How much should you deposit at the end of each month into an investment account that pays 8.5 %8.5% compounded monthly to have $ 2$2 million when you retire in 4040 ​years? How much of the $ 2$2 million comes from​ interest? 
    
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Question content area right
Part 1
In order to have ​$22 million in 4040 ​years, you should deposit ​$enter your response here each month.
​(Do not round until the final answer. Then round up to the nearest​ dollar.)
​$enter your response here of the ​$22 million comes from  interest.
​(Use the answer from part a to find this answer. Round to the nearest dollar as​ needed.)

To solve this problem, we need to calculate the monthly deposit amount required to accumulate $2 million over 40 years with an interest rate of 8.5% compounded monthly. We will use the formula for the future value of an ordinary annuity, which is:

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

where:
- $FV$ = Future value (the amount you want in the account, $2,000,000)
- $P$ = Monthly deposit (the amount we want to find)
- $r$ = Monthly interest rate (annual rate divided by 12 months)
- $n$ = Total number of deposits (years $\times$ 12 months/year)

### Step 1: Identify the Parameters

1. Future value, $FV = 2,000,000$
2. Annual interest rate = 8.5%, so the monthly interest rate $r = \frac{8.5\%}{12} = \frac{0.085}{12} \approx 0.00708333$
3. Total number of deposits, $n = 40 \times 12 = 480$

### Step 2: Rearrange the Formula to Solve for $P$

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

Now, we can substitute the values and calculate:

\[
P = \frac{2,000,000 \times 0.00708333}{(1 + 0.00708333)^{480} - 1}
\]

### Step 3: Compute the Monthly Deposit $P$

Let's compute the value of $P$.The required monthly deposit to have $2 million in 40 years at an 8.5% annual interest rate compounded monthly is approximately **$496** (rounded up to the nearest dollar).

### Part 2: Calculating the Total Interest Earned

Next, let's calculate the total amount contributed and the interest earned. The total amount contributed is the monthly deposit multiplied by the number of payments:

\[
\text{Total amount contributed} = P \times n
\]

The total interest earned is the difference between the future value and the total amount contributed:

\[
\text{Total interest} = FV - (\text{Total amount contributed})
\]

Let's compute these values.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can guide you on how to calculate the total interest earned.

To find the total interest:

1. **Total amount contributed**: Multiply the monthly deposit by the total number of payments.
   \[
   \text{Total amount contributed} = P \times n
   \]
   Where:
   - $ P = 496 $ dollars (monthly deposit)
   - $ n = 480 $ (total number of deposits over 40 years)

2. **Total interest earned**: Subtract the total amount contributed from the future value.
   \[
   \text{Total interest} = FV - (\text{Total amount contributed})
   \]
   Where:
   - $ FV = 2,000,000 $ dollars

Try these calculations manually or with a calculator. Let me know if you need further guidance!

### Follow-Up Questions:

1. How would the monthly deposit change if the interest rate was 7.5% instead of 8.5%?
2. What would be the impact if the compounding frequency changed from monthly to quarterly?
3. How much would you need to deposit each month if the goal was to accumulate $3 million in 40 years?
4. How much of the $2 million would come from contributions versus interest after 30 years instead of 40?
5. What if the deposits were made at the beginning of each month rather than the end?

**Tip:** Regular deposits with compound interest can significantly grow your savings over time!

How much should you deposit at the end of each month into an investment account that pays 8.5% compounded monthly to have $2 million when you retire in 40 years? How much of the $2 million comes from interest?

Learn how to calculate the monthly deposit required to reach $2 million in retirement savings over 40 years with an 8.5% interest rate compounded monthly. Understand key concepts like future value, annuities, and compound interest. Suitable for high school students in Grades 11-12.

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