If the different between the compound interset and simple interest on a certain sum of money for three years at 10 % p.a. is Rs 558, then the sum is :

Options

A. Rs 18, 500
B. Rs 15,000
C. Rs 16,000
D. Rs 18,000

Answer

To solve the problem, we need to find the principal amount (P) given that the difference between the compound interest (CI) and simple interest (SI) for three years at a rate of 10% per annum is Rs 558.

### Step-by-Step Solution:

1. **Understand the Formulas**:
- The formula for Compound Interest (CI) for 3 years is:
\[ CI = P \left(1 + \frac{r}{100}\right)^t - P \]
- The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times r \times t}{100} \]

2. **Substitute the Known Values**:
- Here, \( r = 10\% \) and \( t = 3 \) years.
- Therefore, for Compound Interest:
\[ CI = P \left(1 + \frac{10}{100}\right)^3 - P = P \left(1.1^3\right) - P \]

3. **Calculate \( 1.1^3 \)**:
- \( 1.1^3 = 1.1 \times 1.1 \times 1.1 = 1.331 \)
- Thus, we can write:
\[ CI = P \times 1.331 - P = 0.331P \]

4. **Calculate Simple Interest**:
- For Simple Interest:
\[ SI = \frac{P \times 10 \times 3}{100} = \frac{30P}{100} = 0.3P \]

5. **Set Up the Equation for the Difference**:
- The difference between CI and SI is given as Rs 558:
\[ CI - SI = 558 \]
- Substituting the values we calculated:
\[ 0.331P - 0.3P = 558 \]

6. **Simplify the Equation**:
- Combine the terms:
\[ 0.031P = 558 \]

7. **Solve for P**:
- To find P, divide both sides by 0.031:
\[ P = \frac{558}{0.031} \]
- Calculate:
\[ P = 18000 \]

### Final Answer:
The sum (principal amount) is Rs 18,000.