To calculate how much the IRA will contain at retirement, we need to use the future value formula for a series of monthly contributions, which is given by the formula for the future value of an annuity due:

$FV = P \times \left( \frac{{(1 + r)^n - 1}}{r} \right)$

Where:

• $FV$ is the future value of the IRA at retirement
• $P$ is the monthly deposit ($45)
• $r$ is the monthly interest rate (APR divided by 12) = 8% annual interest rate → $r = \frac{8\%}{12} = \frac{0.08}{12} = 0.006667$
• $n$ is the number of monthly deposits (since the person starts at 24 and retires at 65, the number of months is $n = (65 - 24) \times 12 = 41 \times 12 = 492$)

Now, let's plug in the values into the formula:

$FV = 45 \times \left( \frac{{(1 + 0.006667)^{492} - 1}}{0.006667} \right)$

We can calculate the future value step by step.

Let’s calculate this value:

$FV = 45 \times \left( \frac{{(1.006667)^{492} - 1}}{0.006667} \right)$

First, calculate $(1.006667)^{492}$, then subtract 1, divide by 0.006667, and multiply by 45.

Let me compute that for you. Given values P = 45 # monthly deposit in dollars r = 0.006667 # monthly interest rate n = 492 # number of months (41 years * 12 months)

Calculating the future value of the IRA

FV = P * (((1 + r) ** n - 1) / r) FVThe IRA will contain approximately $170,714.81 when the person retires at age 65.

Total Deposits:

To calculate the total deposits made over the period, we simply multiply the monthly deposit by the number of months:

$\text{Total Deposits} = P \times n = 45 \times 492 = 22,140$

So, the total deposits over the 41 years will be $22,140.

Comparison:

• Total Deposits: $22,140
• Future Value at Retirement: $170,714.81

The difference, or the growth due to interest, is approximately $148,574.81.

Would you like more details or have any questions about this calculation?

Here are a few questions that could expand your understanding:

1. How does compounding interest affect the growth of savings over time?
2. What would the future value be if the interest rate were increased to 10%?
3. How can adjusting the monthly deposit amount affect the final value?
4. What are the key factors in choosing the right retirement account?
5. How does the time horizon impact the total amount saved for retirement?

Tip: Starting to save early and regularly, even with small amounts, can have a huge impact on your retirement savings due to the power of compound interest.