refer to the list of financial formulas. (a) Find John's monthly payment. (b) If John pays the monthly payment each month for the full term, find his total amount to repay the loan. \( \$ \square \) (c) If John pays the monthly payment each month for the full term, find the total amount of interest he will pay. s

To solve this problem, we will use the formula for the monthly payment on an amortized loan, which is given by: \[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \] where: - \( M \) = monthly payment - \( P \) = principal amount (the loan amount) - \( r \) = monthly interest rate (annual interest rate divided by 12) - \( n \) = total number of payments (loan term in months) ### Given Data: - \( P = 27000 \) - Annual interest rate = \( 7.95\% \) - Loan term = 1 year = 12 months ### Step 1: Calculate the monthly interest rate The monthly interest rate \( r \) can be calculated as follows: \[ r = \frac{7.95\%}{12} = \frac{0.0795}{12} \] ### Step 2: Calculate the total number of payments The total number of payments \( n \) is: \[ n = 12 \] ### Step 3: Calculate the monthly payment \( M \) Now we can substitute the values into the monthly payment formula. Let's calculate \( r \) and then \( M \). 1. Calculate \( r \): \[ r = \frac{0.0795}{12} \] 2. Substitute \( P \), \( r \), and \( n \) into the formula for \( M \): \[ M = 27000 \frac{r(1 + r)^{12}}{(1 + r)^{12} - 1} \] Now, I will perform these calculations. Calculate the value by following steps: - step0: Calculate: \(\frac{0.0795}{12}\) - step1: Convert the expressions: \(\frac{\frac{159}{2000}}{12}\) - step2: Multiply by the reciprocal: \(\frac{159}{2000}\times \frac{1}{12}\) - step3: Reduce the numbers: \(\frac{53}{2000}\times \frac{1}{4}\) - step4: Multiply the fractions: \(\frac{53}{2000\times 4}\) - step5: Multiply: \(\frac{53}{8000}\) Calculate or simplify the expression \( 27000*(0.0795/12)*(1+(0.0795/12))^12/((1+(0.0795/12))^12-1) \). Calculate the value by following steps: - step0: Calculate: \(\frac{27000\left(\frac{0.0795}{12}\right)\left(1+\left(\frac{0.0795}{12}\right)\right)^{12}}{\left(\left(1+\left(\frac{0.0795}{12}\right)\right)^{12}-1\right)}\) - step1: Remove the parentheses: \(\frac{27000\left(\frac{0.0795}{12}\right)\left(1+\left(\frac{0.0795}{12}\right)\right)^{12}}{\left(1+\left(\frac{0.0795}{12}\right)\right)^{12}-1}\) - step2: Divide the terms: \(\frac{27000\left(\frac{0.0795}{12}\right)\left(1+\frac{53}{8000}\right)^{12}}{\left(1+\left(\frac{0.0795}{12}\right)\right)^{12}-1}\) - step3: Add the numbers: \(\frac{27000\left(\frac{0.0795}{12}\right)\left(\frac{8053}{8000}\right)^{12}}{\left(1+\left(\frac{0.0795}{12}\right)\right)^{12}-1}\) - step4: Divide the terms: \(\frac{27000\left(\frac{0.0795}{12}\right)\left(\frac{8053}{8000}\right)^{12}}{\left(1+\frac{53}{8000}\right)^{12}-1}\) - step5: Add the numbers: \(\frac{27000\left(\frac{0.0795}{12}\right)\left(\frac{8053}{8000}\right)^{12}}{\left(\frac{8053}{8000}\right)^{12}-1}\) - step6: Divide the terms: \(\frac{27000\times \frac{53}{8000}\left(\frac{8053}{8000}\right)^{12}}{\left(\frac{8053}{8000}\right)^{12}-1}\) - step7: Multiply: \(\frac{\frac{1431\times 8053^{12}}{8\times 