Rs 6000 when invested at a certain rate at SI for 2 years, it fetches Rs 1200. If same sum is invested at same rate for a year compounded half - yearly then find compound'interest.

Options

A. A)Rs 615
B. B)Rs 600
C. C)Rs 1200
D. D)Rs 5854

Answer

To solve the problem step by step, we will follow the process of calculating the rate of interest from the simple interest and then use that rate to find the compound interest when compounded half-yearly.

### Step 1: Calculate the Rate of Interest from Simple Interest
We know that Simple Interest (SI) is given by the formula:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount (Rs 6000)
- \( R \) = Rate of interest (unknown)
- \( T \) = Time in years (2 years)

From the problem, we know that the Simple Interest earned in 2 years is Rs 1200. So we can set up the equation:
\[ 1200 = \frac{6000 \times R \times 2}{100} \]

### Step 2: Simplify the Equation
First, we can simplify the equation:
\[ 1200 = \frac{12000R}{100} \]
\[ 1200 = 120R \]

### Step 3: Solve for R
Now, divide both sides by 120:
\[ R = \frac{1200}{120} = 10\% \]

### Step 4: Calculate Compound Interest for 1 Year Compounded Half-Yearly
Now that we have the rate of interest \( R = 10\% \), we need to find the compound interest when the same principal is invested for 1 year compounded half-yearly.

#### Step 4.1: Adjust the Rate and Time
When compounded half-yearly:
- The rate for each half-year = \( \frac{10}{2} = 5\% \)
- The time period in half-year cycles = 1 year = 2 half-year cycles

#### Step 4.2: Use the Compound Interest Formula
The formula for Compound Interest (CI) is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( A \) = Amount after time \( n \)
- \( P \) = Principal amount (Rs 6000)
- \( r \) = Rate of interest per period (5%)
- \( n \) = Number of periods (2)

Substituting the values:
\[ A = 6000 \left(1 + \frac{5}{100}\right)^2 \]
\[ A = 6000 \left(1 + 0.05\right)^2 \]
\[ A = 6000 \left(1.05\right)^2 \]

#### Step 4.3: Calculate \( (1.05)^2 \)
Calculating \( (1.05)^2 \):
\[ (1.05)^2 = 1.1025 \]

#### Step 4.4: Calculate the Amount
Now substitute back to find \( A \):
\[ A = 6000 \times 1.1025 = 6615 \]

### Step 5: Calculate the Compound Interest
Now, we find the Compound Interest (CI):
\[ \text{CI} = A - P \]
\[ \text{CI} = 6615 - 6000 = 615 \]

### Final Answer
The compound interest when Rs 6000 is invested at the same rate for 1 year compounded half-yearly is **Rs 615**.

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