principal = 408000 years = 4 amount = 700000 find rate if compound rate is been applied

To find the compound interest rate given the principal amount, the time period, and the final amount, we use the formula for compound interest:

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

where:
- \(A\) is the final amount,
- \(P\) is the principal,
- \(r\) is the annual interest rate,
- \(n\) is the number of times interest is compounded per year,
- \(t\) is the number of years.

If the interest is compounded annually (once per year), then \(n = 1\), and the formula simplifies to:

\[
A = P \times (1 + r)^t
\]

Given:
- \(P = 408,000\)
- \(A = 700,000\)
- \(t = 4\)

We need to find the rate \(r\). Let's rearrange the formula to solve for \(r\):

\[
700,000 = 408,000 \times (1 + r)^4
\]

Now, let's calculate the value of \(r\).It looks like the calculation took too long to complete. Let me break the problem into smaller steps and compute the rate again.

First, we rearrange the formula to isolate \( (1 + r) \):

\[
(1 + r)^4 = \frac{700000}{408000}
\]

Then we take the fourth root of both sides:

\[
1 + r = \left( \frac{700000}{408000} \right)^{\frac{1}{4}}
\]

Finally, we subtract 1 to find \( r \):

\[
r = \left( \frac{700000}{408000} \right)^{\frac{1}{4}} - 1
\]

Let's calculate this step-by-step to find the rate.It seems there is still a timeout issue while performing the calculation. I'll compute the compound interest rate step-by-step manually:

1. **Calculate the ratio of \( A \) to \( P \):**
   \[
   \frac{A}{P} = \frac{700000}{408000} \approx 1.718
   \]

2. **Take the fourth root of this ratio:**
   \[
   (1.718)^{\frac{1}{4}}
   \]

3. **Subtract 1 to find the rate \( r \):**
   \[
   r = (1.718)^{\frac{1}{4}} - 1
   \]

4. **Convert the rate to a percentage:**
   \[
   r \times 100
   \]

Let's continue to find this value step-by-step.It seems the calculations are still timing out. I will provide a manual calculation instead:

1. **Calculate the ratio**:  
   \[
   \frac{700,000}{408,000} \approx 1.7157
   \]

2. **Take the fourth root**:  
   \[
   (1.7157)^{\frac{1}{4}} \approx 1.1423
   \]

3. **Subtract 1 to find \(r\)**:  
   \[
   r = 1.1423 - 1 = 0.1423
   \]

4. **Convert to a percentage**:  
   \[
   r \times 100 \approx 14.23\%
   \]

### Conclusion
The annual compound interest rate is approximately **14.23%**.

---

Would you like to see more details or have any questions about the calculation? Here are some related questions you might find interesting:

1. How do you calculate the compound interest for different compounding frequencies (e.g., monthly, quarterly)?
2. What is the difference between compound interest and simple interest?
3. How does the compounding frequency affect the final amount?
4. How do you calculate the effective annual rate (EAR) from a nominal rate?
5. What are some practical examples where compound interest calculations are used?

**Tip**: When dealing with compound interest problems, always check if the interest is compounded annually or more frequently to ensure accurate calculations.

Learn how to calculate the compound interest rate for a principal of 408,000, a final amount of 700,000 over 4 years. This example uses the compound interest formula and covers key algebraic and exponential growth concepts. Suitable for students in Grades 10-12.

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