Question 23 of 25 Step 1 of 1 No Time Limit James 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 15 years. Assuming an APR of $$ 3.6 \% $$ compounded monthly, how much of the $$ \$ 85,000 $$ will James ultimately deposit in the account, and how much is interest eamed? Round your answers to the nearest cent, if necessary. Formulas AnswerHow to enter your answer (opens in new window) 2 Points Keypad Keyboard Shortcuts Amount James will deposit: $$ \$ $$ $$ \square $$ Interest earned: \$ $$ \square $$ Next

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UpStudy AI · Accepted Answer

Below is the step‐by‐step solution.

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### Step 1. Determine the Monthly Interest Rate and Total Number of Deposits

The Annual Percentage Rate (APR) is given as $$ 3.6\% $$. Since the interest is compounded monthly, the monthly interest rate is:
$$
r = \frac{0.036}{12} = 0.003
$$
James will be making deposits for 15 years. The total number of deposits (months) is:
$$
n = 15 \times 12 = 180
$$

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### Step 2. Use the Future Value of an Annuity Formula

For a series of fixed monthly deposits $$ M $$, the future value $$ FV $$ after $$ n $$ periods is given by:
$$
FV = M \cdot \frac{(1 + r)^n - 1}{r}
$$
We know that James wishes the future value to be:
$$
FV = 85000
$$
Thus,
$$
85000 = M \cdot \frac{(1.003)^{180} - 1}{0.003}
$$

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### Step 3. Solve for the Monthly Deposit $$ M $$

Rearrange the formula to isolate $$ M $$:
$$
M = \frac{85000 \times 0.003}{(1.003)^{180} - 1}
$$

Before proceeding, we need to compute $$(1.003)^{180}$$.

Taking the natural logarithm:
$$
\ln((1.003)^{180}) = 180 \cdot \ln(1.003)
$$
Since $$\ln(1.003) \approx 0.0029955$$, then:
$$
180 \cdot 0.0029955 \approx 0.53919
$$
Thus,
$$
(1.003)^{180} \approx e^{0.53919} \approx 1.714
$$

Now, substitute back:
$$
(1.003)^{180} - 1 \approx 1.714 - 1 = 0.714
$$
So,
$$
M \approx \frac{85000 \times 0.003}{0.714} = \frac{255}{0.714} \approx 357.14
$$
Thus, the monthly deposit is approximately:
$$
M \approx \$357.14
$$

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### Step 4. Calculate the Total Amount Deposited

James makes 180 deposits. Therefore, the total amount he deposits is:
$$
\text{Total Deposited} = M \times n = 357.14 \times 180 \approx 64285.20
$$
So, James deposits approximately:
$$
\$64,\!285.20
$$

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### Step 5. Determine the Interest Earned

The interest earned is the difference between the future value and the total deposited amount:
$$
\text{Interest Earned} = FV - \text{Total Deposited} \approx 85000 - 64285.20 \approx 20714.80
$$
Thus, the interest earned is approximately:
$$
\$20,\!714.80
$$

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### Final Answers

- **Amount James will deposit:** $$ \$64,\!285.20 $$
- **Interest earned:** $$ \$20,\!714.80 $$
James will deposit \$64,285.20 and earn \$20,714.80 in interest.

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