Suppose you want to have $800,000 for retirement in 30 years. Your account earns 5% 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?

To solve this problem, we'll use the **future value of an annuity formula** to determine how much needs to be deposited each month and how much interest will be earned over time.

### a) Monthly Deposit

The formula for the future value of an annuity is:

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

Where:
- $FV$ is the future value (the amount you want in the future, which is $800,000),
- $P$ is the monthly deposit (what we need to find),
- $r$ is the monthly interest rate (annual interest rate divided by 12),
- $n$ is the total number of months (years × 12).

#### Step 1: Determine values for $r$ and $n$
- Annual interest rate = 5%, so the monthly interest rate $r$ is:
  \[
  r = \frac{5\%}{12} = \frac{0.05}{12} \approx 0.004167
  \]
- Number of months $n$ is:
  \[
  n = 30 \times 12 = 360
  \]

#### Step 2: Solve for $P$

We know $FV = 800,000$, so plug everything into the formula and solve for $P$:

\[
800,000 = P \times \frac{(1 + 0.004167)^{360} - 1}{0.004167}
\]

First, calculate the term inside the parentheses:

\[
(1 + 0.004167)^{360} \approx 4.467744314
\]

Now, subtract 1 and divide by 0.004167:

\[
\frac{4.467744314 - 1}{0.004167} \approx 832.426489
\]

Now solve for $P$:

\[
800,000 = P \times 832.426489
\]

\[
P = \frac{800,000}{832.426489} \approx 961.02
\]

So, the required monthly deposit is approximately **$961.02**.

### b) Interest Earned

To calculate the interest earned, we first need to find the total amount deposited over 30 years, then subtract that from the future value.

#### Step 1: Total Amount Deposited

The total amount deposited is:

\[
\text{Total deposited} = P \times n = 961.02 \times 360 = 345,967.20
\]

#### Step 2: Interest Earned

Now subtract the total deposited from the future value to get the interest:

\[
\text{Interest} = FV - \text{Total deposited} = 800,000 - 345,967.20 = 454,032.80
\]

So, the interest earned is approximately **$454,032.80**.

---

Would you like more details on any part of this solution or any specific questions? Here are 5 related questions to consider:

1. What happens if the interest rate changes during the 30 years?
2. How would a different retirement target (e.g., $1,000,000) affect the monthly deposit?
3. What if deposits were made quarterly instead of monthly—how would that affect the calculations?
4. How is the future value of a lump-sum deposit calculated differently from an annuity?
5. What would happen if withdrawals were made during retirement instead of making deposits?

**Tip**: Always ensure you use the same time units for interest rates and payment periods when calculating future values or annuities.

Suppose you want to have $800,000 for retirement in 30 years. Your account earns 5% 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?

Calculate how much you need to deposit each month to have $800,000 in 30 years at a 5% interest rate. This problem uses the annuity formula and compound interest, providing a step-by-step solution. Learn how to find the interest earned over time. Suitable for Grades 11-12.

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