1. **Problem statement:** A $300,000 house requires a 25% down payment, and the rest is financed with monthly payments of $1,500 at an interest rate of 4.15% compounded semi-annually. We want to find: a) How long it will take to pay off the mortgage. b) How much interest will be paid over the entire term. 2. **Given data:** - House price = 300000 - Down payment = 25% of 300000 = $75000 - Loan amount (PV) = 300000 - 75000 = $225000 - Monthly payment (PMT) = 1500 - Annual interest rate (nominal) = 4.15% - Compounding semi-annually means interest is compounded twice a year. 3. **Convert interest rate to effective monthly rate:** Since interest is compounded semi-annually, the effective semi-annual rate is $\frac{4.15}{100} = 0.0415$. The effective monthly interest rate $i$ is given by: $$i = \left(1 + 0.0415\right)^{\frac{1}{6}} - 1$$ Calculate: $$i = (1.0415)^{\frac{1}{6}} - 1 \approx 0.0068$$ So, $i \approx 0.0068$ or 0.68% per month. 4. **Use the amortization formula to find number of months $n$:** The formula for the present value of an annuity is: $$PV = PMT \times \frac{1 - (1 + i)^{-n}}{i}$$ Rearranged to solve for $n$: $$1 - (1 + i)^{-n} = \frac{PV \times i}{PMT}$$ $$ (1 + i)^{-n} = 1 - \frac{PV \times i}{PMT}$$ $$ -n \ln(1 + i) = \ln\left(1 - \frac{PV \times i}{PMT}\right)$$ $$ n = -\frac{\ln\left(1 - \frac{PV \times i}{PMT}\right)}{\ln(1 + i)}$$ Substitute values: $$n = -\frac{\ln\left(1 - \frac{225000 \times 0.0068}{1500}\right)}{\ln(1.0068)}$$ Calculate numerator inside log: $$\frac{225000 \times 0.0068}{1500} = \frac{1530}{1500} = 1.02$$ Since $1 - 1.02 = -0.02$ is negative, this means the payment is too low to pay off the loan at this interest rate (the loan will never be paid off). **Therefore, the monthly payment of 1500 is insufficient to pay off the mortgage at 4.15% compounded semi-annually.** 5. **Conclusion for part (a):** The loan cannot be paid off with $1500 monthly payments at the given interest rate. 6. **Part (b) interest paid:** Since the loan is never paid off, total interest paid is infinite. 7. **Summary:** - The monthly payment is too low to amortize the loan. - No finite payoff time exists. - Interest paid over the term is unbounded. **Final answers:** - a) No finite payoff time; loan will not be paid off. - b) Interest paid is unlimited. --- **Note:** Since the first problem cannot be solved as stated, the other problems are ignored per instructions.