The difference between the compound interest and simple interest on Rs x at 7.5% per annum for 2 years is Rs 45. What is the value of x?

Options

A. 7000
B. 10000
C. 8000
D. 9000

Answer

To solve the problem step by step, we need to find the value of \( x \) given that the difference between the compound interest (CI) and simple interest (SI) on \( Rs. x \) at \( 7.5\% \) per annum for \( 2 \) years is \( Rs. 45 \).

### Step 1: Understand the formula for the difference between CI and SI
The difference between compound interest and simple interest for \( T \) years is given by the formula:
\[ \text{Difference} = \frac{P \cdot R^2 \cdot T}{100^2} \]
where:
- \( P \) is the principal amount (in this case, \( x \)),
- \( R \) is the rate of interest (in this case, \( 7.5\% \)),
- \( T \) is the time period in years (in this case, \( 2 \)).

### Step 2: Substitute the known values into the formula
We know that the difference is \( Rs. 45 \), \( R = 7.5 \), and \( T = 2 \). Substituting these values into the formula gives us:
\[ 45 = \frac{x \cdot (7.5)^2 \cdot 2}{100^2} \]

### Step 3: Calculate \( (7.5)^2 \)
Calculating \( (7.5)^2 \):
\[ (7.5)^2 = 56.25 \]
So the equation becomes:
\[ 45 = \frac{x \cdot 56.25 \cdot 2}{10000} \]

### Step 4: Simplify the equation
Now simplify the equation:
\[ 45 = \frac{x \cdot 112.5}{10000} \]

### Step 5: Cross-multiply to solve for \( x \)
Cross-multiplying gives:
\[ 45 \cdot 10000 = x \cdot 112.5 \]
\[ 450000 = x \cdot 112.5 \]

### Step 6: Solve for \( x \)
Now, divide both sides by \( 112.5 \):
\[ x = \frac{450000}{112.5} \]

### Step 7: Calculate the value of \( x \)
Calculating \( x \):
\[ x = 4000 \]

### Step 8: Conclusion
Thus, the value of \( x \) is \( Rs. 4000 \).