Aldo is taking out a mortgage for \( \$ 246,000 \) to buy a new house and is deciding between the offers from two lenders. He wants to know which one would be the better deal over the life of th mortgage 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) His credit union has offered him a 15 -year mortgage loan at an annual interest rate of \( 4.8 \% \). Find the monthly payment. \( \$ \square \) (b) An online lending company has offered him a \( 30- \) year mortgage loan at an annual interest rate of \( 4.8 \% \). Find the monthly payment. \( \$ \square \) (c) \( \begin{array}{l}\text { Suppose Aldo pays the monthly payment each } \\ \text { month for the full term. Which lender's mortgage } \\ \text { loan would have the lowest total amount to pay off, } \\ \text { and by how much? } \\ \text { o credit union } \\ \text { The total amount paid would be } \$ \square \text { less than to the online lending company. } \\ \text { online lending company } \\ \text { the total amount paid would be } \$ \square \text { less than to the credit union. }\end{array} \)

To solve the problem, we will use the formula for the monthly payment on a fixed-rate mortgage, 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 (the amount borrowed), - \( r \) is the monthly interest rate (annual interest rate divided by 12), - \( n \) is the number of payments (loan term in months). ### (a) Monthly Payment for the 15-Year Mortgage 1. **Loan Amount (P)**: \( 246,000 \) 2. **Annual Interest Rate**: \( 4.8\% \) or \( 0.048 \) 3. **Monthly Interest Rate (r)**: \[ r = \frac{0.048}{12} = 0.004 \] 4. **Loan Term (n)**: \( 15 \) years \( = 15 \times 12 = 180 \) months Now, substituting these values into the formula: \[ M = 246000 \frac{0.004(1 + 0.004)^{180}}{(1 + 0.004)^{180} - 1} \] Calculating \( (1 + 0.004)^{180} \): \[ (1 + 0.004)^{180} \approx 1.8983 \] Now substituting back into the formula: \[ M = 246000 \frac{0.004 \times 1.8983}{1.8983 - 1} \] \[ M = 246000 \frac{0.0075932}{0.8983} \approx 246000 \times 0.008448 \approx 2087.01 \] Thus, the monthly payment for the 15-year mortgage is: \[ \boxed{2087.01} \] ### (b) Monthly Payment for the 30-Year Mortgage 1. **Loan Amount (P)**: \( 246,000 \) 2. **Annual Interest Rate**: \( 4.8\% \) or \( 0.048 \) 3. **Monthly Interest Rate (r)**: \[ r = \frac{0.048}{12} = 0.004 \] 4. **Loan Term (n)**: \( 30 \) years \( = 30 \times 12 = 360 \) months Now, substituting these values into the formula: \[ M = 246000 \frac{0.004(1 + 0.004)^{360}}{(1 + 0.004)^{360} - 1} \] Calculating \( (1 + 0.004)^{360} \): \[ (1 + 0.004)^{360} \approx 3.8495 \] Now substituting back into the formula: \[ M = 246000 \frac{0.004 \times 3.8495}{3.8495 - 1} \] \[ M = 246000 \frac{0.015398}{2.8495} \approx 246000 \times 0.005396 \approx 1327.00 \] Thus, the monthly payment for the 30-year mortgage is: \[ \boxed{1327.00} \] ### (c) Total Amount Paid Over the Life of Each Loan 1. **Total Amount Paid for 15-Year Mortgage**: \[ \text{Total} = M \times n = 2087.01 \times 180 \approx 375660.18 \] 2. **Total Amount Paid for 30-Year Mortgage**: \[ \text{Total} = M \times n = 1327.00 \times 360 \approx 478920.00 \] Now, we can compare the total amounts: - The total amount paid for the 15-year mortgage is \( 375660.18 \). - The total amount paid for the 30-year mortgage is \( 478920.00 \). To find the difference: \[ \text{Difference} = 478920.00 - 375660.18 \approx 103259.82 \] Thus, the 15-year mortgage from the credit union would have the lowest total amount to pay off, and it would be: \[ \text{The total amount paid would be } \$ 103259.82 \text{ less than to the online lending company.} \] Final answers: - Credit Union: \( \boxed{375660.18} \) - Online Lending Company: \( \boxed{478920.00} \) - Difference: \( \boxed{103259.82} \)