A bank offers `15%` compound Interest per half year. A customer deposits `Rs. 2400` each on 1st January and 1st July of a year. At the end of the year, the amount he would have gained by way of Interest is

Options

A. `Rs. 597.81'
B. `Rs. 553.5`
C. `Rs. 557.81 `
D. `Rs. 567.21`

Answer

To solve the problem step by step, we will use the formula for compound interest and calculate the interest earned from two deposits made at different times.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Rate of interest (R) = 15% per half year
- First deposit (P1) = Rs. 2400 on 1st January
- Second deposit (P2) = Rs. 2400 on 1st July

2. **Calculate the Amount for the First Deposit:**
- The first deposit is compounded for 2 half-year periods (from January to July and from July to December).
- Using the formula for compound interest:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
- For the first deposit:
\[ A_1 = 2400 \left(1 + \frac{15}{100}\right)^2 \]
- Simplifying:
\[ A_1 = 2400 \left(1 + 0.15\right)^2 = 2400 \left(1.15\right)^2 \]
- Calculate \( (1.15)^2 \):
\[ (1.15)^2 = 1.3225 \]
- Therefore:
\[ A_1 = 2400 \times 1.3225 = 3174 \]

3. **Calculate the Amount for the Second Deposit:**
- The second deposit is compounded for 1 half-year period (from July to December).
- For the second deposit:
\[ A_2 = 2400 \left(1 + \frac{15}{100}\right)^1 \]
- Simplifying:
\[ A_2 = 2400 \left(1 + 0.15\right) = 2400 \times 1.15 = 2760 \]

4. **Calculate the Total Amount at the End of the Year:**
- Total amount (A) at the end of the year is the sum of both amounts:
\[ A = A_1 + A_2 = 3174 + 2760 = 5934 \]

5. **Calculate the Total Principal:**
- The total principal (P) deposited is:
\[ P = P1 + P2 = 2400 + 2400 = 4800 \]

6. **Calculate the Compound Interest:**
- The compound interest (CI) is given by:
\[ CI = A - P = 5934 - 4800 = 1134 \]

### Final Answer:
The amount gained by way of interest at the end of the year is **Rs. 1134**.