An ordinary annuity pays 8.04​% compounded monthly.
​(A) A person deposits ​$100 monthly for 30 years and then makes equal monthly withdrawals for the next 15​ years, reducing the balance to zero. What are the monthly​ withdrawals? How much interest is earned during the entire​ 45-year process?
​(B) If the person wants to make withdrawals of 2,500 per month for the last 15​ years, how much must be deposited monthly for the first 30​ years?
Question content area bottom
Part 1
​(A) The monthly withdrawals are ​$enter your response here. 
​(Round to the nearest cent as​ needed.) 

To solve this problem, we will approach it step by step.

### Part A:
We need to calculate two things:
1. **The monthly withdrawals** after depositing $100 per month for 30 years at an 8.04% annual interest rate compounded monthly.
2. **The total interest earned** over the 45-year period.

#### Step 1: Calculate the Future Value (FV) of the deposits after 30 years.

The formula for the future value of an ordinary annuity is:
\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]
where:
- $ P = 100 $ is the monthly deposit,
- $ r = \frac{0.0804}{12} $ is the monthly interest rate,
- $ n = 30 \times 12 = 360 $ is the total number of payments.

#### Step 2: Use the future value as the present value (PV) for withdrawals.
We then use this future value to calculate the monthly withdrawals over 15 years. The formula for the present value of an annuity is:
\[
PV = W \times \frac{1 - (1 + r)^{-n}}{r}
\]
where:
- $ W $ is the monthly withdrawal amount,
- $ r = \frac{0.0804}{12} $,
- $ n = 15 \times 12 = 180 $.

We will calculate both the future value and the monthly withdrawal amount now.After 30 years of depositing $100 per month, the future value (the amount accumulated) is approximately **$150,251.75**.

The monthly withdrawals for the next 15 years, assuming the account is reduced to zero, will be approximately **$1,439.36**.

#### Step 3: Calculate the total interest earned.

To find the total interest, we calculate:
\[
\text{Total Interest} = \text{Total Withdrawals} + \text{Total Deposits} - \text{Future Value}
\]
Where:
- Total deposits = $100 \times 360 = 36,000$
- Total withdrawals = $1,439.36 \times 180$ 

Let's calculate the total interest earned.The total interest earned during the entire 45-year process is approximately **$223,084.03**.

### Summary for Part A:
- The monthly withdrawals are **$1,439.36**.
- The total interest earned over the 45-year period is **$223,084.03**.

---

### Part B:
If the person wants to withdraw **$2,500 per month** for the last 15 years, we need to calculate how much must be deposited monthly for the first 30 years.

We use the same process, but now the withdrawal amount is given.

#### Step 1: Calculate the present value (PV) for $2,500 monthly withdrawals.
\[
PV = 2500 \times \frac{1 - (1 + r)^{-n}}{r}
\]
This PV is the future value of the deposits after 30 years.

#### Step 2: Use the PV to calculate the monthly deposit.

We will now calculate how much should be deposited monthly.To achieve monthly withdrawals of **$2,500** for the last 15 years, the person must deposit approximately **$173.69** per month for the first 30 years.

---

### Summary for Part B:
- The required monthly deposit for the first 30 years is **$173.69** to allow for $2,500 withdrawals for the following 15 years.

---

Do you want further details or explanations on any part of this solution? Here are 5 follow-up questions related to this topic:

1. How does changing the interest rate affect the monthly withdrawals?
2. What would happen if the compounding frequency changed to quarterly?
3. How would the outcome differ if the deposits were made at the beginning of the period (annuity due)?
4. Can you show the total amount of withdrawals in Part B?
5. What happens if the withdrawal period is extended beyond 15 years?

**Tip:** The power of compound interest becomes significant over long periods, making even small increases in deposit amounts impactful.

An ordinary annuity pays 8.04% compounded monthly. (A) A person deposits $100 monthly for 30 years and then makes equal monthly withdrawals for the next 15 years, reducing the balance to zero. What are the monthly withdrawals? How much interest is earned during the entire 45-year process? (B) If the person wants to make withdrawals of 2,500 per month for the last 15 years, how much must be deposited monthly for the first 30 years?

Learn how to calculate the monthly withdrawals and total interest earned for an ordinary annuity with an 8.04% interest rate, compounded monthly. A step-by-step solution covers deposits over 30 years and withdrawals over 15 years. Additionally, find out how to determine the required monthly deposits for a $2,500 withdrawal plan.

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