The difference between the compound interest and simple interest earned on a particular sum in the second year alone is Rs. 123, and for the third year of investing is Rs. 369. If the interest rate is "r %" per annum for both simple interest and compound interest, find the value of r

To find the value of \( r \% \) (the rate of interest), let's define some key concepts first.

### Given:
- The difference between compound interest (CI) and simple interest (SI) for the second year alone is \( 123 \).
- The difference between compound interest and simple interest for the third year alone is \( 369 \).

### Step 1: Formula for Simple Interest (SI)
For a principal amount \( P \) and rate of interest \( r \% \), the simple interest for \( n \) years is given by:
\[
\text{SI} = \frac{P \times r \times n}{100}
\]
However, we're not directly using this formula for calculating the difference in interests but for understanding how interest behaves.

### Step 2: Formula for Compound Interest (CI)
For compound interest, the formula for \( n \) years is:
\[
\text{CI} = P \left( 1 + \frac{r}{100} \right)^n - P
\]
The CI in the second year and third year will involve compounding.

### Step 3: Understanding the Difference in the Second and Third Years

The difference between the compound interest and simple interest for the **second year** is due to the compound interest earning interest on the interest accrued from the **first year**.

Similarly, the difference in the third year arises because compound interest continues to compound on the previous interest.

Let’s derive the formula for the difference between CI and SI for each year:

#### For the second year:
The difference between CI and SI for the second year alone is \( \frac{P \times r^2}{100^2} \). We are given:
\[
\frac{P \times r^2}{100^2} = 123
\]
\[
P \times r^2 = 123 \times 100^2 = 1,230,000
\quad \text{(Equation 1)}
\]

#### For the third year:
The difference between CI and SI for the third year alone is \( \frac{P \times r^3}{100^3} \). We are given:
\[
\frac{P \times r^3}{100^3} = 369
\]
\[
P \times r^3 = 369 \times 100^3 = 3,690,000,000
\quad \text{(Equation 2)}
\]

### Step 4: Dividing the two equations
To eliminate \( P \), divide Equation 2 by Equation 1:
\[
\frac{P \times r^3}{P \times r^2} = \frac{3,690,000,000}{1,230,000}
\]
\[
r = \frac{3,690,000,000}{1,230,000} = 3000
\]

Thus, \( r = 30\% \).

### Final Answer:
The rate of interest \( r \% \) is \( \boxed{30\%} \).

Would you like a further breakdown or any clarifications?

Here are some related questions you might explore:
1. How does compound interest differ from simple interest in general?
2. What are the conditions under which compound interest grows exponentially?
3. Can the formula for compound interest be applied to non-annual compounding periods?
4. How do different compounding frequencies (monthly, quarterly) affect compound interest?
5. Can compound interest be negative, and under what circumstances?

**Tip:** Compound interest grows faster than simple interest because it compounds on accumulated interest, not just the principal.

The difference between the compound interest and simple interest earned on a particular sum in the second year alone is Rs. 123, and for the third year of investing is Rs. 369. If the interest rate is 'r %' per annum for both simple interest and compound interest, find the value of r.

Determine the interest rate 'r%' by analyzing the difference between compound interest and simple interest in the second and third years. This problem involves key concepts of compound and simple interest, suitable for students in Grades 11-12.

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