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The problem you're dealing with involves calculating the value of an annuity where regular deposits are made into an account with compounded interest. Let's break down the parts:

**Given:**
- Monthly deposit: $400
- Interest rate: 6% annually (compounded monthly)
- Time period: 30 years

We will use the future value of an ordinary annuity formula:

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

Where:
- $ P $ = monthly deposit ($400)
- $ r $ = annual interest rate (6% or 0.06)
- $ n $ = number of compounding periods per year (12, since it's monthly)
- $ t $ = number of years (30)

### Part (a): Future Value of the Account in 30 Years
Plugging the values into the formula:

\[
FV = 400 \times \left( \frac{(1 + 0.06/12)^{12 \times 30} - 1}{0.06/12} \right)
\]

I'll compute this step-by-step:

1. $ r/n = 0.06/12 = 0.005 $
2. $ nt = 12 \times 30 = 360 $
3. $ (1 + 0.005)^{360} = 1.005^{360} $
4. Subtract 1 from the result.
5. Divide the result by $ 0.005 $.
6. Multiply by $ 400 $.

### Part (b): Total Amount Deposited
Since you're depositing $400 every month for 30 years:

\[
\text{Total deposited} = 400 \times 12 \times 30
\]

### Part (c): Total Interest Earned
The interest earned will be the difference between the future value calculated in part (a) and the total amount deposited:

\[
\text{Interest earned} = \text{Future value} - \text{Total deposited}
\]

I'll calculate all these now.Here are the results based on the annuity formula:

### Part (a): Future Value of the Account in 30 Years
The total amount you will have in the account after 30 years is **$401,806.02**.

### Part (b): Total Amount Deposited
The total amount you will deposit into the account over 30 years is **$144,000**.

### Part (c): Total Interest Earned
The total interest earned over the 30 years will be **$257,806.02**.

Would you like further clarification on any part of this? Here are five related questions to explore:

1. How would the future value change if the interest rate increased to 7%?
2. What happens to the total interest if deposits are made quarterly instead of monthly?
3. How does the future value change if the time period is reduced to 20 years?
4. Can you calculate the present value of the annuity if the future value is known?
5. How does continuous compounding differ from monthly compounding in this scenario?

**Tip:** The more frequently the interest compounds, the more interest you'll earn, even if the nominal interest rate remains the same.

You deposit $400 each month into an account earning 6% interest compounded monthly. a) How much will you have in the account in 30 years? b) How much total money will you put into the account? c) How much total interest will you earn?

Solve an annuities problem involving monthly deposits of $400 into an account with 6% interest compounded monthly over 30 years. Learn how to calculate the future value, total amount deposited, and total interest earned using key algebraic formulas. Suitable for Grades 10-12.

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