You want to be able to withdraw $50,000 each year for 20 years. Your account earns 10% 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?

To solve this problem, we need to calculate the present value of an annuity for part (a), the total amount withdrawn over 20 years for part (b), and the interest earned over that time for part (c).

### Given:
- Annual withdrawal, $ PMT = 50,000 $
- Interest rate, $ r = 10\% = 0.10 $
- Number of years, $ n = 20 $

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

The amount needed initially is the present value of an annuity, which is calculated using the formula:
\[
PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}
\]
Substituting the given values:
\[
PV = 50,000 \times \frac{1 - (1 + 0.10)^{-20}}{0.10}
\]

Calculating this:
1. $ (1 + 0.10)^{-20} = (1.10)^{-20} \approx 0.1486 $
2. $ 1 - 0.1486 \approx 0.8514 $
3. $ 0.8514 / 0.10 \approx 8.514 $
4. $ PV = 50,000 \times 8.514 \approx 425,700 $

So, you need **$425,700** in your account at the beginning.

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

The total amount withdrawn over 20 years is simply:
\[
\text{Total Withdrawn} = PMT \times n = 50,000 \times 20 = 1,000,000
\]

Thus, the total money pulled out is **$1,000,000**.

### (c) How much of that money is interest?

The interest earned is the difference between the total amount withdrawn and the initial amount needed.
\[
\text{Interest} = \text{Total Withdrawn} - PV = 1,000,000 - 425,700 = 574,300
\]

Therefore, the total interest is **$574,300**.

---

### Summary of Answers:
- **(a)** Amount needed at the beginning: **$425,700**
- **(b)** Total money pulled out: **$1,000,000**
- **(c)** Total interest: **$574,300**

Would you like further details or have any questions?

### Related Questions:
1. How would the initial amount change if the interest rate were higher?
2. What is the future value of the withdrawals if calculated as a growing annuity?
3. How would the total interest differ with a shorter withdrawal period?
4. What happens if you increase the annual withdrawal amount?
5. How would compounding monthly instead of annually affect the calculations?

### Tip:
When dealing with annuities, remember that the present value calculation is useful for understanding how much is needed to meet future withdrawal goals.

You want to be able to withdraw $50,000 each year for 20 years. Your account earns 10% interest.

Discover how to calculate the necessary initial amount to withdraw $50,000 each year for 20 years at a 10% interest rate. This guide covers present value of annuities, total amount withdrawn, and interest earned. Essential for finance students and professionals.

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