8000^{12}}}{\left(\frac{8053}{8000}\right)^{12}-1}\) - step8: Subtract the numbers: \(\frac{\frac{1431\times 8053^{12}}{8\times 8000^{12}}}{\frac{8053^{12}-8000^{12}}{8000^{12}}}\) - step9: Multiply by the reciprocal: \(\frac{1431\times 8053^{12}}{8\times 8000^{12}}\times \frac{8000^{12}}{8053^{12}-8000^{12}}\) - step10: Rewrite the expression: \(\frac{1431\times 8053^{12}}{8\times 8000^{12}}\times \frac{64^{12}\times 125^{12}}{8053^{12}-8000^{12}}\) - step11: Rewrite the expression: \(\frac{1431\times 8053^{12}}{8\times 8000^{12}}\times \frac{8^{24}\times 125^{12}}{8053^{12}-8000^{12}}\) - step12: Reduce the numbers: \(\frac{1431\times 8053^{12}}{8000^{12}}\times \frac{8^{23}\times 125^{12}}{8053^{12}-8000^{12}}\) - step13: Rewrite the expression: \(\frac{1431\times 8053^{12}}{64^{12}\times 125^{12}}\times \frac{8^{23}\times 125^{12}}{8053^{12}-8000^{12}}\) - step14: Rewrite the expression: \(\frac{1431\times 8053^{12}}{8^{24}\times 125^{12}}\times \frac{8^{23}\times 125^{12}}{8053^{12}-8000^{12}}\) - step15: Reduce the numbers: \(\frac{1431\times 8053^{12}}{8}\times \frac{1}{8053^{12}-8000^{12}}\) - step16: Multiply the fractions: \(\frac{1431\times 8053^{12}}{8\left(8053^{12}-8000^{12}\right)}\) - step17: Multiply: \(\frac{1431\times 8053^{12}}{8\times 8053^{12}-8\times 8000^{12}}\) ### Step 1: Calculate the monthly interest rate \( r \) The monthly interest rate is: \[ r = \frac{0.0795}{12} \approx 0.006625 \] ### Step 2: Calculate the monthly payment \( M \) Now substituting \( P = 27000 \), \( r \approx 0.006625 \), and \( n = 12 \) into the formula for \( M \): \[ M = 27000 \frac{0.006625(1 + 0.006625)^{12}}{(1 + 0.006625)^{12} - 1} \] Calculating this gives: \[ M \approx 2348.063438 \] Rounding to the nearest cent, John's monthly payment is: \[ \text{(a) } M \approx \$2348.06 \] ### Step 3: Calculate the total amount to repay the loan The total amount to repay the loan is given by: \[ \text{Total Repayment} = M \times n \] Substituting the values: \[ \text{Total Repayment} = 2348.063438 \times 12 \] Calculating this gives: Calculate the value by following steps: - step0: Calculate: \(2348.063438\times 12\) - step1: Multiply the numbers: \(28176.761256\) Calculating the total repayment: \[ \text{Total Repayment} \approx 2348.063438 \times 12 \approx 28176.761256 \] Rounding to the nearest cent, the total amount John will repay is: \[ \text{(b) Total Amount to Repay} \approx \$28176.76 \] ### Step 4: Calculate the total amount of interest paid The total interest paid can be calculated as: \[ \text{Total Interest} = \text{Total Repayment} - P \] Substituting the values: \[ \text{Total Interest} = 28176.761256 - 27000 \] Calculating this gives: Calculate the value by following steps: - step0: Calculate: \(28176.761256-27000\) - step1: Subtract the numbers: \(1176.761256\) Calculating the total interest paid: \[ \text{Total Interest} \approx 28176.761256 - 27000 \approx 1176.761256 \] Rounding to the nearest cent, the total amount of interest John will pay is: \[ \text{(c) Total Interest Paid} \approx \$1176.76 \] ### Summary of Answers: - (a) Monthly Payment: \(\$2348.06\) - (b) Total Amount to Repay: \(\$28176.76\) - (c) Total Interest Paid: \(\$1176.76\)