A man invested Rs. X and Rs. 2X at 15% simple interest and at 8% compound interest (compounding annually) for two years respectively. Difference between simple interest and compound interest received after two years is Rs.820. Find the value of X.

Options

A. Rs.25000
B. Rs.30600
C. Rs.28600
D. Rs.22200

Answer

To solve the problem step by step, we will calculate the simple interest and compound interest for the given investments and then find the value of \( X \).

### Step 1: Calculate Simple Interest (SI)
The formula for Simple Interest is given by:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest
- \( T \) = Time in years

For the investment of Rs. \( X \) at 15% for 2 years:
\[ SI = \frac{X \times 15 \times 2}{100} = \frac{30X}{100} = 0.3X \]

### Step 2: Calculate Compound Interest (CI)
The formula for Compound Interest is given by:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]

For the investment of Rs. \( 2X \) at 8% for 2 years:
\[ CI = 2X \left(1 + \frac{8}{100}\right)^2 - 2X \]
Calculating \( \left(1 + \frac{8}{100}\right)^2 \):
\[ = 2X \left(1.08\right)^2 - 2X = 2X \left(1.1664\right) - 2X = 2X \cdot 1.1664 - 2X \]
\[ = 2X \cdot 1.1664 - 2X = 2X(1.1664 - 1) = 2X \cdot 0.1664 = 0.3328X \]

### Step 3: Set up the equation for the difference
According to the problem, the difference between the Compound Interest and Simple Interest is Rs. 820:
\[ CI - SI = 820 \]
Substituting the values we calculated:
\[ 0.3328X - 0.3X = 820 \]
\[ (0.3328 - 0.3)X = 820 \]
\[ 0.0328X = 820 \]

### Step 4: Solve for \( X \)
To find \( X \), divide both sides by 0.0328:
\[ X = \frac{820}{0.0328} \]
Calculating the value:
\[ X = 25000 \]

### Conclusion
The value of \( X \) is Rs. 25,000.