Suppose you want to have $500,000 for retirement in 20 years. Your account earns 10% interest. How much would you need to deposit in the account each month?

To determine how much needs to be deposited monthly in order to reach $500,000 in 20 years, we can use the formula for the future value of an annuity. The formula is:

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

Where:
- $ FV $ is the future value (in this case, $500,000),
- $ P $ is the monthly payment (what we are solving for),
- $ r $ is the monthly interest rate (annual rate divided by 12),
- $ n $ is the total number of payments (number of months).

### Step 1: Convert the interest rate and time period
- The annual interest rate is 10%, so the monthly interest rate $ r = \frac{10\%}{12} = \frac{0.10}{12} = 0.00833 $.
- The number of months $ n = 20 \times 12 = 240 $.

### Step 2: Use the future value annuity formula
Rearranging the formula to solve for $ P $ (the monthly deposit), we get:

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

Now, we can substitute the values:

\[
P = \frac{500,000}{\frac{(1 + 0.00833)^{240} - 1}{0.00833}}
\]

I'll compute the exact value for $ P $.The amount you would need to deposit each month to accumulate $500,000 in 20 years, with an account earning 10% interest annually, is approximately **$658.44**.

Would you like more details or have any questions?

Here are 5 related questions you might find useful:
1. How would the required deposit change if the interest rate were 8% instead of 10%?
2. How much would you accumulate if you deposited $500 monthly for 20 years at a 10% interest rate?
3. What happens if the interest compounds quarterly instead of monthly?
4. How would the time period affect the required monthly deposit if you only had 15 years instead of 20?
5. What would be the total amount of interest earned after 20 years?

**Tip:** The sooner you start saving, the lower your monthly contributions need to be, thanks to the power of compound interest.

Find out how much you need to deposit each month to accumulate $500,000 for retirement in 20 years with an account earning 10% interest. This solution uses the Future Value Annuity Formula, providing a detailed step-by-step calculation, suitable for Grades 11-12 and college students.

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