The difference between the simple interest on a certain sum at the rate of 10% per annum for 2 years and the compound interest on the same sum, rate and time is ₹ 124.05. If the compound interest is semi-annually. The sum is

Options

A. ₹ 5,000
B. ₹ 10,000
C. ₹ 12,000
D. ₹ 8,000

Answer

To solve the problem step by step, we will follow the outlined approach from the video transcript.

### Step 1: Define the Variables
Let the principal amount be \( P \).

### Step 2: Calculate Simple Interest (SI)
The formula for Simple Interest is given by:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( R = 10\% \) (rate of interest)
- \( T = 2 \) years

Substituting the values:
\[ SI = \frac{P \times 10 \times 2}{100} = \frac{20P}{100} = \frac{P}{5} \]

### Step 3: Calculate Compound Interest (CI)
Since the compound interest is compounded semi-annually, we need to adjust the rate and time:
- The rate per half-year = \( \frac{10}{2} = 5\% \)
- The number of half-years = \( 2 \times 2 = 4 \)

The formula for Compound Interest is:
\[ CI = P \left(1 + \frac{R}{100}\right)^n - P \]
Where:
- \( n = 4 \) (number of compounding periods)

Substituting the values:
\[ CI = P \left(1 + \frac{5}{100}\right)^4 - P = P \left(1.05^4\right) - P \]

Calculating \( 1.05^4 \):
\[ 1.05^4 = 1.21550625 \quad (\text{approximately } 1.2155) \]
Thus,
\[ CI = P \times 1.2155 - P = P(1.2155 - 1) = 0.2155P \]

### Step 4: Set Up the Equation
According to the problem, the difference between the Compound Interest and Simple Interest is ₹124.05:
\[ CI - SI = 124.05 \]
Substituting the values of CI and SI:
\[ 0.2155P - \frac{P}{5} = 124.05 \]

### Step 5: Simplify the Equation
Convert \( \frac{P}{5} \) to a decimal:
\[ \frac{P}{5} = 0.2P \]
Now the equation becomes:
\[ 0.2155P - 0.2P = 124.05 \]
\[ 0.0155P = 124.05 \]

### Step 6: Solve for P
To find \( P \), divide both sides by \( 0.0155 \):
\[ P = \frac{124.05}{0.0155} \approx 8000 \]

### Conclusion
The principal sum \( P \) is ₹8000.

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