If the difference of the compound Interest and the simple interest on a sum of money for 3 years is `Rs. 186`. Find the sum of money, if the rate of interest in both cases be `10%`.

Options

A. `Rs. 5500`
B. `Rs. 7200 `
C. `Rs. 6500`
D. ` Rs. 6000 `

Answer

To solve the problem step by step, we will use the formula for the difference between compound interest (CI) and simple interest (SI) over a period of time.

### Step-by-step Solution:

1. **Understand the Given Information**:
- The difference between Compound Interest and Simple Interest for 3 years is given as Rs. 186.
- The rate of interest (r) is 10%.

2. **Use the Formula for the Difference**:
The formula for the difference (D) between Compound Interest and Simple Interest over 3 years is:
\[ D = \frac{P \cdot r^2}{100^2} \cdot \left(3 + \frac{r}{100}\right) \]
where:
- \( P \) = Principal amount (the sum of money we need to find)
- \( r \) = Rate of interest

3. **Substitute the Known Values**:
We know \( D = 186 \) and \( r = 10 \). Substitute these values into the formula:
\[ 186 = \frac{P \cdot 10^2}{100^2} \cdot \left(3 + \frac{10}{100}\right) \]

4. **Simplify the Equation**:
Calculate \( 10^2 \) and \( 100^2 \):
\[ 10^2 = 100 \quad \text{and} \quad 100^2 = 10000 \]
Now, substitute these values into the equation:
\[ 186 = \frac{P \cdot 100}{10000} \cdot \left(3 + 0.1\right) \]
Simplifying further:
\[ 186 = \frac{P}{100} \cdot 3.1 \]

5. **Rearranging the Equation**:
Multiply both sides by 100 to eliminate the fraction:
\[ 18600 = P \cdot 3.1 \]

6. **Solve for Principal (P)**:
Now, divide both sides by 3.1 to find \( P \):
\[ P = \frac{18600}{3.1} \]
Calculate the value:
\[ P = 6000 \]

7. **Conclusion**:
The sum of money (Principal) is Rs. 6000.