The compound interest on a sum of ₹15625 for `2 (3)/(4)` years at 16% per annum, The interest compound annually is :

Options

A. ₹6661
B. ₹5400
C. ₹7932
D. ₹7923

Answer

To calculate the compound interest on a sum of ₹15625 for \(2 \frac{3}{4}\) years at 16% per annum, we can follow these steps:

### Step 1: Convert the time into years
The time given is \(2 \frac{3}{4}\) years. This can be converted into an improper fraction:
\[ 2 \frac{3}{4} = 2 + \frac{3}{4} = \frac{8}{4} + \frac{3}{4} = \frac{11}{4} \text{ years} \]

**Hint:** Always convert mixed numbers into improper fractions for easier calculations.

### Step 2: Identify the principal, rate, and time
- Principal (P) = ₹15625
- Rate (R) = 16% per annum
- Time (T) = \( \frac{11}{4} \) years

### Step 3: Use the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- \(A\) is the amount after time \(T\),
- \(P\) is the principal,
- \(R\) is the rate of interest,
- \(T\) is the time in years.

### Step 4: Substitute the values into the formula
Substituting the values into the formula:
\[ A = 15625 \left(1 + \frac{16}{100}\right)^{\frac{11}{4}} \]
\[ A = 15625 \left(1 + 0.16\right)^{\frac{11}{4}} = 15625 \left(1.16\right)^{\frac{11}{4}} \]

### Step 5: Calculate \( (1.16)^{\frac{11}{4}} \)
To calculate \( (1.16)^{\frac{11}{4}} \), we can find \( (1.16)^2 \) and then raise it to the power of \( \frac{11}{4} \):
\[ (1.16)^2 = 1.3456 \]
Now we need to calculate \( (1.3456)^{\frac{11}{4}} \).

For simplicity, we can use a calculator or approximate it:
\[ (1.16)^{2.75} \approx 1.487 \]

### Step 6: Calculate the amount \(A\)
Now substituting back:
\[ A \approx 15625 \times 1.487 \approx 23205.625 \]

### Step 7: Calculate the compound interest
The compound interest (CI) can be calculated as:
\[ CI = A - P \]
\[ CI = 23205.625 - 15625 = 7580.625 \]

### Final Answer
The compound interest on a sum of ₹15625 for \(2 \frac{3}{4}\) years at 16% per annum is approximately ₹7580.63.

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