A sum of money amounts to Rs. 2400 in 3 years and Rs. 2520 in 4 years on C.I. Find the rate of interest.

Answer

To solve the problem of finding the rate of interest when a sum of money amounts to Rs. 2400 in 3 years and Rs. 2520 in 4 years on compound interest, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Amount after 3 years (A1) = Rs. 2400
- Amount after 4 years (A2) = Rs. 2520
- Let the principal amount be \( P \) and the rate of interest be \( R \).

2. **Set Up the Equations:**
- The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
- For 3 years:
\[ 2400 = P \left(1 + \frac{R}{100}\right)^3 \quad \text{(Equation 1)} \]
- For 4 years:
\[ 2520 = P \left(1 + \frac{R}{100}\right)^4 \quad \text{(Equation 2)} \]

3. **Divide Equation 2 by Equation 1:**
\[ \frac{2520}{2400} = \frac{P \left(1 + \frac{R}{100}\right)^4}{P \left(1 + \frac{R}{100}\right)^3} \]
- The \( P \) cancels out:
\[ \frac{2520}{2400} = \frac{\left(1 + \frac{R}{100}\right)^4}{\left(1 + \frac{R}{100}\right)^3} \]
- Simplifying the left side:
\[ \frac{2520}{2400} = \frac{21}{20} \]
- Therefore, we have:
\[ \frac{21}{20} = 1 + \frac{R}{100} \]

4. **Solve for \( R \):**
- Rearranging the equation:
\[ \frac{R}{100} = \frac{21}{20} - 1 \]
- Simplifying the right side:
\[ \frac{R}{100} = \frac{21 - 20}{20} = \frac{1}{20} \]
- Multiplying both sides by 100:
\[ R = 100 \times \frac{1}{20} = 5 \]

5. **Conclusion:**
- The rate of interest \( R \) is 5% per annum.