Question 4 of 18, Step 1 of 1 James deposits a fixed quarterly amount into an annuity account for his child's college fund. He wishes to accumulate a future value of \( \$ 60,000 \) in 17 years. Assuming an APR of \( 3.7 \% \) compounded quarterly, how much of the \( \$ 60,000 \) will james ultimately deposit in the account, and how much is interest earned? Round your answers to the nearest cent, if necessary, \( \begin{array}{l}\text { Answer } \\ \text { How to enter your answer (opens in new window) }\end{array} \)

To solve this problem, we need to determine how much James will deposit into the annuity account and how much interest he will earn by the time he accumulates \( \$60,000 \) in 17 years with an APR of \( 3.7\% \) compounded quarterly. ### Step 1: Extract Known Conditions - Future Value (FV) = \( 60,000 \) - Annual Percentage Rate (APR) = \( 3.7\% \) - Compounding frequency = quarterly - Total time (t) = 17 years ### Step 2: Calculate the Quarterly Interest Rate and Total Number of Deposits 1. **Quarterly Interest Rate (r)**: \[ r = \frac{APR}{4} = \frac{3.7\%}{4} = \frac{0.037}{4} = 0.00925 \] 2. **Total Number of Deposits (n)**: \[ n = 17 \text{ years} \times 4 \text{ quarters/year} = 68 \text{ quarters} \] ### Step 3: Use the Future Value of an Annuity Formula The future value of an annuity formula is given by: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] Where: - \( FV \) = future value - \( P \) = payment per period (quarterly deposit) - \( r \) = interest rate per period - \( n \) = total number of periods We need to rearrange this formula to solve for \( P \): \[ P = \frac{FV \times r}{(1 + r)^n - 1} \] ### Step 4: Substitute Known Values and Calculate \( P \) Now we will substitute the known values into the formula to find \( P \): - \( FV = 60000 \) - \( r = 0.00925 \) - \( n = 68 \) Let's calculate \( P \). Calculate the value by following steps: - step0: Calculate: \(\frac{60000\times 0.00925}{\left(\left(1+0.00925\right)^{68}-1\right)}\) - step1: Remove the parentheses: \(\frac{60000\times 0.00925}{\left(1+0.00925\right)^{68}-1}\) - step2: Add the numbers: \(\frac{60000\times 0.00925}{1.00925^{68}-1}\) - step3: Convert the expressions: \(\frac{60000\times 0.00925}{\left(\frac{4037}{4000}\right)^{68}-1}\) - step4: Multiply the numbers: \(\frac{555}{\left(\frac{4037}{4000}\right)^{68}-1}\) - step5: Subtract the numbers: \(\frac{555}{\frac{4037^{68}-4000^{68}}{4000^{68}}}\) - step6: Multiply by the reciprocal: \(555\times \frac{4000^{68}}{4037^{68}-4000^{68}}\) - step7: Multiply: \(\frac{555\times 4000^{68}}{4037^{68}-4000^{68}}\) The calculation gives us the quarterly deposit amount \( P \): \[ P \approx 637.70 \] ### Step 5: Calculate Total Deposits and Interest Earned 1. **Total Deposits**: \[ \text{Total Deposits} = P \times n = 637.70 \times 68 \] 2. **Interest Earned**: \[ \text{Interest Earned} = FV - \text{Total Deposits} \] Now, let's calculate the total deposits and the interest earned. Calculate the value by following steps: - step0: Calculate: \(637.7\times 68\) - step1: Multiply the numbers: \(43363.6\) Calculate or simplify the expression \( 60000 - (637.70 * 68) \). Calculate the value by following steps: - step0: Calculate: \(60000-\left(637.7\times 68\right)\) - step1: Multiply the numbers: \(60000-43363.6\) - step2: Subtract the numbers: \(16636.4\) The calculations yield the following results: 1. **Total Deposits**: \[ \text{Total Deposits} \approx 43,363.60 \] 2. **Interest Earned**: \[ \text{Interest Earned} \approx 16,636.40 \] ### Final Answers - **Total Amount Deposited by James**: \( \approx 43,363.60 \) - **Total Interest Earned**: \( \approx 16,636.40 \) These amounts are rounded to the nearest cent as required.