Anthony invested $250 in an account paying an interest rate of 6 1/8 % compounded monthly. Landon invested $250 in an account paying an interest rate of 5 3/4% compounded continuously. After 6 years, how much more money would Anthony have in his account than Landon, to the nearest dollar?

To solve this problem, we need to calculate the future value of both Anthony's and Landon's investments and then find the difference between them.

### Anthony's Investment (Compounded Monthly)
The formula for compound interest compounded $ n $ times per year is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- $ P $ is the principal amount ($250),
- $ r $ is the annual interest rate (6 1/8% or 6.125% = 0.06125 as a decimal),
- $ n $ is the number of times the interest is compounded per year (12 times for monthly compounding),
- $ t $ is the time the money is invested for (6 years).

Substituting the values:
\[
A = 250 \left(1 + \frac{0.06125}{12}\right)^{12 \times 6}
\]

### Landon's Investment (Compounded Continuously)
The formula for continuous compounding is:
\[
A = P e^{rt}
\]
where:
- $ P $ is the principal amount ($250),
- $ r $ is the annual interest rate (5 3/4% or 5.75% = 0.0575 as a decimal),
- $ t $ is the time the money is invested for (6 years),
- $ e $ is the base of the natural logarithm, approximately equal to 2.71828.

Substituting the values:
\[
A = 250 \times e^{0.0575 \times 6}
\]

Now, let's calculate both amounts and find the difference.After 6 years, Anthony would have $8 more in his account than Landon, to the nearest dollar.

Would you like more details on how this was calculated, or do you have any other questions?

Here are some related questions you might find interesting:
1. How does the frequency of compounding affect the final amount in an account?
2. What is the formula for calculating simple interest and how does it compare to compound interest?
3. What would happen if both accounts were compounded continuously?
4. How would the difference change if the interest rates were slightly different?
5. What would the outcome be if the investment period was shorter or longer?
6. How do taxes affect the growth of investments with compound interest?
7. Can you explain the significance of the natural logarithm base $ e $ in continuous compounding?
8. How does inflation impact the real value of these investments over time?

**Tip:** When comparing investments, always consider both the interest rate and the compounding frequency to get an accurate picture of potential returns.

Find out how much more money Anthony would have after 6 years compared to Landon, using different compounding methods. Includes step-by-step calculations for compound interest and continuous compounding. Explore the impact of interest rates and compounding frequency on investment returns.

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