A 34 -year old woman has \( \$ 388,115.00 \) in an IRA account. Due to medical concerns of a family member, she decided to make no additional contributions to the account but expect it to grow at \( 7 \% \) compounded every 4 months. How much does she expect to have in the account when she retires at age 62 ? Round value to the nearest cent. The IRA should be worth \( \$ \)

To determine how much the IRA account will be worth when the woman retires at age 62, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the number of years the money is invested or borrowed. ### Given: - \( P = 388115 \) - \( r = 0.07 \) (7% as a decimal) - \( n = 3 \) (since interest is compounded every 4 months, there are 3 compounding periods in a year) - \( t = 62 - 34 = 28 \) years (from age 34 to age 62) Now, we can substitute these values into the formula to find \( A \): \[ A = 388115 \left(1 + \frac{0.07}{3}\right)^{3 \times 28} \] Let's calculate this step by step. 1. Calculate \( \frac{r}{n} \): \[ \frac{0.07}{3} \] 2. Calculate \( nt \): \[ 3 \times 28 \] 3. Substitute these values into the formula and calculate \( A \). Let's perform these calculations. Calculate the value by following steps: - step0: Calculate: \(3\times 28\) - step1: Multiply the numbers: \(84\) Calculate or simplify the expression \( 0.07/3 \). Calculate the value by following steps: - step0: Calculate: \(\frac{0.07}{3}\) - step1: Convert the expressions: \(\frac{\frac{7}{100}}{3}\) - step2: Multiply by the reciprocal: \(\frac{7}{100}\times \frac{1}{3}\) - step3: Multiply the fractions: \(\frac{7}{100\times 3}\) - step4: Multiply: \(\frac{7}{300}\) Calculate or simplify the expression \( 388115*(1+(0.07/3))^(3*28) \). Calculate the value by following steps: - step0: Calculate: \(388115\left(1+\left(\frac{0.07}{3}\right)\right)^{3\times 28}\) - step1: Divide the terms: \(388115\left(1+\frac{7}{300}\right)^{3\times 28}\) - step2: Add the numbers: \(388115\left(\frac{307}{300}\right)^{3\times 28}\) - step3: Multiply the numbers: \(388115\left(\frac{307}{300}\right)^{84}\) - step4: Simplify: \(388115\times \frac{307^{84}}{300^{84}}\) - step5: Rewrite the expression: \(5\times 77623\times \frac{307^{84}}{300^{84}}\) - step6: Rewrite the expression: \(5\times 77623\times \frac{307^{84}}{25^{84}\times 12^{84}}\) - step7: Rewrite the expression: \(5\times 77623\times \frac{307^{84}}{5^{168}\times 12^{84}}\) - step8: Reduce the numbers: \(77623\times \frac{307^{84}}{5^{167}\times 12^{84}}\) - step9: Multiply the fractions: \(\frac{77623\times 307^{84}}{5^{167}\times 12^{84}}\) Now that we have the necessary calculations: 1. The value of \( \frac{r}{n} = \frac{0.07}{3} \approx 0.02333 \). 2. The value of \( nt = 3 \times 28 = 84 \). Now substituting these values into the compound interest formula: \[ A = 388115 \left(1 + 0.02333\right)^{84} \] Calculating this gives us: \[ A \approx 2694006.08 \] Thus, the IRA should be worth approximately \( \$ 2,694,006.08 \) when she retires at age 62. So, the final answer is: \[ \text{The IRA should be worth } \$ 2,694,006.08 \]