1. **State the problem:** We have an annuity with a present value (PV) of 80000, an interest rate of 5.1% compounded quarterly, and a term of 4.5 years. We want to find the quarterly payment amount. 2. **Identify the formula:** The present value of an ordinary annuity formula is: $$PV = P \times \frac{1 - (1 + r)^{-n}}{r}$$ where: - $P$ is the payment per period, - $r$ is the interest rate per period, - $n$ is the total number of payments. 3. **Calculate the parameters:** - Annual interest rate = 5.1% = 0.051 - Compounded quarterly means 4 periods per year, so quarterly interest rate: $$r = \frac{0.051}{4} = 0.01275$$ - Number of quarters in 4.5 years: $$n = 4.5 \times 4 = 18$$ 4. **Rearrange the formula to solve for $P$:** $$P = PV \times \frac{r}{1 - (1 + r)^{-n}}$$ 5. **Substitute the values:** $$P = 80000 \times \frac{0.01275}{1 - (1 + 0.01275)^{-18}}$$ 6. **Calculate the denominator:** $$1 - (1 + 0.01275)^{-18} = 1 - (1.01275)^{-18}$$ Calculate $(1.01275)^{-18}$: $$ (1.01275)^{18} \approx 1.2527 \Rightarrow (1.01275)^{-18} = \frac{1}{1.2527} \approx 0.7985$$ So denominator: $$1 - 0.7985 = 0.2015$$ 7. **Calculate $P$:** $$P = 80000 \times \frac{0.01275}{0.2015} = 80000 \times 0.06328 = 5062.40$$ **Answer:** The quarterly payment is approximately $5062.40$.