This question involves calculating the present value of an annuity because you want to withdraw a fixed amount each month over a certain period, with the account earning interest.

Let's break down the solution:

Given:

• Monthly Withdrawal ($PMT$) = $1,500
• Interest Rate = 6% annually, or 0.5% monthly (since 6% / 12 months = 0.5% per month)
• Time Period = 25 years (300 months, since $25 \times 12 = 300$ months)

(a) How much do you need in your account at the beginning?

This question is asking for the present value of an annuity. The present value formula for an annuity is:

$PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}$

where:

• $PMT$ = monthly payment (withdrawal) = $1,500
• $r$ = monthly interest rate = 0.5% = 0.005
• $n$ = total number of payments = 300

Plugging in the values:

$PV = 1500 \times \frac{1 - (1 + 0.005)^{-300}}{0.005}$

Calculating this gives:

$PV \approx 1,500 \times 166.7916 \approx 250,187.40$

So, you need approximately $250,187.40 in your account at the beginning.

(b) How much total money will you pull out of the account?

Total money withdrawn over 25 years:

$\text{Total Withdrawals} = PMT \times n = 1,500 \times 300 = 450,000$

So, you will pull out a total of $450,000.

(c) How much of that money is interest?

To find the total interest, subtract the initial amount needed from the total amount withdrawn:

$\text{Total Interest} = \text{Total Withdrawals} - PV = 450,000 - 250,187.40 = 199,812.60$

So, the total interest earned is approximately $199,812.60.

Would you like a deeper explanation or have any questions about these calculations?

Related Questions to Explore:

1. What would the initial amount be if the interest rate were different, say 5% annually?
2. How would the required initial balance change if you wanted to withdraw $2,000 each month instead?
3. If you extended the withdrawals to 30 years, how would that impact the initial balance needed?
4. What if you made withdrawals every six months instead of monthly—how would that affect the interest calculations?
5. How can compound interest affect the growth of this account if no withdrawals were made?

Tip:

When dealing with interest rates in annuities, always convert the annual rate to the appropriate period (e.g., monthly, quarterly) for accurate calculations.