If the differnce between the simple interest and compound interests on some principal amont at 20% for 3 years is ₹48. then the principal amount is

Options

A. 636
B. 650
C. 375
D. 400

Explanation

To solve the problem of finding the principal amount given that the difference between the simple interest (SI) and compound interest (CI) at a rate of 20% for 3 years is ₹48, we can follow these steps:

### Step 1: Understand the Formula for Simple Interest
The formula for calculating Simple Interest is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
where:
- \( P \) = Principal amount
- \( R \) = Rate of interest (in %)
- \( T \) = Time period (in years)

### Step 2: Calculate Simple Interest
Assume the principal amount \( P \) is ₹100 for simplicity. Using the formula:
\[ \text{SI} = \frac{100 \times 20 \times 3}{100} = 60 \]
So, the Simple Interest for ₹100 at 20% for 3 years is ₹60.

### Step 3: Understand the Formula for Compound Interest
The formula for calculating Compound Interest is:
\[ \text{CI} = A - P \]
where \( A \) is the amount after time \( T \):
\[ A = P \left(1 + \frac{R}{100}\right)^T \]

### Step 4: Calculate Compound Interest
Using the assumed principal amount of ₹100:
\[ A = 100 \left(1 + \frac{20}{100}\right)^3 = 100 \left(1.2\right)^3 \]
Calculating \( (1.2)^3 \):
\[ (1.2)^3 = 1.728 \]
So,
\[ A = 100 \times 1.728 = 172.8 \]
Now, calculate the Compound Interest:
\[ \text{CI} = A - P = 172.8 - 100 = 72.8 \]

### Step 5: Find the Difference Between CI and SI
Now, we find the difference:
\[ \text{Difference} = \text{CI} - \text{SI} = 72.8 - 60 = 12.8 \]

### Step 6: Set Up the Equation
We know from the problem statement that the difference between CI and SI is ₹48. Therefore, we can set up the equation:
\[ \frac{P \times 20 \times 3}{100} - \left(P \left(1 + \frac{20}{100}\right)^3 - P\right) = 48 \]

### Step 7: Solve for Principal Amount
From the earlier calculations, we can generalize the difference formula:
\[ \text{Difference} = \frac{P \times R^2}{100^2} \times T(T-1) \]
Substituting \( R = 20 \) and \( T = 3 \):
\[ \text{Difference} = \frac{P \times 20^2}{100^2} \times 3 \times 2 = \frac{P \times 400}{10000} \times 6 = \frac{P \times 2400}{10000} = \frac{P \times 24}{100} \]
Setting this equal to ₹48:
\[ \frac{P \times 24}{100} = 48 \]
Multiplying both sides by 100:
\[ P \times 24 = 4800 \]
Dividing by 24:
\[ P = \frac{4800}{24} = 200 \]

### Final Step: Conclusion
Thus, the principal amount is ₹200.