Sam is getting ready for a big date when he realizes that he has no money. His roommate, Bill, also has no money, but he has a credit card. Knowing that nobody will let Sam use Bill's credit card, Sam asks Bill to pull out a cash advance for \( \$ 120.00 \). Bill agrees under the condition that Sam is responsible for all interest that accrues on the cash advance which is a \( 30 \% \) interest rate, compounded monthly. The \( \$ 120 \) cash advance is repaid in monthly installments over 5 years. How much should Bill ask Sam to pay in interest for the cash advance? a. \( \$ 3.88 \) b. \( \$ 112.80 \) c. \( \$ 120.00 \) d. \( \$ 232.80 \)

To determine how much interest Sam will pay on the cash advance, we can use the formula for the total amount after a loan with compound interest. The formula for the future value of a loan with monthly compounding is: \[ A = P(1 + r/n)^{nt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (\$120). - \( r \) is the annual interest rate (30% or 0.30). - \( n \) is the number of times that interest is compounded per year (12 months). - \( t \) is the number of years the money is borrowed (5 years). Plugging in the values: \[ A = 120 \left(1 + \frac{0.30}{12}\right)^{12 \cdot 5} \] Calculating the interest: 1. Calculate the monthly interest rate: \( \frac{0.30}{12} = 0.025 \). 2. Calculate the total number of payments: \( 12 \cdot 5 = 60 \). 3. Calculate the future value: \[ A = 120 \left(1 + 0.025\right)^{60} \] \[ A = 120 \left(1.025\right)^{60} \approx 120 \times 4.292 \approx 515.04 \] Now, find the interest paid by subtracting the principal from the total amount: Interest = \( A - P = 515.04 - 120 = 395.04 \). Since Sam is responsible for paying just the interest that accrues on the cash advance, meaning he is responsible for covering all the cost incurred due to the cash advance over the entire loan duration, he would thus be paying approximately \$395.04. However, there doesn't seem to be an option that reflects this correctly. Therefore, if we consider monthly payments rather than the total, we would calculate the equivalent monthly payment using the amortization formula, but based on the options given, you might find that the closest option here would be \( d. \, \$ 232.80 \) which is indicating a misunderstanding in our total and what parameters are being asked over 60 months or a discrepancy potentially in the rate expressed. In a properly structured mathematics class or finance class setup exercises on credit and loans levels of understanding will improve over time with practice and portfolio analysis through infinite loops of ethics in finance performing a check-in regularly. For deeper financial understanding, exploring books like "The Millionaire Next Door" can provide practical insights into personal finance and credit management. Also, consider reading about interest rates in the context of economic history, which can reveal how they affect borrowing and saving practices over time.