Questions: Consumer Mathematics Comparing monthly payments and total costs of two loans Charmaine is taking out an amortized loan for 74,000 to open a small business and is deciding between the offers from two lenders. She wants to know which one would be the better deal over the life of the small business 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 her a 8 -year small business loan at an annual interest rate of 10.5%. Find the monthly payment. (b) A savings and loan association has offered her a 9-year small business loan at an annual interest rate of 10.5%. Find the monthly payment. (c) Suppose Charmaine pays the monthly payment each month for the full term. Which lender's small business loan would have the lowest total amount to pay off, and by how much? Bank The total amount paid would be less than to the savings and loan association. Savings and loan association The total amount paid would be less than to the bank.

Solution Steps

Using the amortization formula, we calculate the monthly payment \( M_1 \) for the 8-year loan with an annual interest rate of \( 10.5\% \): \[ M_1 = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1} \] where:

• \( P = 74000 \)
• \( r = \frac{0.105}{12} \)
• \( n = 8 \times 12 = 96 \)

Substituting the values, we find: \[ M_1 \approx 1142.56 \]

Next, we calculate the monthly payment \( M_2 \) for the 9-year loan, also at an annual interest rate of \( 10.5\% \): \[ M_2 = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1} \] where:

• \( P = 74000 \)
• \( r = \frac{0.105}{12} \)
• \( n = 9 \times 12 = 108 \)

Substituting the values, we find: \[ M_2 \approx 1061.96 \]

Now, we calculate the total amount paid for each loan over its term. For the 8-year loan: \[ \text{Total Paid}_1 = M_1 \times n_1 = 1142.56 \times 96 \approx 109685.88 \]

For the 9-year loan: \[ \text{Total Paid}_2 = M_2 \times n_2 = 1061.96 \times 108 \approx 114692.08 \]

Finally, we compare the total amounts paid for both loans:

• Total amount paid for the 8-year loan: \( \text{Total Paid}_1 \approx 109685.88 \)
• Total amount paid for the 9-year loan: \( \text{Total Paid}_2 \approx 114692.08 \)

To determine which loan is the better deal, we calculate the difference: \[ \text{Difference} = \text{Total Paid}_2 - \text{Total Paid}_1 \approx 114692.08 - 109685.88 \approx 5006.20 \]

Thus, the bank's loan is the better deal, with a total amount paid that is approximately \( 5006.20 \) less than that of the savings and loan association.

Final Answer