Suppose you want to have $400,000 for retirement in 35 years. Your account earns 8% interest. Round your answers to the nearest cent.

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

$

b) How much interest will you earn?

$

This problem involves calculating how much you need to deposit each month into a retirement account earning 8% interest in order to have $400,000 in 35 years. This is a typical future value of an annuity problem, where regular monthly deposits accumulate with compound interest.

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

We can use the future value of an annuity formula:

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

Where:
- $ FV $ is the future value ($400,000),
- $ P $ is the monthly deposit,
- $ r $ is the monthly interest rate,
- $ n $ is the total number of deposits.

#### Step 1: Identify the known values.
- Annual interest rate = 8%, so the monthly interest rate is $ \frac{8\%}{12} = \frac{0.08}{12} = 0.0066667 $.
- Time = 35 years, so the number of months is $ 35 \times 12 = 420 $.

We need to solve for $ P $. Rearranging the formula:

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

Substituting the known values:

\[
P = \frac{400,000 \times 0.0066667}{(1 + 0.0066667)^{420} - 1}
\]

Let me calculate that for you.

The monthly deposit you would need to make in order to accumulate $400,000 in 35 years is approximately **$174.38**.

### b) How much interest will you earn?

To find the total interest earned, we can first calculate the total amount deposited and then subtract this from the future value.

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

Let's calculate this next.The total interest you will earn over the 35 years is approximately **$326,761.72**.

### Summary:
- **Monthly deposit**: $174.38
- **Total interest earned**: $326,761.72

Would you like more details on the process or have any questions?

Here are 5 related questions you might want to explore:
1. How would the monthly deposit change if the interest rate were 7% instead of 8%?
2. What if you wanted to accumulate $500,000 instead of $400,000?
3. How would the calculation differ if you made quarterly deposits instead of monthly?
4. What if the interest was compounded yearly instead of monthly?
5. How would your total interest change if you deposited a lump sum at the start?

**Tip:** When planning for retirement, even small changes in interest rates or contribution amounts can significantly affect your future savings.

Suppose you want to have $400,000 for retirement in 35 years. Your account earns 8% interest. Round your answers to the nearest cent.

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

b) How much interest will you earn?

Learn how to calculate the monthly deposit required to accumulate $400,000 in 35 years with an 8% interest rate. This example involves the future value of an annuity formula and demonstrates the compound interest calculation. Suitable for high school and college-level students.

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