1. **Problem Statement:** Shayn purchases a MacBook with installment payments of 4443.45 every quarter for 3 years, with an interest rate of 6.5% compounded semi-annually. We need to find the cash price (present value) of the MacBook. 2. **Formula Used:** The present value of an annuity formula is $$PV = P \times \frac{1 - (1 + i)^{-n}}{i}$$ where $P$ is the payment per period, $i$ is the interest rate per period, and $n$ is the total number of payments. 3. **Important Rules:** - Interest is compounded semi-annually, but payments are quarterly, so we must find the effective quarterly interest rate. - The nominal annual interest rate is 6.5%, compounded semi-annually means 2 compounding periods per year. 4. **Calculate the effective quarterly interest rate $i_2$:** Given nominal annual rate $r = 0.065$, compounded semi-annually (2 times a year), the semi-annual rate is $$i_{semi} = \frac{0.065}{2} = 0.0325$$ The effective quarterly rate is $$i_2 = (1 + i_{semi})^{\frac{1}{2}} - 1 = (1 + 0.0325)^{0.5} - 1$$ Calculate: $$i_2 = 1.0325^{0.5} - 1 = 1.016113 - 1 = 0.016113$$ Rounded to 6 decimal places: $$i_2 = 0.016113$$ 5. **Calculate total number of payments $n$:** Payments are quarterly for 3 years: $$n = 3 \times 4 = 12$$ 6. **Calculate present value $PV$:** $$PV = 4443.45 \times \frac{1 - (1 + 0.016113)^{-12}}{0.016113}$$ Calculate: $$1 + 0.016113 = 1.016113$$ $$1.016113^{-12} = \frac{1}{1.016113^{12}} = \frac{1}{1.219} = 0.820$$ $$1 - 0.820 = 0.180$$ $$\frac{0.180}{0.016113} = 11.171$$ $$PV = 4443.45 \times 11.171 = 49618.68$$ 7. **Final Answer:** The cash price of the MacBook is $$\boxed{49618.68}$$