Yusuf invested $4,600 in an account paying an interest rate of 6, one eigth6 
8
1
​
 % compounded continuously. David invested $4,600 in an account paying an interest rate of 6, one quarter6 
4
1
​
 % compounded quarterly. After 13 years, how much more money would David have in his account than Yusuf, to the nearest dollar?

We have two different types of compound interest to deal with here: continuous compounding for Yusuf and quarterly compounding for David.

### For Yusuf:
Yusuf's investment is compounded continuously. The formula for continuous compounding is:

\[
A = P \cdot e^{rt}
\]

Where:
- $A$ is the amount of money accumulated after $t$ years, including interest.
- $P$ is the principal amount (initial investment).
- $r$ is the annual interest rate (as a decimal).
- $t$ is the time the money is invested for (in years).
- $e$ is Euler's number (approximately 2.71828).

Given:
- $P = 4600$
- $r = \frac{49}{800} = 0.06125$ (converting $6\frac{1}{8}\%$)
- $t = 13$

So the formula for Yusuf's investment becomes:

\[
A = 4600 \cdot e^{0.06125 \cdot 13}
\]

### For David:
David's investment is compounded quarterly. The formula for quarterly compounding is:

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

Where:
- $A$ is the amount of money accumulated after $t$ years, including interest.
- $P$ is the principal amount.
- $r$ is the annual interest rate (as a decimal).
- $n$ is the number of times the interest is compounded per year (for quarterly compounding, $n = 4$).
- $t$ is the time the money is invested for.

Given:
- $P = 4600$
- $r = \frac{25}{400} = 0.0625$ (converting $6\frac{1}{4}\%$)
- $n = 4$
- $t = 13$

So the formula for David's investment becomes:

\[
A = 4600 \left(1 + \frac{0.0625}{4}\right)^{4 \cdot 13}
\]

Now, let's calculate both amounts and find the difference.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can help guide you through the calculations manually, or we can revisit this when the tool is available again. Let me know how you'd like to proceed.

Here are some related questions:
1. What is the impact of compounding frequency on investment growth?
2. How does continuous compounding differ from quarterly compounding?
3. How would the results change if the interest rates were swapped?
4. How does the investment duration (years) affect the total growth?
5. Can different compounding strategies affect retirement planning?

**Tip**: Compounded continuously, interest grows at a faster rate compared to periodic compounding, but the differences diminish over time.

Yusuf invested $4,600 in an account paying an interest rate of 6 1/8% compounded continuously. David invested $4,600 in an account paying an interest rate of 6 1/4% compounded quarterly. After 13 years, how much more money would David have in his account than Yusuf, to the nearest dollar?

Discover how to calculate the difference in investment growth between Yusuf’s account, compounded continuously, and David’s, compounded quarterly, after 13 years. This example explains key concepts in compound interest and exponential growth, including the formulas for continuous and quarterly compounding.

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