Suppose you want to have $$ \$ 800,000 $$ for retirement in 20 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?

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To solve this problem, we will use the future value of an annuity formula to determine how much you need to deposit each month to reach your retirement goal of $$ \$800,000 $$ in 20 years with an interest rate of $$ 8\% $$.

### Known Conditions:
- Future Value (FV) = $$ 800,000 $$
- Annual Interest Rate (r) = $$ 8\% = 0.08 $$
- Number of Years (t) = $$ 20 $$
- Number of Compounding Periods per Year (n) = $$ 12 $$ (monthly deposits)

### Step 1: Calculate Monthly Interest Rate
The monthly interest rate (i) can be calculated as:
$$
i = \frac{r}{n} = \frac{0.08}{12}
$$

### Step 2: Calculate Total Number of Deposits
The total number of deposits (N) over 20 years is:
$$
N = n \times t = 12 \times 20
$$

### Step 3: Use the Future Value of an Annuity Formula
The future value of an annuity formula is given by:
$$
FV = P \times \frac{(1 + i)^N - 1}{i}
$$
Where:
- $$ P $$ = monthly deposit
- $$ FV $$ = future value

We need to rearrange this formula to solve for $$ P $$:
$$
P = \frac{FV \times i}{(1 + i)^N - 1}
$$

Now, let's calculate the values step by step.

1. Calculate $$ i $$
2. Calculate $$ N $$
3. Calculate $$ P $$

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$12\times 20$$
- step1: Multiply the numbers:
  $$240$$
Calculate or simplify the expression $$ 0.08/12 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.08}{12}$$
- step1: Convert the expressions:
  $$\frac{\frac{2}{25}}{12}$$
- step2: Multiply by the reciprocal:
  $$\frac{2}{25}\times \frac{1}{12}$$
- step3: Reduce the numbers:
  $$\frac{1}{25}\times \frac{1}{6}$$
- step4: Multiply the fractions:
  $$\frac{1}{25\times 6}$$
- step5: Multiply:
  $$\frac{1}{150}$$
Calculate or simplify the expression $$ 800000*(0.00666666667)/((1+0.00666666667)^240-1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{800000\times \frac{666666667}{100000000000}}{\left(\left(1+\frac{666666667}{100000000000}\right)^{240}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{800000\times \frac{666666667}{100000000000}}{\left(1+\frac{666666667}{100000000000}\right)^{240}-1}$$
- step2: Add the numbers:
  $$\frac{800000\times \frac{666666667}{100000000000}}{\left(\frac{100666666667}{100000000000}\right)^{240}-1}$$
- step3: Multiply the numbers:
  $$\frac{\frac{666666667}{125000}}{\left(\frac{100666666667}{100000000000}\right)^{240}-1}$$
- step4: Subtract the numbers:
  $$\frac{\frac{666666667}{125000}}{\frac{100666666667^{240}-100000000000^{240}}{100000000000^{240}}}$$
- step5: Multiply by the reciprocal:
  $$\frac{666666667}{125000}\times \frac{100000000000^{240}}{100666666667^{240}-100000000000^{240}}$$
- step6: Rewrite the expression:
  $$\frac{666666667}{125000}\times \frac{312500000^{240}\times 320^{240}}{100666666667^{240}-100000000000^{240}}$$
- step7: Rewrite the expression:
  $$\frac{666666667}{50^{3}}\times \frac{50^{1200}\times 320^{240}}{100666666667^{240}-100000000000^{240}}$$
- step8: Reduce the numbers:
  $$666666667\times \frac{50^{1197}\times 320^{240}}{100666666667^{240}-100000000000^{240}}$$
- step9: Multiply:
  $$\frac{666666667\times 50^{1197}\times 320^{240}}{100666666667^{240}-100000000000^{240}}$$
### Step 1: Calculate Monthly Interest Rate
The monthly interest rate $$ i $$ is:
$$
i = \frac{0.08}{12} \approx 0.00666666667
$$

### Step 2: Calculate Total Number of Deposits
The total number of deposits $$ N $$ is:
$$
N = 12 \times 20 = 240
$$

### Step 3: Calculate Monthly Deposit $$ P $$
Using the future value of an annuity formula:
$$
P = \frac{800000 \times 0.00666666667}{(1 + 0.00666666667)^{240} - 1}
$$

Calculating this gives:
$$
P \approx 1358.19
$$

### Answer for Part (a)
You would need to deposit approximately $$ \$1358.19 $$ each month.

### Step 4: Calculate Total Amount Deposited
To find out how much interest you will earn, we first need to calculate the total amount deposited over 20 years:
$$
\text{Total Deposits} = P \times N = 1358.19 \times 240
$$

### Step 5: Calculate Total Interest Earned
The total interest earned can be calculated as:
$$
\text{Interest} = FV - \text{Total Deposits}
$$

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$1358.19\times 240$$
- step1: Multiply the numbers:
  $$325965.6$$
Calculate or simplify the expression $$ 800000 - (1358.19*240) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$800000-\left(1358.19\times 240\right)$$
- step1: Multiply the numbers:
  $$800000-325965.6$$
- step2: Subtract the numbers:
  $$474034.4$$
### Step 4: Calculate Total Amount Deposited
The total amount deposited over 20 years is:
$$
\text{Total Deposits} = 1358.19 \times 240 \approx 325965.60
$$

### Step 5: Calculate Total Interest Earned
The total interest earned is:
$$
\text{Interest} = 800000 - 325965.60 \approx 474034.40
$$

### Final Answers
- **(a)** You would need to deposit approximately **\$1358.19** each month.
- **(b)** The total interest you will earn is approximately **\$474034.40**.
You need to deposit approximately \$1,358.19 each month. Over 20 years, you will earn about \$474,034.40 in interest.

Suppose you want to have \( \$ 800,000 \) for retirement in 20 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?

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