Omar is taking out an amortized loan for \( \$ 22,000 \) to buy a new car and is deciding between the offers from two lenders. He wants to know which one would be the better deal over the life of the car loan, and by how much. Answer each part. Do not round intermediate computations, and round your answers to the nearest cent. If necessary, refer to the list of financial formulas. (a) A bank has offered him a 6-year car loan at an annual interest rate of \( 7.9 \% \). Find the monthly payment. (b) A savings and loan association has offered him a 6-year car loan at an annual interest rate of \( 8.1 \% \). Find the monthly payment. (c) Suppose Omar pays the monthly payment each month for the full term. Which lender's car loan would have the lowest total amount to pay off, and by how much? Check 5 Sabe For Later

To find the monthly payment for each loan, we can use the formula for monthly payments on a 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 (initial loan amount) - \( r \) is the monthly interest rate - \( n \) is the number of payments (months) Given: - Principal amount (\( P \)) = $22,000 - Annual interest rate for bank = 7.9% - Annual interest rate for savings and loan association = 8.1% - Loan term = 6 years = 72 months (a) For the bank's loan: - Annual interest rate = 7.9% - Monthly interest rate = \( \frac{7.9\%}{12} = 0.006583 \) Using the formula, we can calculate the monthly payment for the bank's loan. (b) For the savings and loan association's loan: - Annual interest rate = 8.1% - Monthly interest rate = \( \frac{8.1\%}{12} = 0.00675 \) Using the formula, we can calculate the monthly payment for the savings and loan association's loan. (c) To find which lender's car loan would have the lowest total amount to pay off, we need to calculate the total amount paid for each loan by multiplying the monthly payment by the number of payments and then comparing the results. Let's calculate the monthly payments and the total amount to pay off for each loan. Calculate the value by following steps: - step0: Calculate: \(\frac{22000\times 0.006583\left(1+0.006583\right)^{72}}{\left(1+0.006583\right)^{72}-1}\) - step1: Add the numbers: \(\frac{22000\times 0.006583\times 1.006583^{72}}{\left(1+0.006583\right)^{72}-1}\) - step2: Add the numbers: \(\frac{22000\times 0.006583\times 1.006583^{72}}{1.006583^{72}-1}\) - step3: Convert the expressions: \(\frac{22000\times 0.006583\left(\frac{1006583}{1000000}\right)^{72}}{1.006583^{72}-1}\) - step4: Convert the expressions: \(\frac{22000\times 0.006583\left(\frac{1006583}{1000000}\right)^{72}}{\left(\frac{1006583}{1000000}\right)^{72}-1}\) - step5: Multiply: \(\frac{\frac{72413\times 1006583^{72}}{500\times 1000000^{72}}}{\left(\frac{1006583}{1000000}\right)^{72}-1}\) - step6: Subtract the numbers: \(\frac{\frac{72413\times 1006583^{72}}{500\times 1000000^{72}}}{\frac{1006583^{72}-1000000^{72}}{1000000^{72}}}\) - step7: Multiply by the reciprocal: \(\frac{72413\times 1006583^{72}}{500\times 1000000^{72}}\times \frac{1000000^{72}}{1006583^{72}-1000000^{72}}\) - step8: Rewrite the expression: \(\frac{72413\times 1006583^{72}}{500\times 1000000^{72}}\times \frac{250000^{72}\times 4^{72}}{1006583^{72}-1000000^{72}}\) - step9: Rewrite the expression: \(\frac{72413\times 1006583^{72}}{500\times 1000000^{72}}\times \frac{500^{144}\times 4^{72}}{1006583^{72}-1000000^{72}}\) - step10: Reduce the numbers: \(\frac{72413\times 1006583^{72}}{1000000^{72}}\times \frac{500^{143}\times 4^{72}}{1006583^{72}-1000000^{72}}\) - step11: Rewrite the expression: \(\frac{72413\times 