The compound interest on a certain sum for 1 year at `8%` p.a. interest compounded half-yearly. In Rs. 652.80 . What will be the simple interest on the same sum for `2(1)/(2)` years at the same rate of interest ?

Options

A. Rs. 1,680
B. Rs. 1,720
C. Rs. 1, 740
D. Rs. 1,600

Answer

To solve the problem step by step, we will first determine the principal amount using the given compound interest, and then we will calculate the simple interest for the specified duration.

### Step 1: Understand the given information
We know that:
- Compound Interest (CI) for 1 year = Rs. 652.80
- Rate of interest = 8% per annum
- Compounding is half-yearly.

### Step 2: Calculate the half-yearly interest rate
Since the interest is compounded half-yearly, we need to divide the annual interest rate by 2:
\[ \text{Half-yearly interest rate} = \frac{8\%}{2} = 4\% \]

### Step 3: Determine the number of compounding periods
For 1 year, there are 2 compounding periods (half-yearly):
\[ \text{Number of periods} = 2 \]

### Step 4: Use the formula for compound interest
The formula for compound interest is:
\[ CI = P \left(1 + \frac{r}{100}\right)^n - P \]
Where:
- \(CI\) = Compound Interest
- \(P\) = Principal
- \(r\) = Rate of interest per period
- \(n\) = Number of periods

Rearranging the formula to find \(P\):
\[ 652.80 = P \left(1 + \frac{4}{100}\right)^2 - P \]
\[ 652.80 = P \left(1.04^2 - 1\right) \]

### Step 5: Calculate \(1.04^2\)
Calculating \(1.04^2\):
\[ 1.04^2 = 1.0816 \]
Thus,
\[ 652.80 = P (1.0816 - 1) = P (0.0816) \]

### Step 6: Solve for \(P\)
Now, we can solve for \(P\):
\[ P = \frac{652.80}{0.0816} = 8000 \]

### Step 7: Calculate Simple Interest for \(2\frac{1}{2}\) years
Now that we have the principal, we can calculate the simple interest (SI) for \(2\frac{1}{2}\) years (which is \(2.5\) years):
\[ \text{SI} = \frac{P \times r \times t}{100} \]
Where:
- \(P = 8000\)
- \(r = 8\%\)
- \(t = 2.5\)

Substituting the values:
\[ \text{SI} = \frac{8000 \times 8 \times 2.5}{100} \]
\[ \text{SI} = \frac{8000 \times 20}{100} = 1600 \]

### Final Answer
The simple interest on the same sum for \(2\frac{1}{2}\) years at the same rate of interest is **Rs. 1600**.

---