The difference between the compound and the simple interest on a sum for 2 years at `10%` per annum, when the interest is compounded annually, is 28. If the interest were compounded halfyearly, the difference in the two interests will be

Options

A. `Rs.44 `
B. `Rs. 28.35 `
C. `Rs. 43.41`
D. `Rs. 43.29 `

Answer

To solve the problem step by step, we will first find the principal amount using the information provided about the difference between compound interest (CI) and simple interest (SI) for 2 years at 10% per annum, and then we will calculate the difference when the interest is compounded half-yearly.

### Step 1: Understand the given information
The difference between CI and SI for 2 years at 10% per annum is given as ₹28.

### Step 2: Use the formula for the difference between CI and SI
The formula for the difference between compound interest and simple interest for 2 years is:
\[ \text{Difference} = \text{SI for 2 years} - \text{CI for 2 years} = \left(\frac{P \times R \times T}{100}\right) - \left(P \left(1 + \frac{R}{100}\right)^T - P\right) \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

For 2 years at 10%:
\[ \text{Difference} = \frac{P \times 10 \times 2}{100} - \left(P \left(1 + \frac{10}{100}\right)^2 - P\right) \]
This simplifies to:
\[ \text{Difference} = \frac{P \times 20}{100} - \left(P \left(1.1^2 - 1\right)\right) \]
Calculating \( 1.1^2 \):
\[ 1.1^2 = 1.21 \implies \text{Difference} = \frac{P \times 20}{100} - P \times 0.21 \]
\[ \text{Difference} = \frac{P \times 20}{100} - \frac{P \times 21}{100} = \frac{P \times (20 - 21)}{100} = -\frac{P}{100} \]

### Step 3: Set the difference equal to ₹28
Since the difference is given as ₹28, we have:
\[ -\frac{P}{100} = 28 \implies P = -28 \times 100 = 2800 \]

### Step 4: Calculate Simple Interest (SI) for 2 years
Using the principal amount \( P = 2800 \):
\[ \text{SI} = \frac{P \times R \times T}{100} = \frac{2800 \times 10 \times 2}{100} = \frac{56000}{100} = 560 \]

### Step 5: Calculate Compound Interest (CI) for 2 years (annually)
\[ \text{CI} = P \left(1 + \frac{R}{100}\right)^T - P = 2800 \left(1 + \frac{10}{100}\right)^2 - 2800 \]
Calculating:
\[ = 2800 \times 1.21 - 2800 = 3388 - 2800 = 588 \]

### Step 6: Find the difference between CI and SI
\[ \text{Difference} = CI - SI = 588 - 560 = 28 \]
This confirms our calculations.

### Step 7: Calculate the difference when compounded half-yearly
When compounded half-yearly, the rate becomes \( \frac{10}{2} = 5\% \) and the time becomes \( 2 \times 2 = 4 \) half-years.

#### Step 7.1: Calculate CI for half-yearly compounding
\[ \text{CI} = P \left(1 + \frac{5}{100}\right)^4 - P \]
Calculating:
\[ = 2800 \left(1.05^4\right) - 2800 \]
Calculating \( 1.05^4 \):
\[ 1.05^4 = 1.21550625 \implies \text{CI} = 2800 \times 1.21550625 - 2800 \]
Calculating:
\[ = 3403.418 - 2800 = 603.418 \]

#### Step 7.2: Calculate the difference in interests
Now, we find the difference between CI and SI:
\[ \text{Difference} = CI - SI = 603.418 - 560 = 43.418 \]

### Final Answer
The difference in the two interests when compounded half-yearly is approximately ₹43.42.

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