Ingrid wants to buy a $20,000 car in 8 years. How much money must she deposit at the end quarter paying 5.5% compounded quarterly so that she will have enough to pay for her car?

To solve this problem, we can use the formula for the future value of an ordinary annuity, since Ingrid is making regular deposits at the end of each quarter. The formula for the future value (FV) of an annuity compounded periodically is:

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

Where:
- $ FV $ is the future value, which is $20,000 (the amount Ingrid needs).
- $ P $ is the regular deposit amount (what we're solving for).
- $ r $ is the periodic interest rate.
- $ n $ is the total number of compounding periods.

### Step 1: Identify the variables

- **Interest rate**: 5.5% annually compounded quarterly means the quarterly interest rate $ r $ is:
\[
r = \frac{5.5\%}{4} = \frac{0.055}{4} = 0.01375
\]

- **Number of quarters**: Since she wants to save for 8 years, the number of quarters $ n $ is:
\[
n = 8 \times 4 = 32
\]

- **Future Value**: $ FV = 20,000 $

### Step 2: Plug the values into the formula

We now rearrange the formula to solve for $ P $ (the deposit amount):

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

Let's calculate this amount.Ingrid must deposit approximately **$501.77** at the end of each quarter in order to have enough money to buy the $20,000 car in 8 years.

Would you like further details on the steps, or do you have any other questions?

Here are some related questions you might find useful:
1. How would the deposit change if the interest rate were different?
2. What if Ingrid wanted to buy the car in 6 years instead of 8?
3. How would the situation change if the compounding was monthly instead of quarterly?
4. What is the total amount of money Ingrid will have deposited by the end of the 8 years?
5. How would inflation affect Ingrid's savings plan?

**Tip:** Compounding more frequently (monthly, weekly, etc.) can lead to larger savings over time due to interest accumulating more often.

Ingrid wants to buy a $20,000 car in 8 years. How much money must she deposit at the end of each quarter paying 5.5% compounded quarterly so that she will have enough to pay for her car?

Find out how much Ingrid must deposit each quarter to afford a $20,000 car in 8 years with a 5.5% interest rate compounded quarterly. This problem demonstrates compound interest calculations, annuities, and the time value of money. Suitable for Grades 10-12.

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