Matilda wishes to retire at age 67
with $1,600,000
in her retirement account. When she turns 28
, she decides to begin depositing money into an account with an APR of 9%
compounded monthly. What is the monthly deposit that Matilda must make in order to reach her goal? Round your answer to the nearest cent, if necessary.

To solve this problem, we will use the **future value of an annuity formula** for monthly compounded interest, since Matilda is making monthly deposits. The formula is:

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

Where:
- $ FV $ is the future value (Matilda's goal: $1,600,000).
- $ P $ is the monthly deposit (what we're solving for).
- $ r $ is the annual interest rate (9% or 0.09).
- $ n $ is the number of compounding periods per year (12, since it's compounded monthly).
- $ t $ is the number of years Matilda will make deposits (from age 28 to 67, so 67 - 28 = 39 years).

Let's break it down:

1. $ FV = 1,600,000 $
2. $ r = 0.09 $
3. $ n = 12 $
4. $ t = 39 $

Now we plug everything into the formula to solve for $ P $, the monthly deposit. We rearrange the formula to solve for $ P $:

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

I'll calculate that now.Matilda must make a monthly deposit of approximately **$374.85** in order to reach her goal of $1,600,000 by the time she retires at age 67.

Do you want further details or have any questions about the solution?

Here are 5 related questions to expand on this:
1. What if the interest rate was 7% instead of 9%?
2. How much interest would Matilda earn over the entire period?
3. What happens if Matilda starts saving at age 35 instead of 28?
4. What is the impact of changing the compounding frequency to yearly instead of monthly?
5. How long would it take Matilda to reach her goal if she increases her monthly deposit?

**Tip:** Always ensure that you are clear about whether the interest is compounded monthly, yearly, or with another frequency, as it affects the calculation significantly.

Matilda wishes to retire at age 67 with $1,600,000 in her retirement account. When she turns 28, she decides to begin depositing money into an account with an APR of 9% compounded monthly. What is the monthly deposit that Matilda must make in order to reach her goal? Round your answer to the nearest cent, if necessary.

Discover how to calculate the monthly deposit required for Matilda to retire with $1.6 million at age 67, starting at age 28. This problem involves the future value of an annuity formula, compounded interest, and financial planning principles. Suitable for advanced high school and college students.

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