Amadou invested $760 in an account paying an interest rate of 8 1/2% compounded continuously. Mia invested $760 in an account paying an interest rate of 8 3/8% compounded annually. After 20 years, how much more money would Amadou have in his account than Mia, to the nearest dollar?

To solve this problem, we'll calculate the future value of both Amadou's and Mia's investments after 20 years and then find the difference.

### Amadou's Investment (Compounded Continuously)
The formula for continuous compounding is:
\[
A = P e^{rt}
\]
where:
- $ P $ is the principal amount ($760),
- $ r $ is the annual interest rate (8 1/2% or 8.5% = 0.085 as a decimal),
- $ t $ is the time the money is invested for (20 years),
- $ e $ is the base of the natural logarithm, approximately equal to 2.71828.

Substituting the values:
\[
A = 760 \times e^{0.085 \times 20}
\]

### Mia's Investment (Compounded Annually)
The formula for compound interest compounded annually is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- $ P $ is the principal amount ($760),
- $ r $ is the annual interest rate (8 3/8% or 8.375% = 0.08375 as a decimal),
- $ n $ is the number of times the interest is compounded per year (1 for annual compounding),
- $ t $ is the time the money is invested for (20 years).

Substituting the values:
\[
A = 760 \left(1 + \frac{0.08375}{1}\right)^{1 \times 20}
\]

Now, let's calculate both amounts and find the difference.After 20 years, Amadou would have $364 more in his account than Mia, to the nearest dollar.

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

Here are some related questions you might find interesting:
1. How does continuous compounding generally compare to annual compounding over long periods?
2. What would happen if both interest rates were the same but the compounding methods differed?
3. How would the difference change if the investment period was shorter, say 10 years?
4. What are the advantages of choosing an investment with continuous compounding?
5. How does the initial principal amount affect the growth of investments with different compounding frequencies?
6. What would be the impact of inflation on these investments over 20 years?
7. Can the continuous compounding formula be applied to other areas in finance?
8. How does changing the compounding frequency (e.g., monthly or quarterly) impact the future value?

**Tip:** Continuous compounding often results in higher returns than periodic compounding, especially over long investment periods, due to the exponential nature of growth.

Discover how different compounding methods affect investments over 20 years. Calculate the future values of investments with continuous and annual compounding rates. Understand the impact of interest rates and compounding frequencies on investment growth.

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