1. **State the problem:** We want to find the purchase price of a property given a down payment and a series of payments with interest compounded monthly. 2. **Given data:** - Down payment: 8656 - Payment amount: 971 every 6 months - Number of years: 10 - Interest rate: 12% per annum compounded monthly 3. **Formula used:** The purchase price is the sum of the down payment and the present value of the annuity payments. The present value of an annuity formula is: $$PV = P \times \frac{1 - (1 + i)^{-n}}{i}$$ where: - $P$ is the payment amount - $i$ is the interest rate per period - $n$ is the total number of payments 4. **Calculate the interest rate per period:** Since interest is compounded monthly but payments are every 6 months, we find the effective 6-month interest rate. Monthly interest rate: $$i_m = \frac{12\%}{12} = 0.01$$ Effective 6-month interest rate: $$i = (1 + i_m)^6 - 1 = (1 + 0.01)^6 - 1 = 1.061520 - 1 = 0.061520$$ 5. **Calculate the total number of payments:** Payments every 6 months for 10 years means: $$n = 10 \times 2 = 20$$ 6. **Calculate the present value of the annuity payments:** $$PV = 971 \times \frac{1 - (1 + 0.061520)^{-20}}{0.061520}$$ Calculate $(1 + 0.061520)^{-20}$: $$= \frac{1}{(1.061520)^{20}} = \frac{1}{3.281031} = 0.304740$$ So, $$PV = 971 \times \frac{1 - 0.304740}{0.061520} = 971 \times \frac{0.695260}{0.061520}$$ Calculate the fraction: $$\frac{0.695260}{0.061520} = 11.300000$$ Therefore, $$PV = 971 \times 11.300000 = 10972.300000$$ 7. **Calculate the purchase price:** $$\text{Purchase Price} = \text{Down payment} + PV = 8656 + 10972.3 = 19628.3$$ 8. **Calculate the cost of financing:** Cost of financing is the difference between the purchase price and the down payment: $$\text{Cost of financing} = 19628.3 - 8656 = 10972.3$$ **Final answers:** - Purchase price of the property: $19628.30$ - Cost of financing: $10972.30$