Find the amount and the compound interest on Rs 12800 for 1 years `7(1)/(2)%` per annum, compounded semi- annually.

Answer

To find the amount and the compound interest on Rs 12800 for 1 year at a rate of \(7\frac{1}{2}\%\) per annum, compounded semi-annually, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs 12800
- Time (T) = 1 year
- Rate (R) = \(7\frac{1}{2}\%\) per annum = \(7.5\%\) = \(\frac{15}{2}\%\)

### Step 2: Adjust the time and rate for semi-annual compounding
Since the interest is compounded semi-annually:
- The number of compounding periods (n) = 2 (because there are 2 half-years in 1 year).
- The semi-annual rate = \(\frac{R}{2} = \frac{15/2}{2} = \frac{15}{4}\%\)

### Step 3: Convert the semi-annual rate to a decimal
To use the formula for compound interest, convert the percentage to a decimal:
\[ \text{Semi-annual rate} = \frac{15}{4} \div 100 = \frac{15}{400} = 0.0375 \]

### Step 4: Use the compound interest formula
The formula for the amount (A) when compounded is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Substituting the values we have:
\[ A = 12800 \left(1 + 0.0375\right)^2 \]
\[ A = 12800 \left(1.0375\right)^2 \]

### Step 5: Calculate \( (1.0375)^2 \)
Calculating \( (1.0375)^2 \):
\[ (1.0375)^2 = 1.075140625 \]

### Step 6: Calculate the amount
Now substitute this back into the equation for A:
\[ A = 12800 \times 1.075140625 \]
\[ A \approx 13778.80 \]

### Step 7: Calculate the compound interest
The compound interest (CI) can be calculated using the formula:
\[ CI = A - P \]
Substituting the values:
\[ CI = 13778.80 - 12800 = 978.80 \]

### Final Results
- Amount (A) = Rs 13778.80
- Compound Interest (CI) = Rs 978.80

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