At the age of 31 , to save for retirement, you decide to deposit \( \$ 90 \) at the end of each month in an IRA that pays 5.5\% compounded monthly. a. Determine how much you will have in the IRA when you retire at age 65. b. Find the interest. a. You will have approximately \$ \( \square \) in the IRA when you retire. (Do not round until the final answer. Then round to the nearest dollar as needed.) b. The interest is approximately \$ \( \square \) . (Use the answer from part a to find this answer. Round to the nearest dollar as needed.)

To determine the future value of your IRA and the interest earned by the time you retire at age 65, we'll follow these steps: ### **Given:** - **Current Age:** 31 - **Retirement Age:** 65 - **Monthly Deposit (\( PMT \)):** \$90 - **Annual Interest Rate:** 5.5% compounded monthly ### **Calculations:** 1. **Determine the Number of Periods (\( n \)):** \[ n = (65 - 31) \times 12 = 34 \times 12 = 408 \text{ months} \] 2. **Monthly Interest Rate (\( r \)):** \[ r = \frac{5.5\%}{12} = 0.458333\% = 0.0045833333 \text{ (as a decimal)} \] 3. **Future Value of an Ordinary Annuity (since deposits are made at the end of each month):** \[ FV = PMT \times \left( \frac{(1 + r)^n - 1}{r} \right) \] Plugging in the values: \[ FV = 90 \times \left( \frac{(1 + 0.0045833333)^{408} - 1}{0.0045833333} \right) \] 4. **Calculating \((1 + r)^n\):** \[ (1 + 0.0045833333)^{408} \approx 6.44555 \] 5. **Substituting back into the Future Value formula:** \[ FV = 90 \times \left( \frac{6.44555 - 1}{0.0045833333} \right) = 90 \times \left( \frac{5.44555}{0.0045833333} \right) \approx 90 \times 1,187.5625 \approx 106,930.8 \] 6. **Rounding to the Nearest Dollar:** \[ FV \approx \$106,931 \] 7. **Total Contributions:** \[ \text{Total Deposits} = 90 \times 408 = \$36,720 \] 8. **Interest Earned:** \[ \text{Interest} = FV - \text{Total Deposits} = 106,931 - 36,720 = \$70,211 \] ### **Final Answers:** a. You will have approximately **\$106,931** in the IRA when you retire. b. The interest is approximately **\$70,211**.