You would like to have $650,000 when you retire in 30 years. How much should you invest each quarter if you can earn a rate of 7.2% compounded quarterly?

a) How much should you deposit each quarter?

$

b) How much total money will you put into the account?

$

c) How much total interest will you earn?

To solve this problem, we can use the formula for the future value of an annuity, since you will be making regular quarterly investments over time and earning interest. The formula for the future value $ FV $ of an annuity is:

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

where:
- $ FV $ is the future value (the amount you want, $650,000),
- $ P $ is the payment per period (the amount you need to find),
- $ r $ is the interest rate per period,
- $ n $ is the total number of periods.

### a) Finding the quarterly deposit $ P $

#### Step 1: Identify the known values.
- $ FV = 650,000 $ (the future value you want),
- The annual interest rate is 7.2%, but it is compounded quarterly. The quarterly interest rate $ r $ is:
  
\[
r = \frac{7.2\%}{4} = \frac{0.072}{4} = 0.018
\]

- The number of quarters in 30 years is:

\[
n = 30 \times 4 = 120
\]

#### Step 2: Plug the values into the formula and solve for $ P $.

\[
650,000 = P \times \frac{(1 + 0.018)^{120} - 1}{0.018}
\]

Now, we can solve for $ P $.

### b) Total money deposited

The total amount deposited will be the quarterly payment $ P $ times the number of periods $ n $. So, the total amount deposited is:

\[
\text{Total Deposited} = P \times 120
\]

### c) Total interest earned

The total interest earned is the difference between the future value $ FV $ and the total deposits:

\[
\text{Interest Earned} = FV - \text{Total Deposited}
\]

Let me calculate these values for you.### a) The amount you should deposit each quarter is approximately $ \text{\$1,558.72} $.

### b) The total amount of money you will put into the account is approximately $ \text{\$187,046.46} $.

### c) The total interest you will earn is approximately $ \text{\$462,953.54} $.

Do you want more details on any of these steps, or have any questions?

Here are five related questions you might want to explore:
1. How would the quarterly deposit change if the interest rate were lower, say 5%?
2. What if you only wanted to save for 20 years instead of 30 years?
3. How does the compounding frequency (monthly, quarterly, annually) affect the required deposit?
4. What happens if the interest is compounded monthly instead of quarterly?
5. How much would you earn if you only made a one-time lump sum investment today?

**Tip:** Always check the compounding frequency (quarterly, monthly, annually) to ensure your calculations align with the actual investment terms!

You would like to have $650,000 when you retire in 30 years. How much should you invest each quarter if you can earn a rate of 7.2% compounded quarterly?

Learn how to calculate the quarterly deposit required to have $650,000 when you retire in 30 years, assuming a 7.2% interest rate compounded quarterly. The solution covers the Future Value of Annuity formula and compound interest principles, with a detailed step-by-step guide. Suitable for students in Grades 11-12.

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