https://file.aiquickdraw.com/mathbot/file/1731045054052v9jfl4qu.jpg

To solve this problem, we need to use the future value of an annuity formula, which is often applied in retirement savings plans when making regular monthly deposits. The formula is:

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

where:
- $ FV $ is the future value you want to achieve (in this case, $1.4 million).
- $ P $ is the monthly deposit we need to determine.
- $ r $ is the monthly interest rate (annual rate divided by 12).
- $ n $ is the total number of deposits (months).

### Given values
- Desired future value, $ FV = 1,400,000 $
- Annual interest rate, $ 7\% $ (so $ r = \frac{0.07}{12} $ per month)
- Number of years to retirement, $ 26 $ (so $ n = 26 \times 12 = 312 $ months)

### Step-by-Step Solution

1. **Calculate the monthly interest rate**:
   \[
   r = \frac{0.07}{12} = 0.0058333
   \]

2. **Calculate the total number of deposits**:
   \[
   n = 26 \times 12 = 312
   \]

3. **Rearrange the formula to solve for $ P $**:
   \[
   P = \frac{FV \times r}{(1 + r)^n - 1}
   \]

4. **Plug in the values and calculate $ P $**:
   \[
   P = \frac{1,400,000 \times 0.0058333}{(1 + 0.0058333)^{312} - 1}
   \]

Let's compute this to find the required monthly deposit, rounding the final answer to two decimal places as requested.

Calculating...The monthly deposit you should invest is approximately **$1,589.06**.

Would you like a detailed explanation of the steps or have any questions about the solution?

Here are 5 related questions that expand on this topic:
1. How would the required monthly deposit change if the interest rate were higher or lower?
2. What if you wanted to reach the same goal but in a shorter time, say 20 years instead of 26?
3. How does compounding frequency (e.g., monthly vs. quarterly) affect the final savings amount?
4. What would be the impact if you started saving at a different age?
5. How can you calculate the future value if deposits are made yearly instead of monthly?

**Tip:** Consistent investing, even if it's a small amount, can significantly increase the total future value due to compounding interest!

At age 39, you start saving for retirement. If your investment plan pays an APR of 7% and you want to have $1.4 million when you retire in 26 years, how much should you deposit monthly?

Learn how to calculate the monthly deposit required to reach a $1.4 million retirement goal with a 7% APR over 26 years. This solution involves the future value of annuity formula, compound interest, and is suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation