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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UpStudy AI · Accepted Answer

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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