The compound interest on a sum of Rs. 15,000 at 15% p.a. for `2 (2)/(3)` years, interest compounded annually , is (nearest to Rs. 1):

Options

A. Rs. 6,795
B. Rs. 6,815
C. Rs. 6,781
D. Rs. 6,821

Answer

To find the compound interest on a sum of Rs. 15,000 at an interest rate of 15% per annum for \(2 \frac{2}{3}\) years (which is equivalent to 2 years and 8 months), we can follow these steps:

### Step 1: Understand the Time Period
Convert \(2 \frac{2}{3}\) years into a more usable format:
- \(2 \frac{2}{3}\) years = 2 years + \(\frac{2}{3}\) years
- \(\frac{2}{3}\) years = \(\frac{2}{3} \times 12\) months = 8 months

### Step 2: Use the Compound Interest Formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Where:
- \(A\) = Amount after time \(n\)
- \(P\) = Principal amount (Rs. 15,000)
- \(R\) = Rate of interest (15%)
- \(n\) = Time period in years

### Step 3: Calculate for the First 2 Years
For the first 2 years:
\[ A_2 = 15000 \left(1 + \frac{15}{100}\right)^2 \]
Calculating this:
\[ A_2 = 15000 \left(1 + 0.15\right)^2 = 15000 \left(1.15\right)^2 \]
\[ A_2 = 15000 \times 1.3225 = 19837.5 \]

### Step 4: Calculate for the Next 8 Months
Now, we need to calculate the interest for the next 8 months. The effective rate for 8 months is:
\[ \text{Rate for 8 months} = \frac{15}{12} \times 8 = 10\% \]
So, we apply this rate to the amount after 2 years:
\[ A_3 = A_2 \left(1 + \frac{10}{100}\right) = 19837.5 \times 1.10 \]
Calculating this:
\[ A_3 = 19837.5 \times 1.10 = 21821.25 \]

### Step 5: Calculate the Compound Interest
Now, we find the compound interest (CI) by subtracting the principal from the total amount:
\[ CI = A - P = 21821.25 - 15000 = 6821.25 \]

### Step 6: Round the Result
Rounding \(6821.25\) to the nearest whole number gives us:
\[ CI \approx 6821 \]

### Final Answer
The compound interest on the sum of Rs. 15,000 at 15% p.a. for \(2 \frac{2}{3}\) years is approximately Rs. 6821.

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