Since he was 21 years old, Ben has been depositing $200 at the end of each month into a tax-free retirement account earning interest at the rate of 5.5%/year compounded monthly. Larry, who is the same age as Ben, decided to open a tax-free retirement account 5 years after Ben opened his. If Larry's account earns interest at the same rate as Ben's, determine how much Larry should deposit each month into his account so that both men will have the same amount of money in their accounts at age 65. (Round your answer to the nearest cent.)

To solve this problem, we need to determine how much Larry should deposit each month into his retirement account so that he has the same amount of money as Ben when they both reach the age of 65. 

### Step 1: Calculate the amount in Ben's account at age 65

**Given:**
- Ben deposits $200 per month.
- Interest rate = 5.5% per year compounded monthly.
- Ben starts depositing at age 21.
- Ben deposits for $ 65 - 21 = 44 $ years.
- The monthly interest rate $ r = \frac{5.5\%}{12} = \frac{0.055}{12} $.

The formula for the future value of an annuity (which is a series of equal payments made at regular intervals) is:

\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- $ P $ is the monthly payment.
- $ r $ is the monthly interest rate.
- $ n $ is the total number of payments.

For Ben:
- $ P = 200 $
- $ r = \frac{0.055}{12} \approx 0.00458333 $
- $ n = 44 \times 12 = 528 $

Substituting these values:

\[
FV_{\text{Ben}} = 200 \times \frac{(1 + 0.00458333)^{528} - 1}{0.00458333}
\]

### Step 2: Calculate the amount Larry needs to deposit

Larry starts depositing 5 years after Ben, so he will be depositing for $ 44 - 5 = 39 $ years.

Let $ P_{\text{Larry}} $ be the amount Larry needs to deposit each month. For Larry:

- $ n = 39 \times 12 = 468 $

We need to solve for $ P_{\text{Larry}} $ in the equation:

\[
FV_{\text{Larry}} = P_{\text{Larry}} \times \frac{(1 + 0.00458333)^{468} - 1}{0.00458333}
\]

We know that $ FV_{\text{Larry}} $ should equal $ FV_{\text{Ben}} $. Therefore:

\[
P_{\text{Larry}} \times \frac{(1 + 0.00458333)^{468} - 1}{0.00458333} = 200 \times \frac{(1 + 0.00458333)^{528} - 1}{0.00458333}
\]

We can solve this equation for $ P_{\text{Larry}} $:

\[
P_{\text{Larry}} = 200 \times \frac{(1 + 0.00458333)^{528} - 1}{(1 + 0.00458333)^{468} - 1}
\]

### Step 3: Calculate $ P_{\text{Larry}} $

Let's calculate this value now.Larry should deposit **$271.56** per month to ensure that his retirement account will have the same amount as Ben's at age 65, assuming both accounts earn interest at the same rate. 

Would you like more details on any part of this calculation or have any other questions?

Here are five related questions you might find interesting:
1. How would the required monthly deposit change if Larry started 10 years later instead of 5?
2. What would happen if the interest rate were different for Larry’s account?
3. How much money will both Ben and Larry have at age 65?
4. What if the contributions were made at the beginning of each month instead of the end?
5. How would inflation affect the value of their retirement savings over time?

**Tip:** Starting to save early for retirement significantly reduces the amount needed each month due to the power of compound interest.

Learn how to calculate monthly deposits for retirement accounts earning compound interest, ensuring equal savings by age 65. This problem involves the future value of annuities and compound interest calculations. Suitable for college-level students and anyone interested in financial planning.

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