Celine wishes to retire at age 67
67
 with $1,900,000
$
⁢
1,900,000
 in her retirement account. When she turns 22
22
, she decides to begin depositing money into an account with an APR of 7%
7
%
 compounded monthly. What is the monthly deposit that Celine must make in order to reach her goal? Round your answer to the nearest cent, if necessary.

To determine the monthly deposit Celine must make to reach her goal, we can use the **future value of an ordinary annuity formula**. This formula calculates the future value of a series of regular payments (deposits) given an interest rate and a number of compounding periods. The formula is:

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

Where:
- $ FV $ is the future value of the account (Celine's goal of $1,900,000).
- $ P $ is the monthly deposit.
- $ r $ is the monthly interest rate (annual rate divided by 12).
- $ n $ is the total number of payments (compounding periods).

### Step 1: Define the variables
- **Future Value (FV)**: $1,900,000
- **APR (Annual Interest Rate)**: 7% = 0.07 (as a decimal)
- **Compounding frequency**: Monthly, so the monthly interest rate $ r = \frac{0.07}{12} = 0.0058333 $
- **Time until retirement**: 67 - 22 = 45 years
- **Total number of payments (n)**: 45 years × 12 months/year = 540 months

### Step 2: Plug into the formula
We need to solve for $ P $ (the monthly deposit):

\[
1,900,000 = P \times \frac{(1 + 0.0058333)^{540} - 1}{0.0058333}
\]

Simplifying the right-hand side:

\[
1,900,000 = P \times \frac{(1.0058333)^{540} - 1}{0.0058333}
\]

Calculating $ (1.0058333)^{540} $:

\[
(1.0058333)^{540} \approx 24.114
\]

Now, substituting this back into the formula:

\[
1,900,000 = P \times \frac{24.114 - 1}{0.0058333}
\]

\[
1,900,000 = P \times \frac{23.114}{0.0058333}
\]

\[
1,900,000 = P \times 3962.788
\]

### Step 3: Solve for $ P $
\[
P = \frac{1,900,000}{3962.788} \approx 479.46
\]

### Conclusion
Celine must make a monthly deposit of approximately **$479.46** to reach her goal of $1,900,000 by age 67.

Would you like further details on any step, or do you have any questions?

Here are 5 related questions:
1. What if Celine started saving 5 years later, at age 27? How would this affect her monthly deposit?
2. How does changing the interest rate to 6% affect the monthly deposit?
3. What would happen if Celine contributed a lump sum initially? How would that change her monthly payments?
4. How is compound interest different from simple interest in retirement planning?
5. What happens to the future value if the deposits are made at the beginning of each month instead of the end?

**Tip:** The earlier you start saving for retirement, the less you need to save monthly due to the power of compound interest over time.

Celine wishes to retire at age 67 with $1,900,000 in her retirement account. When she turns 22, she decides to begin depositing money into an account with an APR of 7% compounded monthly. What is the monthly deposit that Celine must make in order to reach her goal? Round your answer to the nearest cent, if necessary.

Find out how much Celine needs to deposit monthly to reach a $1,900,000 retirement goal by age 67 with a 7% APR compounded monthly. This problem covers compound interest and the future value of annuities, providing a detailed step-by-step solution. Ideal for students in Grades 10-12.

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