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The problem states:

"Auto purchaser Barbara Morales wants to buy a $24,000 car in 6 years. How much money must she deposit at the end of each quarter in an account paying 4% compounded quarterly so that she will have enough to pay for the car?"

This is a problem that involves calculating the regular payments into a compound interest account to reach a future value.

### Steps to solve the problem:

1. **Identify the parameters:**
   - Future Value (FV): $24,000
   - Interest rate (r): 4% per year compounded quarterly
   - Number of years (t): 6 years
   - Number of compounding periods per year: 4 (quarterly)
   - Total number of periods (n): $ n = 4 \times 6 = 24 $
   - Quarterly interest rate: $ i = \frac{4\%}{4} = 1\% = 0.01 $

2. **Use the future value of an ordinary annuity formula:**

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

Where:
- $ FV $ is the future value ($24,000),
- $ P $ is the regular payment amount,
- $ i $ is the interest rate per period (0.01),
- $ n $ is the total number of periods (24).

3. **Rearrange the formula to solve for $ P $:**

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

4. **Substitute the known values:**

\[
P = \frac{24,000 \times 0.01}{(1 + 0.01)^{24} - 1}
\]

Now, I'll calculate the value of $ P $.The amount Barbara Morales needs to deposit at the end of each quarter is approximately **$889.76** to reach her goal of $24,000 in 6 years.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How would the required deposit change if the interest rate was 5% compounded quarterly?
2. What if Barbara wanted to save for 8 years instead of 6 years; how would that affect the quarterly deposit?
3. How would the situation differ if the interest was compounded monthly instead of quarterly?
4. If Barbara can only afford to deposit $700 per quarter, how long will it take her to save $24,000 with the same interest rate?
5. How much interest will Barbara have earned by the end of the 6 years?

**Tip:** When dealing with compound interest problems, it's important to keep track of the compounding frequency (quarterly, monthly, etc.) as it can significantly impact the amount you need to save or the interest earned.

Auto purchaser Barbara Morales wants to buy a $24,000 car in 6 years. How much money must she deposit at the end of each quarter in an account paying 4% compounded quarterly so that she will have enough to pay for the car?

Find out how much Barbara Morales needs to deposit at the end of each quarter to buy a $24,000 car in 6 years with 4% interest compounded quarterly. This detailed solution uses the Future Value of an Annuity formula to determine the necessary quarterly deposits.

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