Suppose you want to have $500,000 for retirement in 35 years. Your account earns 10% interest.

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

$


b) How much interest will you earn?

$


This problem involves calculating the monthly deposit needed to reach a future value of $500,000 over 35 years with an interest rate of 10%. This type of problem can be solved using the formula for the future value of an annuity:

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

Where:
- $FV$ is the future value (in this case, $500,000),
- $P$ is the monthly deposit (which we need to find),
- $r$ is the monthly interest rate (which is the annual interest rate divided by 12),
- $n$ is the total number of payments (which is the number of years multiplied by 12).

### Step 1: Identify the known values:
- $FV = 500,000$
- Annual interest rate = 10%, so the monthly interest rate $r = \frac{10\%}{12} = \frac{0.10}{12}$
- Number of years = 35, so $n = 35 \times 12 = 420$ months.

### Step 2: Rearrange the formula to solve for $P$:

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

Now, let's calculate this step by step.The monthly deposit you would need to make is approximately **$131.70**.

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

To find the total interest earned, we first calculate the total amount deposited over the 35 years and subtract that from the future value of $500,000.

Total deposits = $P \times n$

Total interest = $FV - \text{Total deposits}$

Let's calculate this next.The total interest earned over the 35 years is approximately **$444,687.91**.

### Summary:
- (a) You would need to deposit approximately **$131.70** each month.
- (b) The total interest earned would be approximately **$444,687.91**.

Do you want more details or have any questions?

Here are 5 related questions:
1. What happens if the interest rate changes to 8%?
2. How does increasing the deposit amount affect the total interest earned?
3. How would shorter time periods (e.g., 20 years) affect the monthly deposit required?
4. What if you made a lump sum deposit instead of monthly contributions?
5. How does the calculation change if you deposit quarterly instead of monthly?

**Tip:** Consistent deposits and compound interest are key factors in building wealth over time.

Suppose you want to have $500,000 for retirement in 35 years. Your account earns 10% interest. How much would you need to deposit in the account each month? How much interest will you earn?

Calculate the monthly deposit needed to save $500,000 for retirement in 35 years with 10% interest. Understand key concepts like compound interest, annuities, and future value with a step-by-step breakdown. Suitable for Grades 10-12.

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