If the difference between simple and compound interest on some principal amount at 20% per annum for three years is Rs. 48, then the principal amount is:

Options

A. Rs. 450
B. Rs. 5375
C. Rs. 390
D. None of these

Answer

To solve the problem step by step, we will use the formulas for Simple Interest (SI) and Compound Interest (CI) and set up an equation based on the information provided.

### Step 1: Understand the Problem
We know that the difference between Compound Interest and Simple Interest for a principal amount (P) at a rate of 20% per annum over 3 years is Rs. 48.

### Step 2: Write the Formulas
- **Simple Interest (SI)** is calculated using the formula:
\[ SI = \frac{P \times R \times T}{100} \]
where \( R \) is the rate of interest and \( T \) is the time in years.

- **Compound Interest (CI)** is calculated using the formula:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]

### Step 3: Substitute the Values for SI
Given \( R = 20\% \) and \( T = 3 \):
\[ SI = \frac{P \times 20 \times 3}{100} = \frac{60P}{100} = \frac{3P}{5} \]

### Step 4: Substitute the Values for CI
Using the formula for CI:
\[ CI = P \left(1 + \frac{20}{100}\right)^3 - P = P \left(1.2^3\right) - P \]
Calculating \( 1.2^3 \):
\[ 1.2^3 = \frac{6^3}{5^3} = \frac{216}{125} \]
Thus,
\[ CI = P \left(\frac{216}{125}\right) - P = P \left(\frac{216}{125} - 1\right) = P \left(\frac{216 - 125}{125}\right) = P \left(\frac{91}{125}\right) \]

### Step 5: Set Up the Equation
According to the problem, the difference between CI and SI is Rs. 48:
\[ CI - SI = 48 \]
Substituting the values we found:
\[ P \left(\frac{91}{125}\right) - \frac{3P}{5} = 48 \]

### Step 6: Find a Common Denominator
To solve the equation, we need a common denominator. The common denominator for 125 and 5 is 125:
\[ \frac{3P}{5} = \frac{3P \times 25}{5 \times 25} = \frac{75P}{125} \]
So the equation becomes:
\[ P \left(\frac{91}{125}\right) - \frac{75P}{125} = 48 \]

### Step 7: Combine the Terms
Combine the terms on the left:
\[ \frac{91P - 75P}{125} = 48 \]
This simplifies to:
\[ \frac{16P}{125} = 48 \]

### Step 8: Solve for P
To find \( P \), multiply both sides by 125:
\[ 16P = 48 \times 125 \]
Calculating \( 48 \times 125 \):
\[ 48 \times 125 = 6000 \]
Thus,
\[ 16P = 6000 \]
Now, divide both sides by 16:
\[ P = \frac{6000}{16} = 375 \]

### Final Answer
The principal amount is Rs. 375.

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