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The problem shown in the image involves calculating the amount James will deposit into an annuity and the interest earned based on a future value goal. Let's break this down step-by-step:

### Problem:
James deposits a fixed monthly amount into an annuity to accumulate $85,000 in 15 years. The interest rate is 3.6% APR compounded monthly. We need to find:
1. The total amount James will deposit over 15 years.
2. The interest earned.

### Formula for Future Value of an Annuity:
The formula to calculate the future value of an annuity is:

\[
FV = P \times \frac{(1 + r/n)^{nt} - 1}{r/n}
\]

Where:
- $ FV $ = Future value ($85,000)
- $ P $ = Monthly payment (amount James deposits per month)
- $ r $ = Annual interest rate (3.6% or 0.036 as a decimal)
- $ n $ = Number of times interest is compounded per year (12 for monthly compounding)
- $ t $ = Time in years (15 years)

We need to solve for $ P $ (the monthly deposit).

### Step 1: Rearrange the formula to solve for $ P $:
\[
P = \frac{FV}{\frac{(1 + r/n)^{nt} - 1}{r/n}}
\]

### Step 2: Plug in the values:
- $ FV = 85,000 $
- $ r = 0.036 $
- $ n = 12 $
- $ t = 15 $

Let me calculate the monthly payment and the total interest earned.The results are as follows:

1. **Amount James will deposit per month**: \$356.83
2. **Total amount James will deposit over 15 years**: \$64,229.92
3. **Interest earned**: \$20,770.08

Would you like more details or have any questions?

Here are some related questions:
1. How would the calculations change if the interest rate was compounded quarterly instead of monthly?
2. What would happen if James wanted to reach the same goal in 10 years instead of 15?
3. How does changing the interest rate affect the monthly payment?
4. Can the formula be adapted for continuous compounding?
5. What is the effect of increasing the monthly contribution by 5% each year?

**Tip:** The more frequently the interest is compounded, the more James will benefit from compound interest.

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 earned?

Learn how to calculate the total amount deposited and interest earned for an annuity with a 3.6% APR compounded monthly over 15 years. This detailed solution covers key financial math concepts like future value and compound interest.

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