To solve this problem, let's break down the information given and use the future value of an annuity formula.

Problem Summary:

• Matthew deposits $910 each month into an account with an annual interest rate of 8%, compounded monthly.
• He will retire when the account balance reaches $1,000,000.
• We need to determine Matthew's age when he can retire.

Key Information and Formula:

The future value of an annuity formula for monthly deposits with compound interest is:

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

where:

• $FV$ is the future value of the annuity (in this case, $1,000,000).
• $P$ is the monthly deposit ($910).
• $r$ is the monthly interest rate, calculated by dividing the annual interest rate by 12 (since it's compounded monthly).
  • Here, $r = \frac{8\%}{12} = 0.0066667$.
• $n$ is the total number of deposits (months), which we need to find.

Rearrange the formula to solve for $n$:

$n = \frac{\ln\left(\frac{FV \cdot r}{P} + 1\right)}{\ln(1 + r)}$

Steps:

1. Substitute the values:

   • $FV = 1,000,000$
   • $P = 910$
   • $r = 0.0066667$

2. Solve for $n$, which will give us the number of months needed.

3. Convert $n$ to years by dividing by 12, then add that to Matthew's current age (25) to determine his retirement age.

I’ll proceed with the calculations to find the result.import math

Given values

FV = 1000000 # future value P = 910 # monthly deposit annual_interest_rate = 0.08 # annual interest rate r = annual_interest_rate / 12 # monthly interest rate current_age = 25 # Matthew's current age

Calculate the number of months needed

n = math.log((FV * r / P) + 1) / math.log(1 + r)

Convert months to years and add to current age

years_until_retirement = n / 12 retirement_age = current_age + years_until_retirement

Round to the nearest year for the answer

retirement_age_rounded = round(retirement_age)

retirement_age_rounded5252successMatthew will be 52 years old when he is able to retire, given his monthly contributions and the specified interest rate.

Would you like further details or have any questions?

Here are some related questions that expand on this topic:

1. How would a change in the interest rate affect Matthew's retirement age?
2. What if Matthew decided to deposit a larger amount monthly? How would that impact his retirement timeline?
3. How can inflation affect the purchasing power of the $1,000,000 retirement goal?
4. What would happen if the compounding frequency changed (e.g., to quarterly or annually)?
5. How does the future value formula apply to different types of annuities, like those with fixed or variable interest?

Tip: Regularly adjusting contributions based on annual reviews can help meet retirement goals faster or adjust for life changes.