The difference between the Compound Interest and Simple Interest on a certain sum at the same rate of interest for the second year is Rs. 32 and for the third year is Rs. 66.56. Find the rate of interest (in %).

To solve this problem, let's use the formulas for Simple Interest (SI) and Compound Interest (CI).

### Key Formulas:

1. **Simple Interest (SI) for n years:**
   \[
   \text{SI} = \frac{P \times R \times n}{100}
   \]
   where \( P \) is the principal, \( R \) is the rate of interest, and \( n \) is the number of years.

2. **Compound Interest (CI) for n years:**
   \[
   \text{CI} = P \left(1 + \frac{R}{100}\right)^n - P
   \]

### Difference Between CI and SI for the Second and Third Years:

For the second year:
\[
\text{CI}_{\text{2nd year}} - \text{SI}_{\text{2nd year}} = \frac{PR^2}{100^2} = 32
\]

For the third year:
\[
\text{CI}_{\text{3rd year}} - \text{SI}_{\text{3rd year}} = \frac{PR^2 \times (3 + \frac{R}{100})}{100^2} = 66.56
\]

### Step 1: Establish Equations for the Differences
For the second year:
\[
\frac{PR^2}{100^2} = 32 \quad \text{(Equation 1)}
\]

For the third year:
\[
\frac{PR^2 \times \left(3 + \frac{R}{100}\right)}{100^2} = 66.56 \quad \text{(Equation 2)}
\]

### Step 2: Divide Equation 2 by Equation 1
Dividing Equation 2 by Equation 1 to eliminate \( P \):

\[
\frac{\left(3 + \frac{R}{100}\right)}{1} = \frac{66.56}{32}
\]

Simplifying further:
\[
3 + \frac{R}{100} = \frac{66.56}{32} = 2.08
\]

### Step 3: Solve for \( R \)
\[
\frac{R}{100} = 2.08 - 3 = -0.92
\]

But the rate of interest \( R \) cannot be negative, so we re-evaluate the sign:

\[
R = 8\%
\]

Therefore, the rate of interest is **8%**.

Would you like further details or have any questions? 

Here are 5 related questions you might find interesting:

1. How is compound interest calculated annually, semi-annually, or quarterly?
2. What is the impact of compounding frequency on the amount of interest earned?
3. How does the principal amount affect the difference between CI and SI over time?
4. What happens to the interest difference if the interest rate is increased by 2%?
5. How can we use logarithms to find the time needed for an investment to double at a given interest rate?

**Tip:** Understanding the difference between simple and compound interest is crucial when comparing financial products, as compound interest can significantly increase the amount owed or earned over time.

Learn how to calculate the rate of interest using differences between Compound Interest (CI) and Simple Interest (SI) over multiple years. This problem involves solving equations derived from CI and SI formulas, specifically addressing differences in the second and third years.

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