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 \( 5.5 \% \) compounded monthly and you begin making payments at 25 years old. Round your answer to the nearest cent, if necessary. Formulas Present Value with Compound \( \quad P V=\frac{A}{\left(1+\frac{r}{n}\right)^{n t}} \) Annuity Payment Amount Future Value of an Annuity \[ \begin{array}{l} P M T=F V \cdot \frac{\left(\frac{r}{n}\right)}{\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]} \\ F V=P M T \cdot \frac{\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]}{\left(\frac{r}{n}\right)} \end{array} \] Answer Keypad Keyboard Shortcuts

We want to accumulate a future value (FV) of $70,000 by the time you retire at 65. You begin making monthly deposits at age 25, so you have 40 years to save. The account earns an annual percentage rate (APR) of 5.5% compounded monthly. Step 1. Determine the number of periods:   Years = 65 – 25 = 40 years   Monthly periods, n = 12 ⋅ 40 = 480 Step 2. Find the monthly interest rate:   r (annual) = 0.055, so monthly rate = 0.055/12 ≈ 0.00458333 Step 3. Use the Future Value of an Annuity formula:   FV = PMT ⋅ [((1 + r/n)^(n⋅t) – 1) / (r/n)] We need to solve for the monthly payment (PMT) given FV = 70,000:   PMT = FV ⋅ (r/n) / [((1 + r/n)^(n⋅t) – 1)] Substitute the numbers:   PMT = 70000 ⋅ (0.055/12) / [(1 + 0.055/12)^(480) – 1] Step 4. Compute the necessary parts:   Monthly rate = 0.055/12 ≈ 0.00458333   (1 + 0.00458333)^(480)     = (1.00458333)^(480) Calculating the exponent:   Take the natural logarithm: ln(1.00458333) ≈ 0.004573   Then, 480 ⋅ 0.004573 ≈ 2.194   So, (1.00458333)^(480) ≈ e^(2.194) ≈ 8.976 Thus, ((1.00458333)^(480) – 1) ≈ 8.976 – 1 = 7.976 Step 5. Now substitute back into the formula:   PMT = 70000 ⋅ 0.00458333 / 7.976     = 320.8333 / 7.976     ≈ 40.24 Rounding to the nearest cent, the monthly payment is approximately $40.24. Therefore, you would need to deposit about $40.24 each month into your IRA for 40 years to accumulate $70,000 at a 5.5% APR compounded monthly.