The compound interest on `Rs.8,000` at `15%` per annum for 2 years 4 months, compounded annually is:

Options

A. `Rs. 2980`
B. `Rs.3091`
C. `Rs.3109`
D. `Rs.3100`

Answer

To find the compound interest on Rs. 8,000 at a rate of 15% per annum for a period of 2 years and 4 months, compounded annually, we can follow these steps:

### Step 1: Convert the time period into years
The total time period is 2 years and 4 months. To convert 4 months into years:
\[ \text{4 months} = \frac{4}{12} = \frac{1}{3} \text{ years} \]
So, the total time period in years is:
\[ 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3} \text{ years} \]

### Step 2: Calculate the amount after 2 years and 4 months
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \(A\) = the amount after time \(n\)
- \(P\) = principal amount (Rs. 8,000)
- \(r\) = rate of interest (15%)
- \(n\) = time in years (\(\frac{7}{3}\))

Substituting the values into the formula:
\[ A = 8000 \left(1 + \frac{15}{100}\right)^{\frac{7}{3}} \]
\[ A = 8000 \left(1 + 0.15\right)^{\frac{7}{3}} = 8000 \left(1.15\right)^{\frac{7}{3}} \]

### Step 3: Calculate \( (1.15)^{\frac{7}{3}} \)
To calculate \( (1.15)^{\frac{7}{3}} \), we can first find \( (1.15)^7 \) and then take the cube root:
\[ (1.15)^7 \approx 2.5023 \quad (\text{using a calculator}) \]
Now, we need to find the cube root of \( 2.5023 \):
\[ (2.5023)^{\frac{1}{3}} \approx 1.397 \]

### Step 4: Calculate the total amount
Now substituting back into the amount formula:
\[ A \approx 8000 \times 1.397 \approx 11177.60 \]

### Step 5: Calculate the compound interest
Compound Interest (CI) is given by:
\[ CI = A - P \]
Where \(A\) is the amount after the time period and \(P\) is the principal.
\[ CI \approx 11177.60 - 8000 = 3177.60 \]

Thus, the compound interest on Rs. 8,000 at 15% per annum for 2 years and 4 months is approximately **Rs. 3177.60**.

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