Suppose you want to have $400,000 for retirement in 20 years. Your account earns 9% interest.

a) How much would you need to deposit in the account each month?

$

b) How much interest will you earn?


To solve this problem, we will use the **future value of an annuity formula** since you are making regular deposits (monthly) and the account earns interest. The formula is:

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

Where:
- $ FV $ is the future value, which is $400,000.
- $ P $ is the monthly deposit (what we are trying to find).
- $ r $ is the monthly interest rate.
- $ n $ is the total number of payments (months).

### Part (a): How much would you need to deposit each month?

1. **Convert the annual interest rate to a monthly rate**:
   \[
   r = \frac{9\%}{12} = \frac{0.09}{12} = 0.0075
   \]
   
2. **Determine the number of months**:
   Since you want to save for 20 years, and there are 12 months in a year:
   \[
   n = 20 \times 12 = 240 \text{ months}
   \]

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

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

Substituting the known values:

\[
P = \frac{400,000 \times 0.0075}{(1 + 0.0075)^{240} - 1}
\]

Let me calculate this.The monthly deposit you need to make is approximately **$598.90**.

### Part (b): How much interest will you earn?

To find the total interest earned, we first calculate the total amount of money deposited over the 20 years, and then subtract this from the future value.

1. **Total amount deposited**:
   \[
   \text{Total Deposits} = P \times n = 598.90 \times 240
   \]
   
2. **Total interest earned**:
   \[
   \text{Interest Earned} = FV - \text{Total Deposits}
   \]

Let me calculate the interest for you.The total amount you will deposit over the 20 years is approximately **$143,736.92**.

The total interest earned will be approximately **$256,263.08**.

### Summary:
- **Monthly deposit:** $598.90
- **Total interest earned:** $256,263.08

Would you like further clarification or additional details?

Here are 5 related questions:
1. What if the interest rate were 7% instead of 9%?
2. How would the monthly deposit change if the goal was $500,000?
3. What if the time period was 30 years instead of 20 years?
4. How does compounding frequency affect the total interest earned?
5. What happens if you make one-time lump sum deposits instead of monthly ones?

**Tip:** Regularly reviewing your retirement plan and adjusting contributions can significantly impact your savings over time.

Suppose you want to have $400,000 for retirement in 20 years. Your account earns 9% interest.

a) How much would you need to deposit in the account each month?

b) How much interest will you earn?

Discover how to calculate the monthly deposit required to save $400,000 for retirement in 20 years with a 9% interest rate. Learn how to use the future value of annuity formula, calculate total interest earned, and understand compound interest. Ideal for Grades 11-12 and college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation