The difference of SI and CI on a sum at 5% per annum for 3 years is Rs. 122. Find the sum.

Answer

To solve the problem of finding the principal sum based on the difference between Simple Interest (SI) and Compound Interest (CI) over 3 years at a rate of 5% per annum, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Formulae**:
- Simple Interest (SI) for 3 years can be calculated using the formula:
\[ SI = \frac{P \times R \times T}{100} \]
where \( P \) is the principal, \( R \) is the rate of interest, and \( T \) is the time in years.
- Compound Interest (CI) can be calculated using the formula:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
where \( A \) is the amount after time \( T \).

2. **Finding the Difference**:
- The difference between CI and SI for 3 years is given by:
\[ CI - SI = \frac{P \times R^2}{100^2} \]
- For \( R = 5\% \) and \( T = 3 \):
\[ CI - SI = \frac{P \times 5^2}{100^2} \times 3 = \frac{P \times 25 \times 3}{10000} = \frac{75P}{10000} = \frac{3P}{400} \]

3. **Setting Up the Equation**:
- We know from the problem that the difference is Rs. 122:
\[ \frac{3P}{400} = 122 \]

4. **Solving for Principal (P)**:
- To find \( P \), we multiply both sides by 400:
\[ 3P = 122 \times 400 \]
- Calculate \( 122 \times 400 \):
\[ 122 \times 400 = 48800 \]
- Now divide by 3:
\[ P = \frac{48800}{3} = 16266.67 \]

5. **Final Answer**:
- The principal sum \( P \) is approximately Rs. 16266.67.