A sum of Rs. 2500 is invested for 2 years at 20% per annum, interest compounded half-yearly. Find the compound interest.

Options

A. Rs. 3660.25
B. Rs. 1660.25
C. Rs. 1160.25
D. Rs. 1330

Explanation

To find the compound interest on a sum of Rs. 2500 invested for 2 years at an interest rate of 20% per annum, compounded half-yearly, we will follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs. 2500
- Rate of interest (R) = 20% per annum
- Time (T) = 2 years
- Since the interest is compounded half-yearly, we need to adjust the rate and time accordingly.

### Step 2: Adjust the rate and time for half-yearly compounding
- The rate for half-yearly compounding = R/2 = 20%/2 = 10%
- The time period in half-years = T × 2 = 2 × 2 = 4 half-years

### Step 3: Use the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Where:
- A = the amount after time T
- P = principal amount
- R = rate of interest per period
- n = number of compounding periods

Substituting the values:
\[ A = 2500 \left(1 + \frac{10}{100}\right)^4 \]

### Step 4: Calculate the amount (A)
\[ A = 2500 \left(1 + 0.10\right)^4 \]
\[ A = 2500 \left(1.10\right)^4 \]

Now, calculate \( (1.10)^4 \):
\[ (1.10)^4 = 1.4641 \]

So,
\[ A = 2500 \times 1.4641 \]
\[ A = 3660.25 \]

### Step 5: Calculate the compound interest (CI)
Compound Interest (CI) is given by:
\[ CI = A - P \]
\[ CI = 3660.25 - 2500 \]
\[ CI = 1160.25 \]

### Final Answer
The compound interest is Rs. 1160.25.

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