The difference between the simple and the compound interest compounded every six months at the rate of 10 per cent per annum at the end of two years is Rs. 124·05. What is the sum ?

Options

A. Rs.10000
B. Rs.6000
C. Rs.12000
D. Rs.8000

Answer

To solve the problem, we need to find the principal amount (sum) given the difference between the simple interest (SI) and compound interest (CI) compounded semi-annually at a rate of 10% per annum over 2 years is Rs. 124.05.

### Step-by-Step Solution:

**Step 1: Calculate the Simple Interest (SI)**
The formula for Simple Interest is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount (sum)
- \( R \) = Rate of interest per annum (10%)
- \( T \) = Time in years (2 years)

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

**Step 2: Calculate the Compound Interest (CI)**
The formula for Compound Interest compounded semi-annually is:
\[ \text{CI} = P \left(1 + \frac{R}{n}\right)^{nt} - P \]
Where:
- \( n \) = Number of times interest is compounded per year (2 for semi-annual)
- \( R \) = Rate of interest per annum (10%)
- \( t \) = Time in years (2 years)

Substituting the values:
\[ \text{CI} = P \left(1 + \frac{10}{200}\right)^{2 \times 2} - P \]
\[ \text{CI} = P \left(1 + 0.05\right)^{4} - P \]
\[ \text{CI} = P \left(1.05\right)^{4} - P \]
Calculating \( (1.05)^{4} \):
\[ (1.05)^{4} = 1.21550625 \]
Thus,
\[ \text{CI} = P \times 1.21550625 - P \]
\[ \text{CI} = P(1.21550625 - 1) = P \times 0.21550625 \]

**Step 3: Find the Difference between CI and SI**
According to the problem, the difference between CI and SI is Rs. 124.05:
\[ \text{CI} - \text{SI} = 124.05 \]
Substituting the values we found:
\[ P \times 0.21550625 - \frac{P}{5} = 124.05 \]
To make calculations easier, convert \( \frac{P}{5} \) to a decimal:
\[ \frac{P}{5} = 0.2P \]
So, the equation becomes:
\[ P \times 0.21550625 - 0.2P = 124.05 \]
Combining like terms:
\[ P(0.21550625 - 0.2) = 124.05 \]
Calculating \( 0.21550625 - 0.2 = 0.01550625 \):
\[ P \times 0.01550625 = 124.05 \]

**Step 4: Solve for P**
Now, divide both sides by \( 0.01550625 \):
\[ P = \frac{124.05}{0.01550625} \]
Calculating the right side gives:
\[ P \approx 8000 \]

### Final Answer:
The sum (principal amount) is approximately Rs. 8000.

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