Frank plans to buy a used car that costs \( \$ 26,000 \). The dealer requires a \( 5 \% \) down payment. The rest of the cost is financed with a 3 -year, fixed-rate amortize auto loan at \( 9 \% \) annual interest with monthly payments. Complete the parts below. Do not round any intermediate computations. Round your final answers to the nearest cent if necessary. If necessary, refer to the list of financial formulas. (a) Find the required down payment. (b) Find the amount of the auto loan. \( \$ \) (c) Find the monthly payment. (

Given: - Cost of the used car = $26,000 - Down payment = 5% of the cost of the car - Interest rate = 9% per annum - Loan duration = 3 years - Monthly payments (a) Find the required down payment: The down payment is 5% of the cost of the car. To find the down payment, we can calculate 5% of $26,000. Down payment = 5% of $26,000 = 0.05 * $26,000 = $1,300 (b) Find the amount of the auto loan: The amount of the auto loan is the cost of the car minus the down payment. Amount of the auto loan = Cost of the car - Down payment Amount of the auto loan = $26,000 - $1,300 = $24,700 (c) Find the monthly payment: To find the monthly payment, we can use the formula for monthly payments on a fixed-rate amortize loan: \[ M = \frac{P \times r \times (1 + r)^n}{(1 + r)^n - 1} \] Where: - \( M \) is the monthly payment - \( P \) is the principal amount (amount of the auto loan) - \( r \) is the monthly interest rate (annual interest rate divided by 12) - \( n \) is the number of payments (loan duration in years multiplied by 12) First, we need to calculate the monthly interest rate: \[ r = \frac{9\%}{12} = \frac{0.09}{12} = 0.0075 \] Now, we can calculate the monthly payment using the formula: \[ M = \frac{24700 \times 0.0075 \times (1 + 0.0075)^{36}}{(1 + 0.0075)^{36} - 1} \] Let's calculate the monthly payment. Calculate the value by following steps: - step0: Calculate: \(\frac{24700\times 0.0075\left(1+0.0075\right)^{36}}{\left(\left(1+0.0075\right)^{36}-1\right)}\) - step1: Remove the parentheses: \(\frac{24700\times 0.0075\left(1+0.0075\right)^{36}}{\left(1+0.0075\right)^{36}-1}\) - step2: Add the numbers: \(\frac{24700\times 0.0075\times 1.0075^{36}}{\left(1+0.0075\right)^{36}-1}\) - step3: Add the numbers: \(\frac{24700\times 0.0075\times 1.0075^{36}}{1.0075^{36}-1}\) - step4: Convert the expressions: \(\frac{24700\times 0.0075\left(\frac{403}{400}\right)^{36}}{1.0075^{36}-1}\) - step5: Convert the expressions: \(\frac{24700\times 0.0075\left(\frac{403}{400}\right)^{36}}{\left(\frac{403}{400}\right)^{36}-1}\) - step6: Multiply: \(\frac{\frac{741\times 403^{36}}{4\times 400^{36}}}{\left(\frac{403}{400}\right)^{36}-1}\) - step7: Subtract the numbers: \(\frac{\frac{741\times 403^{36}}{4\times 400^{36}}}{\frac{403^{36}-400^{36}}{400^{36}}}\) - step8: Multiply by the reciprocal: \(\frac{741\times 403^{36}}{4\times 400^{36}}\times \frac{400^{36}}{403^{36}-400^{36}}\) - step9: Rewrite the expression: \(\frac{741\times 403^{36}}{4\times 400^{36}}\times \frac{16^{36}\times 25^{36}}{403^{36}-400^{36}}\) - step10: Rewrite the expression: \(\frac{741\times 403^{36}}{4\times 400^{36}}\times \frac{4^{72}\times 25^{36}}{403^{36}-400^{36}}\) - step11: Reduce the numbers: \(\frac{741\times 403^{36}}{400^{36}}\times \frac{4^{71}\times 25^{36}}{403^{36}-400^{36}}\) - step12: Rewrite the expression: \(\frac{741\times 403^{36}}{16^{36}\times 25^{36}}\times \frac{4^{71}\times 25^{36}}{403^{36}-400^{36}}\) - step13: Rewrite the expression: \(\frac{741\times 403^{36}}{4^{72}\times 25^{36}}\times \frac{4^{71}\times 25^{36}}{403^{36}-400^{36}}\) - step14: Reduce the numbers: \(\frac{741\times 403^{36}}{4}\times \frac{1}{403^{36}-400^{36}}\) - step15: Multiply the fractions: \(\frac{741\times 403^{36}}{4\left(403^{36}-400^{36}\right)}\) - step16: Multiply: \(\frac{741\times 403^{36}}{4\times 403^{36}-4\times 400^{36}}\) The monthly payment is approximately $785.45.