Suppose you want to have $500,000 for retirement in 30 years. Your account earns 7% interest.

a) How much would you need to deposit in the account each month?

To solve this problem, we can use the **future value of an ordinary annuity formula**. The formula to calculate the future value of monthly deposits with compound interest is:

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

Where:

- $ FV $ is the future value (in this case, \$500,000),
- $ P $ is the monthly deposit,
- $ r $ is the monthly interest rate (annual interest rate divided by 12),
- $ n $ is the total number of payments (months).

### Step 1: Identify the given values

- Future Value, $ FV = 500,000 $
- Annual interest rate = 7%, so the monthly interest rate, $ r = \frac{7\%}{12} = 0.005833 $
- Time period = 30 years, so the total number of months, $ n = 30 \times 12 = 360 $

### Step 2: Rearrange the formula to solve for $ P $

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

### Step 3: Substitute the values

\[
P = \frac{500,000}{\left( \frac{(1 + 0.005833)^{360} - 1}{0.005833} \right)}
\]

Let me calculate the result.The monthly deposit you would need to make to accumulate $500,000 in 30 years, with a 7% annual interest rate, is approximately **$409.85**.

Would you like more details or have any further questions?

Here are some related questions:
1. How does changing the interest rate to 8% affect the monthly deposit?
2. What if you only want $300,000 for retirement instead of $500,000?
3. What if the deposit period is reduced to 20 years instead of 30 years?
4. How does inflation impact the value of the future amount?
5. What would the total amount of deposits be over 30 years?

**Tip:** When planning for retirement, consider adjusting the interest rate to account for possible changes in the economy over time.

Suppose you want to have $500,000 for retirement in 30 years. Your account earns 7% interest. How much would you need to deposit in the account each month?

Discover how much you need to deposit monthly to accumulate $500,000 for retirement in 30 years, given a 7% interest rate. Learn the formula for future value of annuity and how compound interest works over time. Suitable for high school students and personal finance learners.

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