Mr. Gordon plans to invest $400 at the end of each month in an account that pays 6%, compounded monthly. After how many months will the account be worth $50,000?

To solve this problem, we can use the formula for the future value of an annuity (since Mr. Gordon is making monthly payments into the account):

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

Where:
- $FV$ is the future value of the annuity (which is $50,000),
- $P$ is the monthly payment (which is $400),
- $r$ is the monthly interest rate (which is the annual rate divided by 12, so $ \frac{6\%}{12} = 0.005$),
- $n$ is the number of months (which we are trying to find).

We need to solve for $n$, so let's rearrange the formula:

\[
50,000 = 400 \times \frac{(1 + 0.005)^n - 1}{0.005}
\]

Now let's calculate $n$.The number of months required for the account to be worth $50,000 is approximately 97.34 months, which means Mr. Gordon will reach this goal in about 98 months.

Would you like more details on any part of the solution, or do you have other questions?

Here are 5 related questions for further exploration:
1. What if the interest rate was 5% instead of 6%? How many months would it take?
2. What is the total amount of money Mr. Gordon will invest over these 98 months?
3. How would the formula change if the payments were made at the beginning of each month instead of the end?
4. What if Mr. Gordon wanted to reach $100,000 instead of $50,000?
5. How much interest will Mr. Gordon earn after 98 months?

**Tip**: Compounding interest grows faster when contributions are regular, and more frequent compounding increases the growth rate.

Find out how long it will take to reach $50,000 with a monthly investment of $400 in an account with 6% interest, compounded monthly. This solution uses the future value of annuities formula, showing step-by-step calculations suitable for Grades 11-12.

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