Benjamin invested $20,000 in an account paying an interest rate of 7, start fraction, 5, divided by, 8, end fraction7
8
5
​
% compounded monthly. Kehlani invested $20,000 in an account paying an interest rate of 7, start fraction, 3, divided by, 4, end fraction7
4
3
​
% compounded quarterly. After 10 years, how much more money would Kehlani have in her account than Benjamin, to the nearest dollar?

To solve this problem, we need to calculate the future value of both Benjamin's and Kehlani's investments using the formula for compound interest:

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

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

### Step 1: Calculate Benjamin's Future Value
- Principal $ P = 20,000 $ dollars
- Annual interest rate $ r = \frac{5}{8}\% = 0.05875 $
- Compounded monthly: $ n = 12 $
- Time $ t = 10 $ years

The future value $ A_B $ for Benjamin is:

\[
A_B = 20000 \left(1 + \frac{0.05875}{12}\right)^{12 \times 10}
\]

### Step 2: Calculate Kehlani's Future Value
- Principal $ P = 20,000 $ dollars
- Annual interest rate $ r = \frac{3}{4}\% = 0.075 $
- Compounded quarterly: $ n = 4 $
- Time $ t = 10 $ years

The future value $ A_K $ for Kehlani is:

\[
A_K = 20000 \left(1 + \frac{0.075}{4}\right)^{4 \times 10}
\]

### Step 3: Compute the Difference
The difference between Kehlani's and Benjamin's future value is:

\[
\text{Difference} = A_K - A_B
\]

Now let's calculate these values.After 10 years, Kehlani would have approximately **$6,109** more in her account than Benjamin, to the nearest dollar.

Would you like more details or have any questions about the calculations? 

Here are 8 related questions you might find useful:
1. What is the formula for continuous compounding interest?
2. How does the compounding frequency affect the future value of an investment?
3. What would be the future value of Benjamin's investment if it were compounded annually instead of monthly?
4. How would the outcome change if the interest rates were swapped between Benjamin and Kehlani?
5. What is the difference between simple interest and compound interest?
6. How would inflation impact the future value of these investments?
7. What if Kehlani's investment period was only 5 years instead of 10 years?
8. How does changing the principal amount affect the final outcome?

**Tip:** When comparing investments, always consider both the interest rate and the compounding frequency. The frequency can significantly impact the total returns.

Learn how to calculate future values of investments using compound interest. Solve the problem of comparing investments over time between Benjamin and Kehlani, considering different compounding frequencies and interest rates.

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