Ingrid wants to buy a ​$22 comma 000 car in 8 years. How much money must she deposit at the end of each quarter in an account paying 5.9​% compounded quarterly so that she will have enough to pay for her​ car?
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Part 1
How much money must she deposit at the end of each​ quarter?


To determine how much Ingrid needs to deposit at the end of each quarter to afford the $22,000 car in 8 years, we can use the **future value of an ordinary annuity formula**. The formula for future value $ FV $ of an ordinary annuity is:

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

Where:
- $ FV $ is the future value (which is $22,000),
- $ P $ is the amount deposited at the end of each period (the unknown),
- $ r $ is the interest rate per period,
- $ n $ is the total number of periods.

### Step 1: Identify the known values.
- **Future value** $ FV = 22,000 $,
- **Interest rate per quarter** $ r = \frac{5.9\%}{4} = \frac{0.059}{4} = 0.01475 $ (since it's compounded quarterly),
- **Total number of periods** $ n = 8 \text{ years} \times 4 \text{ quarters/year} = 32 \text{ quarters} $.

### Step 2: Rearrange the formula to solve for $ P $ (the quarterly deposit):

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

Substitute the known values:

\[
P = \frac{22,000}{\frac{(1 + 0.01475)^{32} - 1}{0.01475}}
\]

Now let's calculate this step by step.

1. $ 1 + 0.01475 = 1.01475 $,
2. $ 1.01475^{32} = 1.558307 $,
3. $ 1.558307 - 1 = 0.558307 $,
4. $ \frac{0.558307}{0.01475} = 37.85 $.

Thus,

\[
P = \frac{22,000}{37.85} = 581.31
\]

### Final Answer:
Ingrid must deposit **$581.31** at the end of each quarter in the account to save enough to buy the car.

Would you like further details or have any questions?

Here are 5 related questions:
1. What if the interest rate was compounded monthly instead of quarterly?
2. How much would Ingrid need to deposit if the car costs $25,000 instead?
3. What happens if Ingrid wants to buy the car in 6 years instead of 8?
4. How does the deposit change if the interest rate increases to 6.5%?
5. What would be the future value if Ingrid deposited $600 each quarter?

**Tip:** Always verify that the compounding frequency (quarterly, monthly, etc.) matches the frequency of the deposits in savings problems.

Ingrid wants to buy a $22,000 car in 8 years. How much money must she deposit at the end of each quarter in an account paying 5.9% compounded quarterly so that she will have enough to pay for her car?

Learn how much Ingrid needs to deposit at the end of each quarter to buy a $22,000 car in 8 years. The problem uses the future value of an ordinary annuity formula, applying an interest rate of 5.9% compounded quarterly over 32 periods. Suitable for high school students, Grades 10-12.

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