1. **Problem statement:** A $300,000 house is purchased with 25% down payment and monthly payments of 1500. The mortgage interest rate is 4.15% compounded semi-annually. We need to find how long it will take to pay off the mortgage. 2. **Calculate the loan amount (PV):** Down payment = 25% of 300000 = 0.25 \times 300000 = 75000 Loan amount (PV) = 300000 - 75000 = 225000 3. **Convert interest rate to monthly effective rate:** Given nominal annual rate compounded semi-annually: 4.15% per year compounded semi-annually means semi-annual rate = 4.15%/2 = 2.075% Monthly interest rate $i$ is found by: $$i = \left(1 + 0.02075\right)^{\frac{1}{6}} - 1 = 0.003417 \text{ (approx)}$$ 4. **Set up the annuity formula for monthly payments:** The formula for present value of an ordinary annuity is: $$PV = PMT \times \frac{1 - (1 + i)^{-n}}{i}$$ Where: - $PV = 225000$ - $PMT = 1500$ - $i = 0.003417$ - $n$ = number of months (unknown) 5. **Solve for $n$:** Rearranged: $$\frac{PV \times i}{PMT} = 1 - (1 + i)^{-n}$$ $$ (1 + i)^{-n} = 1 - \frac{PV \times i}{PMT}$$ Calculate: $$1 - \frac{225000 \times 0.003417}{1500} = 1 - \frac{768.825}{1500} = 1 - 0.51255 = 0.48745$$ Take natural logarithm: $$-n \ln(1 + i) = \ln(0.48745)$$ $$n = - \frac{\ln(0.48745)}{\ln(1.003417)}$$ Calculate: $$\ln(0.48745) = -0.7183, \quad \ln(1.003417) = 0.00341$$ $$n = - \frac{-0.7183}{0.00341} = 210.6 \text{ months}$$ 6. **Convert months to years and months:** $$210.6 \text{ months} = 17 \text{ years and } 6.6 \text{ months} \approx 17 \text{ years and } 7 \text{ months}$$ 7. **Calculate total interest paid:** Total payments = $1500 \times 210.6 = 315900$ Total interest = Total payments - Loan amount = $315900 - 225000 = 90900$ 8. **Financial calculator entries for part (a):** MODE: END N: 210.6 I/Y: 4.15/2 = 2.075% semi-annual rate converted to monthly effective rate 0.3417% PV: -225000 PMT: 1500 FV: 0 P/Y: 12 C/Y: 2 9. **Financial calculator entries for part (b):** Interest paid = 90900 Final answers: - Time to pay off mortgage: **17 years and 7 months** - Total interest paid: **90900**