On his 25th​ birthday, Matthew started an annuity for his retirement. He deposits ​$670 each month into an account earning 5​% annual interest compounded monthly. He determines that he will be able to retire once he has​ $1,000,000 in his retirement account. Determine how old Matthew will be when he is able to retire.

To solve this, we will use the formula for the future value of an annuity compounded monthly:

\[
A = P \times \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}}
\]

Where:
- $A$ is the future value (in this case, $1,000,000),
- $P$ is the monthly deposit ($670),
- $r$ is the annual interest rate (5%, or 0.05),
- $n$ is the number of times the interest is compounded per year (monthly, so $n = 12$),
- $t$ is the number of years.

We need to solve for $t$ (the number of years) Matthew needs to reach $1,000,000.

### Plugging in the known values:

\[
1,000,000 = 670 \times \frac{\left(1 + \frac{0.05}{12}\right)^{12t} - 1}{\frac{0.05}{12}}
\]

I'll calculate $t$, the number of years Matthew needs to reach $1,000,000.

Let me compute that.It seems there was an issue in the computation process. Let me recheck and recompute the formula to find the number of years required for Matthew to reach his goal.It seems there was another issue with the rearrangement of the equation. I'll manually compute the solution for $ t $ using the appropriate logarithmic manipulation. Let me go step by step to get the value of $ t $.Matthew will need approximately 39.62 years to reach his goal of $1,000,000 in his retirement account.

Since he started saving at age 25, he will be:

\[
25 + 39.62 = 64.62
\]

Thus, Matthew will be around 65 years old when he is able to retire.

Would you like more details, or do you have any questions? Here are five related questions:

1. How would a higher interest rate affect the time required for retirement?
2. What if Matthew started saving earlier or later? How would this impact his retirement age?
3. How does the monthly deposit amount influence the retirement timeline?
4. What is the effect of different compounding periods (e.g., quarterly or annually) on the outcome?
5. How does inflation affect the real value of Matthew's retirement savings?

**Tip**: Starting earlier or increasing the monthly deposit can significantly reduce the time required to reach a financial goal!

On his 25th birthday, Matthew started an annuity for his retirement. He deposits $670 each month into an account earning 5% annual interest compounded monthly. He determines that he will be able to retire once he has $1,000,000 in his retirement account. Determine how old Matthew will be when he is able to retire.

Explore how to calculate the time it will take for Matthew to retire with $1,000,000 in savings using compound interest and annuity formulas. Learn the impact of monthly deposits, interest rates, and compounding frequency in determining the retirement timeline.

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