Suppose you wish to retire at the age of 65 with \( \$ 70,000 \) in savings. Determine your monthly payment into an IRA if the APR is \( 8.5 \% \) compounded monthly and you begin making payments at 30 years old. Round your answer to the nearest cent, If necessary. \[ \begin{array}{l}\text { Present Value with Compound } \\ \text { Interest }\end{array} \quad \mathrm{PV}=\frac{A}{\left(1+\frac{r}{n}\right)^{n t}} \] Annuity Payment Amount

To retire with \$70,000 at age 65, starting at age 30 (making it 35 years of investment), and with an APR of 8.5% compounded monthly, we first need to determine the monthly interest rate and the total number of payments. The monthly interest rate (r) is 8.5%/12 = 0.00708333, and the total number of payments (n*t) over 35 years is 35*12 = 420. Using the Present Value of an Annuity formula to determine the annuity payment would look like this: \[ PV = \frac{A \cdot \left(1 - \left(1 + \frac{r}{n}\right)^{-nt}\right)}{\frac{r}{n}} \] Rearranging to solve for A (the annuity payment) gives: \[ A = \frac{PV \cdot \frac{r}{n}}{1 - \left(1 + \frac{r}{n}\right)^{-nt}} \] Substituting in the values: \[ A = \frac{70000 \cdot 0.00708333}{1 - \left(1 + 0.00708333\right)^{-420}} \] Calculating this will result in: \[ A \approx 235.67 \] So, your monthly payment into the IRA would be approximately **$235.67**. The key is that compounded interest can work wonders over time, and starting earlier really pays off! Investing early means your money gets to marinate in that sweet interest over more years, leading to larger savings by retirement. The power of compounding allows even small contributions to blossom into significant sums over decades. Stick to your savings plan, and watch your nest egg grow!