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This image shows two problems involving compound interest calculations. Let's break down the two equations step-by-step to explain how the answers are obtained.

### General Formula for Compound Interest:
The formula used in compound interest calculations is:

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

Where:
- \( A \) is the final amount (including interest).
- \( P \) is the principal (initial investment or loan amount).
- \( e \) is Euler's number (approximately 2.718).
- \( r \) is the annual interest rate (in decimal form).
- \( t \) is the time in years.

### Problem 6:
Given:
- \( A = 680,000 \) (final amount)
- \( r = 12\% = 0.12 \) (annual interest rate)
- \( t = 8 \) years

The goal is to find the principal \( P \).

Using the formula \( A = P \cdot e^{rt} \):

\[
680,000 = P \cdot e^{(0.12 \cdot 8)}
\]
\[
680,000 = P \cdot e^{0.96}
\]

Now, calculate \( e^{0.96} \), which is approximately 2.6117:

\[
680,000 = P \cdot 2.6117
\]

To find \( P \), divide both sides by 2.6117:

\[
P = \frac{680,000}{2.6117} \approx 260,334.95
\]

Thus, \( P \approx 260,334.95 \), which is rounded to **260,335**.

---

### Problem 7:
Given:
- \( A = 1,240,000 \) (final amount)
- \( r = 8\% = 0.08 \) (annual interest rate)
- \( t = 30 \) years

Again, we use the formula \( A = P \cdot e^{rt} \):

\[
1,240,000 = P \cdot e^{(0.08 \cdot 30)}
\]
\[
1,240,000 = P \cdot e^{2.4}
\]

Now, calculate \( e^{2.4} \), which is approximately 11.0232:

\[
1,240,000 = P \cdot 11.0232
\]

To find \( P \), divide both sides by 11.0232:

\[
P = \frac{1,240,000}{11.0232} \approx 112,484.64
\]

Thus, \( P \approx 112,484.64 \), which is rounded to **112,485**.

---

### Summary:
1. Problem 6 uses \( A = 680,000 \), \( r = 12\% \), and \( t = 8 \) years, and the principal \( P \) is approximately **260,335**.
2. Problem 7 uses \( A = 1,240,000 \), \( r = 8\% \), and \( t = 30 \) years, and the principal \( P \) is approximately **112,485**.

Would you like further details or have any questions? Here are 5 related questions you could explore:

1. What is the derivation of the compound interest formula?
2. How does continuously compounded interest differ from other types of compounding?
3. How does the principal change if the time period is doubled but the interest rate remains the same?
4. Can the same formula be applied for non-continuous compounding periods (e.g., monthly compounding)?
5. How would changing the interest rate affect the final amount for a fixed time?

**Tip:** Remember to convert percentages to decimals (e.g., 12% becomes 0.12) before using them in formulas!

How do we find the principal amount for compound interest problems with given final amount, rate, and time?

Learn how to find the principal amount for compound interest problems. The solution uses the formula A = P * e^(rt), with a given final amount, interest rate, and time. Suitable for students in Grades 10-12.

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