\begin{tabular}{l|l|} How much should you deposit at the end & In order to have $$ \$ 3 $$ million in 37 \ of each month into an investment account & years, you should deposit $$ \$ \square $$ \ that pays $$ 8 \% $$ compounded monthly to & each month. \ have $$ \$ 3 $$ million when you retire in 37 & (Do not round until the final \ years? How much of the $$ \$ 3 $$ million comes & answer. Then round up to the \ from interest? & nearest dollar.) \end{tabular}

Question

UpStudy AI · Accepted Answer

To solve this problem, we need to determine how much should be deposited at the end of each month into an investment account that pays $$ 8\% $$ interest compounded monthly, in order to accumulate $$ \$ 3 $$ million in $$ 37 $$ years.

### Step 1: Identify the known values
- Future Value ($$ FV $$): $$ 3,000,000 $$
- Annual Interest Rate ($$ r $$): $$ 8\% $$ or $$ 0.08 $$
- Monthly Interest Rate ($$ r_m $$): $$ \frac{0.08}{12} $$
- Total Number of Deposits ($$ n $$): $$ 37 \times 12 $$

### Step 2: Calculate the monthly interest rate and total number of deposits
1. Calculate the monthly interest rate:
   $$
   r_m = \frac{0.08}{12}
   $$

2. Calculate the total number of deposits:
   $$
   n = 37 \times 12
   $$

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

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

### Step 4: Calculate $$ P $$
Now we will calculate $$ r_m $$, $$ n $$, and then $$ P $$.

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$37\times 12$$
- step1: Multiply the numbers:
  $$444$$
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 $$ 3000000*(0.08/12)/((1+(0.08/12))^(37*12)-1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{3000000\left(\frac{0.08}{12}\right)}{\left(\left(1+\left(\frac{0.08}{12}\right)\right)^{37\times 12}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{3000000\left(\frac{0.08}{12}\right)}{\left(1+\left(\frac{0.08}{12}\right)\right)^{37\times 12}-1}$$
- step2: Divide the terms:
  $$\frac{3000000\left(\frac{0.08}{12}\right)}{\left(1+\frac{1}{150}\right)^{37\times 12}-1}$$
- step3: Add the numbers:
  $$\frac{3000000\left(\frac{0.08}{12}\right)}{\left(\frac{151}{150}\right)^{37\times 12}-1}$$
- step4: Divide the terms:
  $$\frac{3000000\times \frac{1}{150}}{\left(\frac{151}{150}\right)^{37\times 12}-1}$$
- step5: Multiply the numbers:
  $$\frac{3000000\times \frac{1}{150}}{\left(\frac{151}{150}\right)^{444}-1}$$
- step6: Multiply the numbers:
  $$\frac{20000}{\left(\frac{151}{150}\right)^{444}-1}$$
- step7: Subtract the numbers:
  $$\frac{20000}{\frac{151^{444}-150^{444}}{150^{444}}}$$
- step8: Multiply by the reciprocal:
  $$20000\times \frac{150^{444}}{151^{444}-150^{444}}$$
- step9: Multiply:
  $$\frac{20000\times 150^{444}}{151^{444}-150^{444}}$$
### Step 5: Calculate the monthly deposit $$ P $$

1. **Monthly Interest Rate**:
   $$
   r_m = \frac{0.08}{12} \approx 0.00666667
   $$

2. **Total Number of Deposits**:
   $$
   n = 37 \times 12 = 444
   $$

3. **Calculate $$ P $$**:
   Using the rearranged formula:
   $$
   P = \frac{3,000,000 \times 0.00666667}{(1 + 0.00666667)^{444} - 1}
   $$

After performing the calculation, we find:
   $$
   P \approx 1104.40
   $$

### Step 6: Round the final answer
The amount to deposit each month, rounded to the nearest dollar, is:
$$
P \approx 1104
$$

### Step 7: Calculate the total amount deposited and the interest earned
1. **Total Amount Deposited**:
   $$
   \text{Total Deposits} = P \times n = 1104 \times 444
   $$

2. **Total Interest Earned**:
   $$
   \text{Interest} = FV - \text{Total Deposits}
   $$

Let's calculate these values.
Calculate the value by following steps:
- step0: Calculate:
  $$3000000-\left(1104\times 444\right)$$
- step1: Multiply the numbers:
  $$3000000-490176$$
- step2: Subtract the numbers:
  $$2509824$$
Calculate or simplify the expression $$ 1104*444 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1104\times 444$$
- step1: Multiply the numbers:
  $$490176$$
### Step 8: Calculate Total Deposits and Interest Earned

1. **Total Amount Deposited**:
   $$
   \text{Total Deposits} = 1104 \times 444 = 490176
   $$

2. **Total Interest Earned**:
   $$
   \text{Interest} = 3,000,000 - 490176 = 2509824
   $$

### Summary of Results
- **Monthly Deposit**: $$ \$ 1104 $$
- **Total Amount Deposited**: $$ \$ 490176 $$
- **Total Interest Earned**: $$ \$ 2509824 $$

Thus, to accumulate $$ \$ 3 $$ million in $$ 37 $$ years with an $$ 8\% $$ interest rate compounded monthly, you should deposit approximately $$ \$ 1104 $$ at the end of each month. The total interest earned from this investment will be approximately $$ \$ 2,509,824 $$.
To accumulate \$3,000,000 in 37 years with an 8% interest rate compounded monthly, you should deposit approximately \$1,104 at the end of each month. The total interest earned from this investment will be approximately \$2,509,824.

Real Tutor Solution

Answer

Solution

Mind Expander

Related Questions

Latest Economics Questions