Matilda, which is to retire at the age of 67 with 1,100,000 in her retirement account when she turns 28 she decides to begin depositing money into the account with an APR of 9% compounded monthly. What is the monthly deposit that Matilda must make in order to reach her goal?

To determine the monthly deposit Matilda must make in order to reach her goal of having $1,100,000 by the time she retires at age 67, we can use the **Future Value of an Annuity formula** since she will be making regular monthly deposits into an account with interest:

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

Where:
- $FV$ = future value (the target amount, $1,100,000)
- $P$ = the monthly deposit (this is what we need to find)
- $r$ = the monthly interest rate (APR divided by 12 months)
- $n$ = the total number of payments (number of months between age 28 and 67)

### Step 1: Calculate the monthly interest rate
The annual percentage rate (APR) is 9%, so the monthly interest rate $r$ is:

\[
r = \frac{9\%}{12} = \frac{0.09}{12} = 0.0075
\]

### Step 2: Determine the number of months
Matilda will be making monthly deposits from age 28 to age 67. The number of years is:

\[
67 - 28 = 39 \text{ years}
\]

The total number of months $n$ is:

\[
n = 39 \times 12 = 468 \text{ months}
\]

### Step 3: Use the formula to find $P$
We need to solve for $P$, the monthly deposit. Rearranging the formula:

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

Now we can plug in the values:

\[
P = \frac{1,100,000 \times 0.0075}{(1 + 0.0075)^{468} - 1}
\]

I'll calculate the value of $P$.The monthly deposit that Matilda must make in order to reach her goal of $1,100,000 by the time she retires at age 67 is approximately **$257.71**.

Would you like any further details or have any questions?

Here are some related questions for deeper exploration:
1. How would the monthly deposit change if the APR were lower, such as 7%?
2. What if Matilda started saving 5 years earlier? How would that affect her monthly deposit?
3. How would inflation impact the future value of $1,100,000 in today's dollars?
4. What is the total amount of interest earned by the time she reaches her goal?
5. What if the compounding frequency was yearly instead of monthly—how would this affect the required deposit?

**Tip:** Starting to invest earlier, even with smaller amounts, significantly reduces the monthly contribution needed due to the power of compound interest.

Learn how to calculate the monthly deposit needed to reach a retirement goal of $1,100,000 by age 67 with a 9% APR compounded monthly. This problem uses the Future Value of an Annuity formula, explaining compound interest and time value of money concepts. Ideal for students at advanced high school or college levels.

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