How much should you deposit at the end of each month into an investment account that pays $$ 9 \% $$ compounded monthly to have $$ \$ 1 $$ million when you retire in 45 years? How much of the $$ \$ 1 $$ million comes from interest? Click the icon to view some finance formulas. In order to have $$ \$ 1 $$ million in 45 years, you should deposit $$ \$ \square $$ each month. (Round up to the nearest dollar.)

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To determine how much you should deposit at the end of each month into an investment account that pays $$ 9\% $$ compounded monthly to reach $$ \$1,000,000 $$ in 45 years, 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 of the annuity (the amount you want to have, which is $$ \$1,000,000 $$).
- $$ P $$ is the monthly deposit (the amount we want to find).
- $$ r $$ is the monthly interest rate (annual rate divided by 12).
- $$ n $$ is the total number of deposits (number of years multiplied by 12).

### Step 1: Calculate the monthly interest rate and total number of deposits

1. **Annual interest rate**: $$ 9\% = 0.09 $$
2. **Monthly interest rate**: 
   $$
   r = \frac{0.09}{12}
   $$
3. **Total number of deposits**: 
   $$
   n = 45 \times 12
   $$

### Step 2: Substitute values into the formula and solve for $$ P $$

Rearranging the formula to solve for $$ P $$:
$$
P = \frac{FV \times r}{(1 + r)^n - 1}
$$

Now, let's calculate the values for $$ r $$ and $$ n $$ and then find $$ P $$.

### Calculating $$ r $$ and $$ n $$

1. Calculate $$ r $$:
   $$
   r = \frac{0.09}{12}
   $$
2. Calculate $$ n $$:
   $$
   n = 45 \times 12
   $$

Now, I will perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$45\times 12$$
- step1: Multiply the numbers:
  $$540$$
Calculate or simplify the expression $$ 0.09/12 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.09}{12}$$
- step1: Convert the expressions:
  $$\frac{\frac{9}{100}}{12}$$
- step2: Multiply by the reciprocal:
  $$\frac{9}{100}\times \frac{1}{12}$$
- step3: Reduce the numbers:
  $$\frac{3}{100}\times \frac{1}{4}$$
- step4: Multiply the fractions:
  $$\frac{3}{100\times 4}$$
- step5: Multiply:
  $$\frac{3}{400}$$
The calculations yield the following results:

1. **Monthly interest rate** $$ r $$:
   $$
   r = 0.0075
   $$

2. **Total number of deposits** $$ n $$:
   $$
   n = 540
   $$

### Step 3: Substitute values into the formula to find $$ P $$

Now we can substitute $$ FV = 1,000,000 $$, $$ r = 0.0075 $$, and $$ n = 540 $$ into the rearranged formula for $$ P $$:

$$
P = \frac{1,000,000 \times 0.0075}{(1 + 0.0075)^{540} - 1}
$$

Let's calculate $$ (1 + 0.0075)^{540} $$ and then find $$ P $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(1+0.0075\right)^{540}$$
- step1: Add the numbers:
  $$1.0075^{540}$$
- step2: Convert the expressions:
  $$\left(\frac{403}{400}\right)^{540}$$
- step3: Use the properties of exponents:
  $$\frac{403^{540}}{400^{540}}$$
Calculate or simplify the expression $$ 1000000*0.0075/(56.536589 - 1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1000000\times 0.0075}{\left(56.536589-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{1000000\times 0.0075}{56.536589-1}$$
- step2: Multiply the numbers:
  $$\frac{7500}{56.536589-1}$$
- step3: Subtract the numbers:
  $$\frac{7500}{55.536589}$$
- step4: Convert the expressions:
  $$\frac{7500}{\frac{55536589}{1000000}}$$
- step5: Multiply by the reciprocal:
  $$7500\times \frac{1000000}{55536589}$$
- step6: Multiply:
  $$\frac{7500\times 1000000}{55536589}$$
- step7: Multiply:
  $$\frac{7500000000}{55536589}$$
Now we have the necessary calculations:

1. **Calculate $$ (1 + 0.0075)^{540} $$**:
   $$
   (1 + 0.0075)^{540} \approx 56.536589
   $$

2. **Substituting into the formula for $$ P $$**:
   $$
   P \approx \frac{1,000,000 \times 0.0075}{56.536589 - 1} \approx \frac{7500}{55.536589} \approx 135.046104
   $$

### Final Step: Round up to the nearest dollar

Thus, the monthly deposit $$ P $$ should be rounded up to the nearest dollar:
$$
P \approx 136
$$

### Conclusion

In order to have $$ \$1,000,000 $$ in 45 years, you should deposit **\$136** each month.

### Interest Calculation

To find out how much of the $$ \$1,000,000 $$ comes from interest, we first calculate the total amount deposited over 45 years:

$$
\text{Total Deposits} = P \times n = 136 \times 540
$$

Now, let's calculate the total deposits and then the interest earned.
Calculate the value by following steps:
- step0: Calculate:
  $$136\times 540$$
- step1: Multiply the numbers:
  $$73440$$
The total amount deposited over 45 years is:

$$
\text{Total Deposits} = 136 \times 540 = 73440
$$

### Step: Calculate the interest earned

Now, we can find out how much of the $$ \$1,000,000 $$ comes from interest:

$$
\text{Interest} = \text{Future Value} - \text{Total Deposits} = 1,000,000 - 73,440
$$

Let's calculate the interest earned.
Calculate the value by following steps:
- step0: Calculate:
  $$1000000-73440$$
- step1: Subtract the numbers:
  $$926560$$
The interest earned is:

$$
\text{Interest} = 1,000,000 - 73,440 = 926,560
$$

### Summary of Results

- To have $$ \$1,000,000 $$ in 45 years, you should deposit **\$136** each month.
- The amount that comes from interest is **\$926,560**.
You should deposit \$136 each month. The interest earned amounts to \$926,560.

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