Question 12 of 15, Step 1 of 1 Andrew deposits a fixed monthly amount into an annuity account for his child's college fund. He wishes to accumulate a future value of $$ \$ 85,000 $$ in 13 years. Assuming an APR of $$ 3.8 \% $$ compounded monthly, how much of the $$ \$ 85,000 $$ will Andrew ultimately deposit in the account, and how much is interest earned? Round your answers to the nearest cent. if necessary. Formulas

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To determine how much Andrew will ultimately deposit in the account and the interest earned, we'll follow these steps:

1. **Identify the Given Information:**
   - **Future Value (FV):** \$85,000
   - **Annual Percentage Rate (APR):** 3.8% compounded monthly
   - **Time (t):** 13 years
   
2. **Calculate the Monthly Interest Rate (r):**
   $$
   r = \frac{\text{APR}}{12} = \frac{3.8\%}{12} = 0.0031667 \text{ (monthly rate)}
   $$

3. **Determine the Total Number of Monthly Payments (n):**
   $$
   n = 13 \text{ years} \times 12 \text{ months/year} = 156 \text{ months}
   $$

4. **Use the Future Value of an Ordinary Annuity Formula to Find the Monthly Payment (PMT):**
   $$
   \text{FV} = \text{PMT} \times \left( \frac{(1 + r)^n - 1}{r} \right)
   $$
   Rearranging to solve for PMT:
   $$
   \text{PMT} = \frac{\text{FV}}{\left( \frac{(1 + r)^n - 1}{r} \right)}
   $$

5. **Calculate $$(1 + r)^n$$:**
   $$
   (1 + 0.0031667)^{156} \approx 1.637
   $$

6. **Plug the Values Back into the PMT Formula:**
   $$
   \text{PMT} = \frac{85,000}{\left( \frac{1.637 - 1}{0.0031667} \right)} \approx \frac{85,000}{201.205} \approx 422.30
   $$
   So, **Andrew deposits approximately \$422.30 monthly**.

7. **Calculate the Total Deposited Over 13 Years:**
   $$
   \text{Total Deposited} = \text{PMT} \times n = 422.30 \times 156 \approx 65,878.80
   $$

8. **Determine the Interest Earned:**
   $$
   \text{Interest Earned} = \text{FV} - \text{Total Deposited} = 85,000 - 65,878.80 = 19,121.20
   $$

**Final Answers:**
- **Total Deposited:** \$65,878.80
- **Interest Earned:** \$19,121.20
Andrew will deposit approximately \$422.30 each month, totaling \$65,878.80 over 13 years. The interest earned will be \$19,121.20.

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