The difference between simple interest and compound interest on a sum for 2 years at `8%` when the interest is compounded annually is Rs. 16. If the interest were compounded half yearly the difference in two interests would be nearly-

Options

A. Rs.16
B. Rs.16.80
C. Rs.21.85
D. Rs.24.64

Answer

To solve the problem step by step, we will first calculate the principal amount using the given difference between simple interest (SI) and compound interest (CI) when compounded annually. Then, we will find the difference when the interest is compounded half-yearly.

### Step 1: Understand the given information
We know:
- The difference between CI and SI for 2 years at 8% compounded annually is Rs. 16.
- We need to find the difference when compounded half-yearly.

### Step 2: Set up the equations
The formula for the compound interest (CI) when compounded annually is:
\[ CI = P \left(1 + \frac{R}{100}\right)^n - P \]
where:
- \( P \) = Principal amount
- \( R \) = Rate of interest (8%)
- \( n \) = Number of years (2)

The formula for simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
where:
- \( T \) = Time in years (2)

### Step 3: Calculate the difference
From the problem, we have:
\[ CI - SI = 16 \]

Substituting the formulas:
\[ P \left(1 + \frac{8}{100}\right)^2 - P - \left(\frac{P \times 8 \times 2}{100}\right) = 16 \]

### Step 4: Simplify the equation
Calculating \( \left(1 + \frac{8}{100}\right)^2 \):
\[ \left(1 + 0.08\right)^2 = (1.08)^2 = 1.1664 \]

Now substituting back:
\[ P(1.1664) - P - \left(\frac{P \times 16}{100}\right) = 16 \]
\[ P(1.1664 - 1 - 0.16) = 16 \]
\[ P(0.0064) = 16 \]

### Step 5: Solve for \( P \)
\[ P = \frac{16}{0.0064} = 2500 \]

### Step 6: Calculate the difference for half-yearly compounding
When interest is compounded half-yearly, the rate is halved and the number of compounding periods is doubled:
- New rate \( R = \frac{8}{2} = 4\% \)
- New time \( n = 2 \times 2 = 4 \) (because it is compounded every 6 months)

Using the CI formula for half-yearly compounding:
\[ CI_{half-yearly} = P \left(1 + \frac{4}{100}\right)^4 - P \]
\[ = 2500 \left(1.04\right)^4 - 2500 \]

Calculating \( (1.04)^4 \):
\[ (1.04)^4 \approx 1.1699 \]

So,
\[ CI_{half-yearly} = 2500 \times 1.1699 - 2500 \]
\[ = 2924.75 - 2500 = 424.75 \]

### Step 7: Calculate SI for 2 years
Using the SI formula:
\[ SI = \frac{2500 \times 8 \times 2}{100} = 400 \]

### Step 8: Find the difference
Now, we find the difference:
\[ CI_{half-yearly} - SI = 424.75 - 400 = 24.75 \]

### Conclusion
The difference in interest when compounded half-yearly is approximately **Rs. 24.75**.

---