A bank offers 5% compound interest calculated on half- yearly basis. A customer deposits 1600 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. 120
B. 121
C. 122
D. 123

Answer

To solve the problem step by step, we will calculate the compound interest earned by the customer on two separate deposits made at different times during the year.

### Step 1: Understand the Interest Rate
The bank offers a compound interest rate of 5% per annum, which is compounded half-yearly. Therefore, the interest rate for each half-year period is:
\[ \text{Half-yearly interest rate} = \frac{5\%}{2} = 2.5\% \]
In decimal form, this is:
\[ \text{Half-yearly interest rate} = \frac{2.5}{100} = 0.025 \]

### Step 2: Calculate the Amount from the First Deposit
The first deposit of ₹1600 is made on January 1st and will be invested for the entire year (2 half-year periods).

Using the formula for compound interest:
\[ A = P \left(1 + r\right)^n \]
where:
- \(A\) = amount after time \(n\)
- \(P\) = principal amount (initial deposit)
- \(r\) = interest rate per period
- \(n\) = number of periods

For the first deposit:
- \(P = 1600\)
- \(r = 0.025\)
- \(n = 2\)

Calculating the amount:
\[ A_1 = 1600 \left(1 + 0.025\right)^2 \]
\[ A_1 = 1600 \left(1.025\right)^2 \]
\[ A_1 = 1600 \times 1.050625 = 1681 \]

### Step 3: Calculate the Amount from the Second Deposit
The second deposit of ₹1600 is made on July 1st and will be invested for only half a year (1 half-year period).

For the second deposit:
- \(P = 1600\)
- \(r = 0.025\)
- \(n = 1\)

Calculating the amount:
\[ A_2 = 1600 \left(1 + 0.025\right)^1 \]
\[ A_2 = 1600 \times 1.025 = 1640 \]

### Step 4: Calculate the Total Amount at the End of the Year
Now, we will add the amounts from both deposits:
\[ \text{Total Amount} = A_1 + A_2 = 1681 + 1640 = 3321 \]

### Step 5: Calculate the Total Interest Earned
The total amount deposited by the customer is:
\[ \text{Total Deposits} = 1600 + 1600 = 3200 \]

The total interest earned is:
\[ \text{Interest} = \text{Total Amount} - \text{Total Deposits} = 3321 - 3200 = 121 \]

### Final Answer
Thus, the amount gained by way of interest at the end of the year is:
\[ \text{Interest} = \text{₹121} \]

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