To determine how old Mathew will be when he is able to retire, we can use the future value formula for an annuity compounded monthly. The formula is as follows:

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

Where:

• $FV$ = Future value of the annuity (Mathew's goal is $1,000,000)
• $P$ = Monthly deposit ($290)
• $r$ = Annual interest rate (4% or 0.04)
• $n$ = Number of times the interest is compounded per year (12 for monthly compounding)
• $t$ = Number of years Mathew will invest
• $t \times 12$ = Total number of monthly contributions

We need to solve for $t$, the number of years it will take Mathew to accumulate $1,000,000. Let's first rearrange the formula to solve for $t$:

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

I'll calculate $t$, and then add this number of years to Mathew's starting age of 25 to find out how old he will be when he retires. Let's compute it.Mathew will be approximately 88.24 years old when he reaches $1,000,000 in his retirement account.

Would you like to dive deeper into how this was calculated or explore alternative scenarios?

Here are five related questions to consider:

1. What happens if Mathew increases his monthly deposit?
2. How does the interest rate affect the time to retirement?
3. What if Mathew aims for a smaller retirement fund, such as $750,000?
4. How would starting earlier impact the time needed to retire?
5. What if Mathew withdraws money periodically during retirement?

Tip: Small increases in monthly contributions can significantly shorten the time to reach your savings goal due to the compounding effect.