If the difference between the compound interest 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 sum based on the difference between the compound interest (CI) and simple interest (SI) over three years at a rate of 10% per annum.

### Step-by-Step Solution:

1. **Understanding the Difference Between CI and SI**:
The difference between compound interest and simple interest for 3 years can be calculated using the formula:
\[ \text{Difference} = \text{CI} - \text{SI} = P \times \left( \left(1 + \frac{r}{100}\right)^n - 1 - \frac{rn}{100} \right) \]
where \( P \) is the principal, \( r \) is the rate of interest, and \( n \) is the number of years.

2. **Substituting the Values**:
Here, \( r = 10\% \) and \( n = 3 \). We can calculate the difference:
\[ \text{Difference} = P \times \left( \left(1 + \frac{10}{100}\right)^3 - 1 - \frac{10 \times 3}{100} \right) \]

3. **Calculating the Compound Interest Factor**:
\[ \left(1 + \frac{10}{100}\right)^3 = (1.1)^3 = 1.331 \]

4. **Calculating the Simple Interest**:
\[ \frac{10 \times 3}{100} = 0.3 \]

5. **Finding the Difference**:
Now substituting back into the difference formula:
\[ \text{Difference} = P \times \left(1.331 - 1 - 0.3\right) = P \times (0.031) \]

6. **Setting Up the Equation**:
We know from the problem that the difference is Rs. 558:
\[ P \times 0.031 = 558 \]

7. **Solving for Principal (P)**:
\[ P = \frac{558}{0.031} \]
Now calculating this:
\[ P = 18000 \]

### Conclusion:
The principal sum is Rs. 18,000.