The difference between the compound interest and simple interest on ₹x at 9% per annum for 2 years is ₹20.25 . What is the value of x?

Options

A. 2400
B. 2800
C. 2500
D. 2200

Answer

To solve the problem, we need to find the value of \( x \) given that the difference between the compound interest (CI) and simple interest (SI) on \( x \) at a rate of 9% per annum for 2 years is ₹20.25.

### Step 1: Understand the formula for Simple Interest (SI)
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount (in this case, \( x \))
- \( R \) = Rate of interest (9%)
- \( T \) = Time (2 years)

### Step 2: Calculate the Simple Interest
Substituting the values into the formula:
\[ SI = \frac{x \times 9 \times 2}{100} = \frac{18x}{100} = 0.18x \]

### Step 3: Understand the formula for Compound Interest (CI)
The formula for Compound Interest is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]
Substituting the values:
\[ CI = x \left(1 + \frac{9}{100}\right)^2 - x \]
Calculating \( \left(1 + \frac{9}{100}\right)^2 \):
\[ = x \left(1.09\right)^2 - x = x \left(1.1881\right) - x = 0.1881x \]

### Step 4: Calculate the difference between CI and SI
The difference between CI and SI is given as:
\[ CI - SI = 20.25 \]
Substituting the values we calculated:
\[ 0.1881x - 0.18x = 20.25 \]

### Step 5: Simplify the equation
\[ (0.1881 - 0.18)x = 20.25 \]
\[ 0.0081x = 20.25 \]

### Step 6: Solve for \( x \)
To find \( x \), divide both sides by 0.0081:
\[ x = \frac{20.25}{0.0081} \]
Calculating the right-hand side:
\[ x = 2500 \]

### Conclusion
The value of \( x \) is ₹2500.