If the difference between the compound interest and the simple interest earned on a sum of money at the rate of 8% p.a. for two years is Rs. 20. What is the Principal ? .

Options

A. Rs. 4800
B. Rs. 3125
C. Rs. 4500
D. Rs. 5200

Answer

To find the principal amount given the difference between compound interest (CI) and simple interest (SI) for 2 years at a rate of 8% per annum is Rs. 20, we can follow these steps:

### 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
- \( R \) = Rate of interest (8% in this case)
- \( T \) = Time (2 years)

### Step 2: Calculate Simple Interest for 2 years
Substituting the values into the formula:
\[ SI = \frac{P \times 8 \times 2}{100} = \frac{16P}{100} = \frac{4P}{25} \]

### Step 3: Understand the formula for Compound Interest (CI)
The formula for Compound Interest for 2 years is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]
Substituting the values:
\[ CI = P \left(1 + \frac{8}{100}\right)^2 - P \]
\[ = P \left(1.08\right)^2 - P \]
\[ = P \left(1.1664\right) - P \]
\[ = 1.1664P - P = 0.1664P \]

### Step 4: Find the difference between CI and SI
The difference between CI and SI is given as Rs. 20:
\[ CI - SI = 20 \]
Substituting the values we calculated:
\[ 0.1664P - \frac{4P}{25} = 20 \]

### Step 5: Convert \(\frac{4P}{25}\) to a decimal
To make calculations easier, convert \(\frac{4P}{25}\) to decimal:
\[ \frac{4}{25} = 0.16 \]
So, the equation becomes:
\[ 0.1664P - 0.16P = 20 \]

### Step 6: Simplify the equation
Now, simplify the left side:
\[ (0.1664 - 0.16)P = 20 \]
\[ 0.0064P = 20 \]

### Step 7: Solve for Principal (P)
Now, solve for \( P \):
\[ P = \frac{20}{0.0064} \]
Calculating this gives:
\[ P = 3125 \]

### Conclusion
The principal amount is Rs. 3,125.