Since he was 20 years old, Ben has been depositing $225 at the end of each month into a tax-free retirement account earning interest at the rate of 4.5%/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 use the formula for the future value of an annuity. The future value of an annuity formula is:

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

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

### Step 1: Calculate Ben's Future Value
First, let's calculate the future value of Ben's account at age 65.

**Given:**
- Monthly deposit ($P$) = $225
- Annual interest rate = 4.5% → Monthly interest rate $r = \frac{4.5\%}{12} = 0.00375$
- Number of years Ben is investing = 65 - 20 = 45 years
- Total number of deposits ($n$) = 45 years × 12 months/year = 540 months

Using the future value formula:

\[
FV_B = 225 \times \frac{(1 + 0.00375)^{540} - 1}{0.00375}
\]

### Step 2: Determine the Amount Larry Needs to Deposit
Larry starts 5 years later, so he has only 40 years to invest.

**Given:**
- Number of years Larry is investing = 65 - 25 = 40 years
- Total number of deposits ($n_L$) = 40 years × 12 months/year = 480 months

Larry's future value $FV_L$ must be equal to Ben's future value $FV_B$.

So, we need to find $P_L$ such that:

\[
FV_L = P_L \times \frac{(1 + 0.00375)^{480} - 1}{0.00375} = FV_B
\]

### Step 3: Calculate Larry's Required Monthly Deposit
To find $P_L$, we rearrange the equation:

\[
P_L = \frac{FV_B \times 0.00375}{(1 + 0.00375)^{480} - 1}
\]

Now, let's calculate this step-by-step.

First, I'll calculate the future value for Ben, and then find the monthly deposit $P_L$ for Larry.Larry should deposit approximately **$292.92** each month into his retirement account to have the same amount of money as Ben at age 65.

Do you want any more details, or do you have any questions about this calculation?

Here are some related questions you might find interesting:
1. What would happen if Ben increased his monthly contributions by 1% each year?
2. How would the outcome change if the interest rate was compounded annually instead of monthly?
3. What is the effect of delaying the start of retirement savings by 10 years instead of 5 years?
4. How much less would Ben have if he contributed for 40 years instead of 45?
5. What if Larry contributed $225 per month but wanted to catch up by age 70 instead of 65?

**Tip:** Starting retirement savings early significantly impacts the future value of your investments due to compound interest.

Learn how to calculate retirement savings with compound interest using the future value of annuity formula. This problem compares the savings of two individuals depositing into tax-free retirement accounts with monthly contributions and compound interest. Suitable for high school students studying financial mathematics.

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