You want to be able to withdraw $40,000 each year for 30 years. Your account earns 7% interest.

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

$

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

$

c) How much of that money is interest?

This problem involves calculating the present value of an annuity, which represents the amount you need to have in your account today to withdraw a fixed sum each year for a specified period, given an interest rate.

### Given Data:
- Annual withdrawal ($ P $): \$40,000
- Interest rate ($ r $): 7% or 0.07
- Number of years ($ n $): 30 years

#### Part (a): How much do you need in your account at the beginning?
To find the initial amount (present value) you need, we use the formula for the present value of an annuity:

\[
PV = P \times \left(1 - (1 + r)^{-n}\right) \div r
\]

Where:
- $ P $ is the annual withdrawal (\$40,000),
- $ r $ is the interest rate per period (7% or 0.07),
- $ n $ is the number of periods (30 years).

Now, let's calculate the present value.

#### Part (b): How much total money will you pull out of the account?
The total money withdrawn is simply:

\[
\text{Total Withdrawn} = P \times n = 40,000 \times 30
\]

#### Part (c): How much of that money is interest?
To find how much of the total withdrawal is interest, subtract the initial investment (from part a) from the total amount withdrawn:

\[
\text{Interest Earned} = \text{Total Withdrawn} - PV
\]

Let me calculate these values for you.### Results:

**(a)** The amount you need in your account at the beginning is approximately **\$496,361.65**.

**(b)** The total amount of money you will pull out of the account over 30 years is **\$1,200,000**.

**(c)** The amount of that total which is interest is approximately **\$703,638.35**.

Would you like further details on the calculations or have any questions?

Here are five related questions:
1. What if the interest rate were 6% instead of 7%?
2. How would the required initial amount change if you wanted to withdraw for 25 years instead of 30?
3. What if you wanted to withdraw \$50,000 per year instead of \$40,000?
4. How does the present value formula change if the interest compounds more frequently (e.g., monthly)?
5. What is the future value of the account if no withdrawals were made and it compounded for 30 years?

**Tip:** The more frequent the compounding, the higher the amount of interest earned, even at the same annual rate.

You want to be able to withdraw $40,000 each year for 30 years. Your account earns 7% interest. a) How much do you need in your account at the beginning? b) How much total money will you pull out of the account? c) How much of that money is interest?

Learn how to calculate the present value of an account to withdraw $40,000 annually for 30 years with a 7% interest rate. Discover how much money is needed initially, the total amount withdrawn, and how much of it is interest. Ideal for students in Grades 11-12 or finance-related studies.

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