Matilda wishes to retire at age 67 with $1,200,000 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 find the monthly deposit Matilda needs to make in order to retire with $1,200,000 at age 67, we can use the formula for the future value of a series of monthly payments (an ordinary annuity). The formula is:

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

Where:
- $FV$ is the future value ($1,200,000),
- $P$ is the monthly deposit (what we are solving for),
- $r$ is the monthly interest rate (APR divided by 12),
- $n$ is the total number of payments (the number of months between ages 28 and 67).

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

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

### Step 2: Determine the total number of months ($n$)
Matilda will make monthly deposits from age 28 to age 67, which is 39 years. Converting that to months:

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

### Step 3: Plug values into the future value formula
We rearrange the formula to solve for $P$:

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

Substituting the known values:

\[
P = \frac{1,200,000}{\frac{(1 + 0.0075)^{468} - 1}{0.0075}}
\]

Now, I'll calculate this.

Matilda must make a monthly deposit of approximately **$281.14** to reach her goal of $1,200,000 by age 67, assuming an APR of 9% compounded monthly.

Would you like more details on any part of this calculation? Here are five related questions to expand this topic:

1. What happens if the APR changes to a different value, say 8%?
2. How does the number of years of contribution affect the monthly payment?
3. Can we calculate how much interest she would earn over this period?
4. What if Matilda starts saving at a different age?
5. How would bi-weekly payments affect the total savings?

**Tip:** The earlier you start saving for retirement, the less you need to deposit monthly, thanks to the power of compound interest!

Discover how to calculate the monthly deposit Matilda must make to retire with $1,200,000 at age 67. This problem uses the future value of an annuity formula, with an APR of 9% compounded monthly. Learn about compound interest and financial planning for retirement.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation