The difference between CI and SI on a certain sum of money at 10% per annum for 3 years is 620. Find the principal if it is known that the interest is compounded annually.

Options

A. 200, 000
B. 20000
C. 10000
D. 100, 000

Answer

To solve the problem, we need to find the principal amount (P) given that the difference between Compound Interest (CI) and Simple Interest (SI) for a certain sum of money at a rate of 10% per annum for 3 years is 620.

### Step-by-Step Solution:

1. **Understand the Formulas**:
- The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
where \( P \) is the principal, \( R \) is the rate of interest, and \( T \) is the time in years.
- The formula for Compound Interest (CI) is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]
which simplifies to:
\[ CI = P \left(\left(1 + \frac{R}{100}\right)^T - 1\right) \]

2. **Calculate SI for 3 years**:
- Given \( R = 10\% \) and \( T = 3 \):
\[ SI = \frac{P \times 10 \times 3}{100} = \frac{30P}{100} = 0.3P \]

3. **Calculate CI for 3 years**:
- Using the compound interest formula:
\[ CI = P \left(\left(1 + \frac{10}{100}\right)^3 - 1\right) = P \left(\left(1.1\right)^3 - 1\right) \]
- Calculate \( (1.1)^3 \):
\[ (1.1)^3 = 1.331 \]
- Therefore,
\[ CI = P \left(1.331 - 1\right) = P \times 0.331 \]

4. **Find the Difference between CI and SI**:
- The difference between CI and SI is given as 620:
\[ CI - SI = 620 \]
- Substitute the values of CI and SI:
\[ 0.331P - 0.3P = 620 \]
- Simplify:
\[ 0.031P = 620 \]

5. **Solve for Principal (P)**:
- Divide both sides by 0.031:
\[ P = \frac{620}{0.031} \approx 20000 \]

### Final Answer:
The principal amount is **20000**.