A bank offers 8% compound interest calculated on half-yearly basis. A customer deposits Rs 2000 each on `1^(st)` January and `1^(st)` July of a year. At the end of the year, the amount he would have gained as interest, is _______

Options

A. Rs 220. 20
B. Rs 221. 60
C. Rs 222. 50
D. Rs 243. 20

Answer

To solve the problem, we need to calculate the compound interest earned on two deposits made at different times during the year. Here's a step-by-step breakdown of the solution:

### Step 1: Understand the Interest Rate and Time Period
The bank offers an 8% compound interest calculated on a half-yearly basis. This means:
- The annual interest rate is 8%.
- The half-yearly interest rate is \( \frac{8\%}{2} = 4\% \).

### Step 2: Calculate Interest for the First Deposit
The first deposit of Rs 2000 is made on January 1st and is invested for the entire year (12 months), but we will calculate it in two halves (6 months each).

1. **First Half (January to July)**:
- Principal (P) = Rs 2000
- Rate (R) = 4%
- Time (T) = 6 months

The interest for the first half can be calculated using the formula:
\[ \text{Interest} = P \times \frac{R}{100} \times T \]
\[ \text{Interest} = 2000 \times \frac{4}{100} \times 0.5 = 2000 \times 0.04 = 80 \text{ Rs} \]

### Step 3: Calculate the Total Amount After the First Half
After the first half, the total amount in the account will be:
\[ \text{Total Amount} = \text{Principal} + \text{Interest} = 2000 + 80 = 2080 \text{ Rs} \]

### Step 4: Calculate Interest for the Second Deposit
The second deposit of Rs 2000 is made on July 1st and is invested for the second half of the year (6 months).

1. **Second Half (July to December)**:
- Principal (P) = Rs 2000
- Rate (R) = 4%
- Time (T) = 6 months

The interest for the second deposit is calculated similarly:
\[ \text{Interest} = 2000 \times \frac{4}{100} \times 0.5 = 2000 \times 0.04 = 80 \text{ Rs} \]

### Step 5: Calculate Total Amount After the Second Half
Now, we need to calculate the interest on the total amount from the first deposit for the second half of the year.

1. **Interest on Total Amount from First Deposit**:
- Principal (P) = Rs 2080 (total amount after first half)
- Rate (R) = 4%
- Time (T) = 6 months

The interest is calculated as:
\[ \text{Interest} = 2080 \times \frac{4}{100} \times 0.5 = 2080 \times 0.04 = 83.2 \text{ Rs} \]

### Step 6: Calculate Total Interest Earned
Finally, we sum up all the interest earned from both deposits:
- Interest from the first deposit (for the first half) = 80 Rs
- Interest from the second deposit (for the second half) = 80 Rs
- Interest from the first deposit (for the second half) = 83.2 Rs

Total Interest = \( 80 + 80 + 83.2 = 243.2 \text{ Rs} \)

### Final Answer
The total amount gained as interest at the end of the year is **Rs 243.20**.

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