To determine which investment is better, we need to calculate the future value of both accounts after 6 years and compare them.

Account 1: 7% per year, compounded monthly

For this account, we use the compound interest formula: $A = P \left(1 + \frac{r}{n}\right)^{nt}$ where:

• $A$ is the future value of the investment,
• $P$ is the initial principal (investment amount),
• $r$ is the annual interest rate (as a decimal),
• $n$ is the number of times the interest is compounded per year,
• $t$ is the number of years.

For this account:

• $P = 12,000$,
• $r = 0.07$,
• $n = 12$ (since it's compounded monthly),
• $t = 6$.

Now, let's calculate the future value $A$.

Account 2: 6.8% per year, compounded continuously

For this account, we use the continuous compound interest formula: $A = P \cdot e^{rt}$ where:

• $A$ is the future value of the investment,
• $P$ is the initial principal,
• $r$ is the annual interest rate (as a decimal),
• $t$ is the number of years,
• $e$ is Euler's number (approximately 2.71828).

For this account:

• $P = 12,000$,
• $r = 0.068$,
• $t = 6$.

Now, let's calculate the future value $A$ for both accounts.After 6 years:

• The future value of the first account (7% compounded monthly) is $18,241.27.
• The future value of the second account (6.8% compounded continuously) is $18,045.69.

Therefore, the first account (7% compounded monthly) is the better investment, as it yields a higher future value.

Would you like further details on the calculations or have any other questions?

Here are 5 related questions:

1. How would the results change if both accounts had the same interest rate but different compounding periods?
2. What would happen if the investment period was extended to 10 years?
3. How does continuous compounding generally compare to monthly compounding?
4. How much would need to be invested in each account to reach $25,000 in 6 years?
5. What is the difference in total interest earned between the two accounts?

Tip: The more frequently interest is compounded, the higher the future value, all other factors being equal.