James deposits a fixed quarterly amount into an annuity account for his child's college fund. He wishes to accumulate a future value of $70,000 in 15 years. Assuming an APR of 3.7% compounded quarterly, how much of the $70,000 will James ultimately deposit in the account, and how much is interest earned? Round your answers to the nearest cent, if necessary.

We can solve this problem using the formula for the future value of an ordinary annuity, where deposits are made at regular intervals and interest is compounded. The future value $ FV $ of an ordinary annuity is given by:

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

Where:
- $ FV $ is the future value of the annuity ($70,000 in this case).
- $ P $ is the regular deposit per quarter (what we want to find).
- $ r $ is the quarterly interest rate, which is the annual percentage rate (APR) divided by the number of compounding periods per year. For 3.7% APR compounded quarterly, $ r = \frac{3.7\%}{4} = 0.00925 $.
- $ n $ is the total number of deposits, which is the number of compounding periods (quarters) in 15 years. So, $ n = 15 \times 4 = 60 $.

### Step 1: Solving for the quarterly deposit $ P $

Rearranging the formula to solve for $ P $:

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

Substituting the given values:

\[
P = \frac{70,000 \times 0.00925}{(1 + 0.00925)^{60} - 1}
\]

Let me calculate this now.

### Step 2: Calculate the total deposits

The total amount deposited by James is simply the quarterly deposit $ P $ multiplied by the number of quarters $ n $.

### Step 3: Calculate the interest earned

The interest earned is the difference between the future value (\$70,000) and the total amount deposited.

Let me calculate these results.### Results:
- **Quarterly deposit $ P $:** \$877.96
- **Total amount deposited over 15 years:** \$52,677.77
- **Interest earned:** \$17,322.23

James will deposit \$52,677.77 over 15 years, and the remaining \$17,322.23 will come from the interest earned.

Would you like more details or have any other questions?

Here are 5 related questions you might be interested in:
1. What if the APR changes to 4%? How would that affect the total deposit?
2. How much would James need to deposit monthly instead of quarterly to reach the same goal?
3. If James wants to deposit \$1,000 per quarter, how much will he have after 15 years?
4. What is the difference in future value if interest is compounded monthly instead of quarterly?
5. How long would it take to reach \$70,000 if James could only afford to deposit \$750 per quarter?

**Tip:** Increasing the compounding frequency (e.g., monthly vs. quarterly) generally increases the interest earned, but the effect diminishes with very frequent compounding.

James deposits a fixed quarterly amount into an annuity account for his child's college fund. He wishes to accumulate a future value of $70,000 in 15 years. Assuming an APR of 3.7% compounded quarterly, how much of the $70,000 will James ultimately deposit in the account, and how much is interest earned? Round your answers to the nearest cent, if necessary.

Find out how much James needs to deposit quarterly to accumulate $70,000 in 15 years with a 3.7% APR compounded quarterly. This detailed solution includes key financial concepts such as compound interest and annuities, making it suitable for students in Grades 10-12.

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