What is the compound interest on a sum of Rs12,600 for `1(1)/2`years at 20% p.a., if the interest is compounded half yearly? (Nearest to Rs)

Options

A. Rs 4.251
B. Rs 4,171
C. Rs 4,169
D. Rs 4,269

Answer

To find the compound interest on a sum of Rs 12,600 for 1.5 years at an interest rate of 20% per annum, compounded half-yearly, we can follow these steps:

### Step 1: Identify the Principal Amount and the Rate of Interest
- **Principal (P)** = Rs 12,600
- **Rate of Interest (R)** = 20% per annum

### Step 2: Convert the Time Period into Half-Yearly Terms
- The time period is given as 1.5 years, which can be converted into half-year periods:
- \(1.5 \text{ years} = 1 \text{ year} + 0.5 \text{ year} = 2 \text{ half-years}\)

### Step 3: Adjust the Rate of Interest for Half-Yearly Compounding
- Since the interest is compounded half-yearly, we need to divide the annual interest rate by 2:
- \(R = \frac{20}{2} = 10\% \text{ per half-year}\)

### Step 4: 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
- \(R\) = Rate of interest per period
- \(n\) = Number of compounding periods

Substituting the values:
- \(P = 12,600\)
- \(R = 10\%\)
- \(n = 3\) (since 1.5 years equals 3 half-year periods)

### Step 5: Calculate the Amount
\[ A = 12,600 \left(1 + \frac{10}{100}\right)^3 \]
\[ A = 12,600 \left(1 + 0.10\right)^3 \]
\[ A = 12,600 \left(1.10\right)^3 \]
Calculating \( (1.10)^3 \):
\[ (1.10)^3 = 1.331 \]
Now substituting back:
\[ A = 12,600 \times 1.331 = 16,770.6 \]

### Step 6: Calculate the Compound Interest
The compound interest (CI) is given by:
\[ CI = A - P \]
Substituting the values:
\[ CI = 16,770.6 - 12,600 = 4,170.6 \]
Rounding to the nearest rupee:
\[ CI \approx 4,171 \]

### Final Answer
The compound interest on the sum of Rs 12,600 for 1.5 years at 20% p.a., compounded half-yearly, is approximately Rs 4,171.

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