Find CI paid when a sum of Rs. 1,60,000 is invested for 2 years at 10% per annum compounded half yearly :

Question

Accepted Answer

To determine the compound interest (CI) paid when a sum of Rs. 1,60,000 is invested for 2 years at 10% per annum compounded half-yearly, we need to follow these steps:

1. **Understanding the Compounding Frequency**: Since the interest is compounded half-yearly, the annual interest rate of 10% is divided into two periods per year, making it 5% per half-year.

2. **Number of Compounding Periods**: Over 2 years, there are 4 half-yearly periods (2 years x 2 periods per year).

3. **Compound Interest Formula**: The formula for compound interest is:
   $$
   A = P \left(1 + \frac{r}{n}\right)^{nt}
   $$
   where:
   - $A$ is the amount after time $t$,
   - $P$ is the principal amount (Rs. 1,60,000),
   - $r$ is the annual interest rate (10% or 0.10),
   - $n$ is the number of times interest is compounded per year (2 for half-yearly),
   - $t$ is the time in years (2 years).

4. **Calculation**:
   $$
   A = 1,60,000 \left(1 + \frac{0.10}{2}\right)^{2 \times 2}
   $$
   $$
   A = 1,60,000 \left(1 + 0.05\right)^{4}
   $$
   $$
   A = 1,60,000 \times (1.21550625)
   $$
   $$
   A = 1,94,481
   $$

5. **Compound Interest**: The compound interest is the total amount minus the principal:
   $$
   \text{CI} = A - P = 1,94,481 - 1,60,000 = 34,481
   $$

Thus, the compound interest paid is Rs. 34,481, which corresponds to Option 2.

**Explanation for Other Options**:

- **Option 1 (Rs. 34,846)**: This option is incorrect because the calculated compound interest using the correct formula and values results in Rs. 34,481, not Rs. 34,846.

- **Option 3 (Rs. 19,448)**: This option is incorrect as it significantly underestimates the compound interest. The calculation clearly shows that the interest accrued over 2 years with half-yearly compounding is much higher.

- **Option 4 (Rs. 37,946)**: This option is incorrect because it overestimates the compound interest. The correct calculation yields Rs. 34,481, which is less than Rs. 37,946.

In conclusion, the correct compound interest paid is Rs. 34,481, making Option 2 the correct choice.