1006583^{72}}{250000^{72}\times 4^{72}}\times \frac{500^{143}\times 4^{72}}{1006583^{72}-1000000^{72}}\) - step12: Rewrite the expression: \(\frac{72413\times 1006583^{72}}{500^{144}\times 4^{72}}\times \frac{500^{143}\times 4^{72}}{1006583^{72}-1000000^{72}}\) - step13: Reduce the numbers: \(\frac{72413\times 1006583^{72}}{500\times 4^{72}}\times \frac{4^{72}}{1006583^{72}-1000000^{72}}\) - step14: Rewrite the expression: \(\frac{72413\times 1006583^{72}}{4\times 125\times 4^{72}}\times \frac{4^{72}}{1006583^{72}-1000000^{72}}\) - step15: Reduce the numbers: \(\frac{72413\times 1006583^{72}}{125\times 4}\times \frac{1}{1006583^{72}-1000000^{72}}\) - step16: Multiply the fractions: \(\frac{72413\times 1006583^{72}}{500\left(1006583^{72}-1000000^{72}\right)}\) - step17: Multiply: \(\frac{72413\times 1006583^{72}}{500\times 1006583^{72}-500\times 1000000^{72}}\) Calculate or simplify the expression \( \\frac{22000*0.00675*(1+0.00675)^72}{(1+0.00675)^72-1} \). Calculate the value by following steps: - step0: Calculate: \(\frac{22000\times 0.00675\left(1+0.00675\right)^{72}}{\left(1+0.00675\right)^{72}-1}\) - step1: Add the numbers: \(\frac{22000\times 0.00675\times 1.00675^{72}}{\left(1+0.00675\right)^{72}-1}\) - step2: Add the numbers: \(\frac{22000\times 0.00675\times 1.00675^{72}}{1.00675^{72}-1}\) - step3: Convert the expressions: \(\frac{22000\times 0.00675\left(\frac{4027}{4000}\right)^{72}}{1.00675^{72}-1}\) - step4: Convert the expressions: \(\frac{22000\times 0.00675\left(\frac{4027}{4000}\right)^{72}}{\left(\frac{4027}{4000}\right)^{72}-1}\) - step5: Multiply: \(\frac{\frac{297\times 4027^{72}}{2\times 4000^{72}}}{\left(\frac{4027}{4000}\right)^{72}-1}\) - step6: Subtract the numbers: \(\frac{\frac{297\times 4027^{72}}{2\times 4000^{72}}}{\frac{4027^{72}-4000^{72}}{4000^{72}}}\) - step7: Multiply by the reciprocal: \(\frac{297\times 4027^{72}}{2\times 4000^{72}}\times \frac{4000^{72}}{4027^{72}-4000^{72}}\) - step8: Rewrite the expression: \(\frac{297\times 4027^{72}}{2\times 4000^{72}}\times \frac{32^{72}\times 125^{72}}{4027^{72}-4000^{72}}\) - step9: Rewrite the expression: \(\frac{297\times 4027^{72}}{2\times 4000^{72}}\times \frac{2^{360}\times 125^{72}}{4027^{72}-4000^{72}}\) - step10: Reduce the numbers: \(\frac{297\times 4027^{72}}{4000^{72}}\times \frac{2^{359}\times 125^{72}}{4027^{72}-4000^{72}}\) - step11: Rewrite the expression: \(\frac{297\times 4027^{72}}{32^{72}\times 125^{72}}\times \frac{2^{359}\times 125^{72}}{4027^{72}-4000^{72}}\) - step12: Rewrite the expression: \(\frac{297\times 4027^{72}}{2^{360}\times 125^{72}}\times \frac{2^{359}\times 125^{72}}{4027^{72}-4000^{72}}\) - step13: Reduce the numbers: \(\frac{297\times 4027^{72}}{2}\times \frac{1}{4027^{72}-4000^{72}}\) - step14: Multiply the fractions: \(\frac{297\times 4027^{72}}{2\left(4027^{72}-4000^{72}\right)}\) - step15: Multiply: \(\frac{297\times 4027^{72}}{2\times 4027^{72}-2\times 4000^{72}}\) The monthly payment for the bank's loan is approximately $384.65, and the monthly payment for the savings and loan association's loan is approximately $386.81. (c) To find which lender's car loan would have the lowest total amount to pay off, we need to calculate the total amount paid for each loan by multiplying the monthly payment by the number of payments and then comparing the results. For the bank's loan: Total amount = $384.65 * 72 ≈ $27,674.20 For the savings and loan association's loan: Total amount = $386.81 * 72 ≈ $27,799.52 Therefore, the bank's loan would have the lowest total amount to pay off, with a difference of approximately $125.32.