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To solve this problem, we can use the **future value of an annuity formula**, since Amelia wants to make regular monthly deposits into her retirement account.

The formula for the future value of an annuity is:

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

Where:
- $ FV $ is the future value (the amount Amelia wants to have in the account when she retires) = \$1,800,000.
- $ P $ is the monthly deposit (this is what we need to find).
- $ r $ is the monthly interest rate, which is the annual rate divided by 12. Given an APR of 8%, $ r = \frac{8\%}{12} = 0.00667 $.
- $ n $ is the number of deposits. Amelia is currently 25 and plans to retire at 67, so the number of years is $ 67 - 25 = 42 $. Since she is depositing monthly, $ n = 42 \times 12 = 504 $ months.

Now, we can rearrange the formula to solve for $ P $ (the monthly deposit):

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

Let me calculate this for you.The monthly deposit that Amelia must make to reach her retirement goal of \$1,800,000 is approximately **\$436.84**.

Would you like more details or have any questions about this calculation?

Here are five related questions to consider:
1. How does the future value change if the interest rate is higher or lower?
2. What happens if Amelia starts depositing at a later age, say at 30?
3. How much more would Amelia need to deposit if her goal was \$2,000,000 instead of \$1,800,000?
4. What is the total amount Amelia deposits over the 42 years?
5. What is the effect of switching from monthly to quarterly deposits?

**Tip:** Starting retirement contributions early significantly reduces the required monthly deposit due to compound interest over time.

Amelia wishes to retire at age 67 with $1,800,000 in her retirement account. When she turns 25, she starts depositing money into an account with an APR of 8% compounded monthly. What is the monthly deposit that Amelia must make to reach her goal?

Learn how to calculate the monthly deposits Amelia must make to reach $1.8 million in retirement with an 8% APR compounded monthly. This financial planning problem explores key mathematical concepts such as compound interest and the future value of annuities.

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