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This is a compound interest problem involving a monthly deposit into a Roth IRA. Here's the breakdown of the given information:

- Monthly deposit ($ P $): \$450
- Annual interest rate ($ r $): 4.5% = 0.045 (compounded monthly)
- Number of years ($ t $): 12
- Compounding periods per year ($ n $): 12 (since it's compounded monthly)

We can use the formula for the future value of an annuity to calculate the total value of the Roth IRA:

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

Where:
- $ A $ is the future value of the annuity (the value of the Roth IRA),
- $ P $ is the monthly deposit,
- $ r $ is the annual interest rate (in decimal form),
- $ n $ is the number of times interest is compounded per year (12 for monthly),
- $ t $ is the time in years.

Let me calculate the future value of this Roth IRA.The value of the Roth IRA after 12 years, with monthly deposits of $450 and an annual interest rate of 4.5% compounded monthly, will be approximately **$86,034.38**.

Do you have any questions or would you like further details?

Here are 5 related questions to expand your understanding:
1. How would the future value change if the interest rate were 5% instead of 4.5%?
2. What happens to the future value if the deposits are made at the end of each month rather than the beginning?
3. How would the value differ if the interest were compounded quarterly instead of monthly?
4. What is the total amount of money contributed over the 12 years, and how much of the future value is from interest?
5. How does the time (number of years) affect the final value of the Roth IRA?

**Tip:** When comparing different savings options, pay attention to the compounding frequency. More frequent compounding typically results in greater returns over time.

The timpanist for an orchestra deposits $450 at the beginning of each month in a Roth IRA. The IRA has an annual interest rate of 4.5% compounded monthly. Find the value of the Roth IRA (in dollars) after 12 years. (Round your answer to the nearest cent.)

Find the future value of a Roth IRA with monthly deposits of $450, an annual interest rate of 4.5%, and interest compounded monthly over 12 years. Learn how to calculate compound interest with detailed steps.

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