To calculate the required values for a loan of $P = 80,000$, annual interest rate $r = 6\%$, compounded monthly over $t = 3$ years, the following formulas and values will be used:

---

1. Number of Payments in a Year

Since the loan is compounded monthly: $n = 12 \text{ payments per year.}$

---

2. Monthly Payment

The monthly payment is calculated using the formula: $M = \frac{P \cdot r_{\text{monthly}} \cdot (1 + r_{\text{monthly}})^{n \cdot t}}{(1 + r_{\text{monthly}})^{n \cdot t} - 1}$ where:

• $r_{\text{monthly}} = \frac{r}{n} = \frac{0.06}{12} = 0.005$.

Substituting: $M = \frac{80,000 \cdot 0.005 \cdot (1 + 0.005)^{12 \cdot 3}}{(1 + 0.005)^{12 \cdot 3} - 1}$

---

3. Total Amount Paid After 3 Years

The total payments made: $\text{Total Paid} = M \cdot n \cdot t$

---

4. Total Interest Paid in 3 Years

The total interest paid: $\text{Interest Paid} = \text{Total Paid} - P$

Let me compute these values.# Variables P = 80000 # Principal amount annual_rate = 0.06 # Annual interest rate n = 12 # Number of payments per year t = 3 # Time in years

Monthly interest rate

r_monthly = annual_rate / n

Total number of payments

total_payments = n * t

Monthly payment calculation using the formula

M = P * r_monthly * (1 + r_monthly)**total_payments / ((1 + r_monthly)**total_payments - 1)

Total amount paid

total_paid = M * total_payments

Total interest paid

interest_paid = total_paid - P

Results

M, total_paid, interest_paid(2433.754996124457, 87615.17986048046, 7615.179860480464)(2433.754996124457, 87615.17986048046, 7615.179860480464)success### Results:

1. Number of Payments in a Year: $12$ (monthly payments).
2. Monthly Payment: $2,433.75$ (approximately).
3. Total Amount Paid After 3 Years: $87,615.18$ (approximately).
4. Total Interest Paid in 3 Years: $7,615.18$ (approximately).

---

Would you like a detailed explanation of how these values were calculated, or do you have any other questions?

5 Related Questions:

1. What would be the monthly payment if the interest rate was 5% instead of 6%?
2. How does the total interest paid change if the loan term is extended to 5 years?
3. What happens to the monthly payment if the loan principal is reduced to $70,000$?
4. Can you show the amortization schedule for the first 12 months?
5. How would switching to semi-annual compounding affect the results?

Tip:

When calculating interest payments over time, always double-check the compounding frequency, as it significantly impacts the total interest paid.