The difference between simple interest and compound interest on a principal for 2 years is 76.8. The simple interest on the same principal for 4 years is 3,840. Find the principal and the rate of interest

Let's solve the problem step-by-step to find the principal (P) and the rate of interest (R).

### Step 1: Understanding the Problem

1. **Simple Interest (SI)** for 4 years is given as \(3840\).
   
   Simple Interest formula:
   \[
   SI = \frac{P \times R \times T}{100}
   \]
   Where:
   - \(P\) is the principal
   - \(R\) is the rate of interest
   - \(T\) is the time period in years
   
   For 4 years, we have:
   \[
   SI = \frac{P \times R \times 4}{100} = 3840
   \]

2. **Difference between Compound Interest (CI) and Simple Interest (SI) for 2 years** is given as \(76.8\).

   The formula for Compound Interest compounded annually is:
   \[
   CI = P \left(1 + \frac{R}{100}\right)^T - P
   \]
   
   For 2 years:
   \[
   CI = P \left(1 + \frac{R}{100}\right)^2 - P
   \]

   The difference between CI and SI for 2 years is:
   \[
   \text{Difference} = CI - SI
   \]
   The Simple Interest for 2 years is:
   \[
   SI = \frac{P \times R \times 2}{100}
   \]
   So:
   \[
   P \left(1 + \frac{R}{100}\right)^2 - P - \frac{P \times R \times 2}{100} = 76.8
   \]

### Step 2: Solving for Principal (P) and Rate of Interest (R)

1. **From the Simple Interest equation for 4 years:**
   \[
   \frac{P \times R \times 4}{100} = 3840
   \]
   Simplifying for \(P \times R\):
   \[
   P \times R = 96000
   \]

2. **From the Compound Interest and Simple Interest difference equation:**
   \[
   P \left(1 + \frac{R}{100}\right)^2 - P - \frac{P \times R \times 2}{100} = 76.8
   \]

Let's substitute \(P \times R = 96000\) into the equations and solve for \(P\) and \(R\).The solutions for the principal (P) and the rate of interest (R) are:

- **Principal (P) = 12,000**
- **Rate of Interest (R) = 8%**

### Verification

1. **Simple Interest for 4 years:**
   \[
   SI = \frac{P \times R \times 4}{100} = \frac{12000 \times 8 \times 4}{100} = 3840
   \]
   This matches the given simple interest.

2. **Difference between Compound Interest and Simple Interest for 2 years:**
   \[
   \text{CI} = 12000 \left(1 + \frac{8}{100}\right)^2 - 12000 = 12000 \times 1.1664 - 12000 = 199.68
   \]
   \[
   \text{SI for 2 years} = \frac{12000 \times 8 \times 2}{100} = 1920
   \]
   \[
   \text{Difference} = 199.68 - 1920 = 76.8
   \]
   This confirms the given difference.

Thus, the answers are correct.

Would you like more details, or do you have any questions? Here are some related questions for further exploration:

1. How would the solution change if the interest was compounded semi-annually?
2. What is the formula for compound interest if compounded quarterly?
3. How does the difference between simple and compound interest vary with different time periods?
4. What are the implications of different compounding frequencies on the effective interest rate?
5. How do you calculate continuous compound interest?

**Tip:** For longer investment durations, compound interest grows significantly compared to simple interest due to compounding effects.

Learn how to calculate the principal and rate of interest using simple interest and compound interest formulas. Solve a problem where the difference between simple and compound interest over specific periods is given. Verify the solution step-by-step with detailed calculations.

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