What is the rate of interest if the difference between the compound interest & simple interest for 2 years on Rs 54,000 is Rs 1,215?

Options

A. 15% per annum
B. 10% per annum
C. 20% per annum
D. 5% per annum

Answer

To solve the problem of finding the rate of interest given that the difference between compound interest (CI) and simple interest (SI) for 2 years on Rs 54,000 is Rs 1,215, we can follow these steps:

### Step 1: Understand the relationship between CI and SI
The difference between the compound interest and simple interest for 2 years can be calculated using the formula:
\[ \text{Difference} = \frac{P \times R^2}{100^2} \]
where \( P \) is the principal amount, and \( R \) is the rate of interest.

### Step 2: Substitute the known values
Here, we know:
- \( P = 54,000 \)
- Difference = Rs 1,215

We can substitute these values into the formula:
\[ 1215 = \frac{54000 \times R^2}{100^2} \]

### Step 3: Simplify the equation
Rearranging the equation gives:
\[ 1215 = \frac{54000 \times R^2}{10000} \]
Multiplying both sides by 10000 to eliminate the fraction:
\[ 1215 \times 10000 = 54000 \times R^2 \]
\[ 12150000 = 54000 \times R^2 \]

### Step 4: Solve for \( R^2 \)
Now, divide both sides by 54000:
\[ R^2 = \frac{12150000}{54000} \]

### Step 5: Calculate \( R^2 \)
Calculating the right side:
\[ R^2 = 225 \]

### Step 6: Find \( R \)
Taking the square root of both sides:
\[ R = \sqrt{225} = 15 \]

### Step 7: Conclusion
Thus, the rate of interest \( R \) is:
\[ R = 15\% \]

### Summary
The rate of interest is **15% per annum**.