Let's solve **Part d** step by step, given the previous information and correct answers from earlier parts. --- ## **Step-by-Step Solution** ### **Given Data** - Principal (PV): $28,500 - Annual interest rate: 15% compounded monthly - Number of monthly payments (n): 48 - Monthly payment (PMT): $793.18 (rounded from 793.176) - Time to payoff: 3.5 years (3 years and 6 months) - Payments made: 3.5 years × 12 months/year = 42 payments - Payments remaining: 48 - 42 = **6 payments** You have **not** made the 42nd payment yet, so the remaining balance should be calculated after **41 payments**. ### **Step 1: Find Monthly Interest Rate** \[ i = \frac{15\%}{12} = 1.25\% = .0125 \] ### **Step 2: Find Remaining Balance After 41 Payments** The remaining balance after \( n \) payments is given by: \[ \text{Remaining Balance after } n \text{ payments} = PMT \times \frac{1 - (1 + i)^{-(N-n)}}{i} \] Where: - \( N = 48 \) (total payments) - \( n = 41 \) (payments made) - \( PMT = 793.18 \) - \( i = .0125 \) - \( N-n = 48-41 = 7 \) (payments remaining **including** the 42nd payment) However, since you have **not made** the 42nd payment yet, you'll owe the present value of the remaining 7 payments. ### **Step 3: Calculate Payoff Amount** \[ \text{Payoff Amount} = PMT \times \frac{1 - (1 + i)^{-7}}{i} \] Plug in the values: \[ = 793.18 \times \frac{1 - (1 + .0125)^{-7}}{.0125} \] First, calculate \( (1 + .0125)^{-7} \): \[ (1 + .0125)^{-7} = (1.0125)^{-7} \approx .9139 \] Now, continue: \[ \frac{1 - .9139}{.0125} = \frac{.0861}{.0125} \approx 6.888 \] Multiply by the monthly payment: \[ 793.18 \times 6.888 \approx 5464.47 \] --- ## **Final Answer** ### **The payoff amount for the loan after 3.5 years (41 payments) is:** \[ \boxed{5464.47} \] **(Rounded to two decimal places, and within the tolerance specified.)** --- ## **Summary Table** | Step | Calculation | Value | |-----------------|-------------------------------------------------------------------------------|----------| | Monthly rate | \( .15 / 12 \) | .0125 | | Payments made | \( 3.5 \times 12 \) | 42 | | Payments left | \( 48 - 41 \) (since the 42nd not paid yet) | 7 | | Payoff amount | \( 793.18 \times \frac{1 - (1.0125)^{-7}}{.0125} \approx 793.18 \times 6.888 \) | 5464.47 | --- **If you have any more questions or need more explanation, let me know!**