To complete the table and find the unknowns, we need to use the compound interest formula:

$F = P \left(1 + \frac{r}{n}\right)^{nt}$

Where:

• $F$ is the future value,
• $P$ is the principal,
• $r$ is the annual interest rate (expressed as a decimal),
• $t$ is the time in years,
• $n$ is the number of times interest is compounded per year (for simplicity, we'll assume it's compounded annually, i.e., $n = 1$),
• $I_c = F - P$ is the compound interest.

Now let's solve the missing values step by step:

First Row:

Given:

• Principal $P = 8,000$,
• Time $t = 5$ years,
• Future Value $F = 12,593.91$.

We need to find the Rate $r$ and Compound Interest $I_c$.

Step 1: Find Rate $r$

Using the formula:

$12,593.91 = 8,000 \left(1 + r\right)^5$

$\frac{12,593.91}{8,000} = (1 + r)^5$

$1.57423875 = (1 + r)^5$

Taking the 5th root of both sides:

$1 + r = \left(1.57423875\right)^{\frac{1}{5}} \approx 1.095$

$r \approx 0.095 \text{ or } 9.5\%$

Step 2: Find Compound Interest $I_c$

$I_c = F - P = 12,593.91 - 8,000 = 4,593.91$

Second Row:

Given:

• Rate $r = 7.6\%$ (or 0.076),
• Time $t = 6$ years,
• Future Value $F = 23,279.03$.

We need to find the Principal $P$ and Compound Interest $I_c$.

Step 3: Find Principal $P$

Using the formula:

$23,279.03 = P \left(1 + 0.076\right)^6$

$23,279.03 = P \left(1.076\right)^6$

$23,279.03 = P \times 1.5289$

$P = \frac{23,279.03}{1.5289} \approx 15,220$

Step 4: Find Compound Interest $I_c$

$I_c = F - P = 23,279.03 - 15,220 \approx 8,059.03$

Summary of the missing values:

• Row 1: Rate $r = 9.5\%$, Compound Interest $I_c = 4,593.91$,
• Row 2: Principal $P \approx 15,220$, Compound Interest $I_c \approx 8,059.03$.

Would you like further clarification or details? Here are five questions you might consider:

1. How is the compound interest formula adjusted for different compounding frequencies?
2. What is the difference between simple and compound interest?
3. How does the interest rate affect the growth of the principal over time?
4. What happens if the interest is compounded monthly instead of annually?
5. How can you solve for the principal if only the compound interest is known?

Tip: To solve for the rate $r$, taking roots often requires logarithms if the time period isn't straightforward.