A bank offers compound interest at 20 percent per half year. A customer deposits Rs. 7600 on 1 January and another Rs. 7600 on 1 July of the same year. By the end of the year, how much total interest in rupees will the customer have earned from these two deposits under this half yearly compounding scheme?

Introduction / Context:
This question focuses on compound interest with a compounding period of half a year. A customer makes two equal deposits at different times in the year, and the bank compounds the interest every six months at 20 percent per half year. We are asked to determine the total interest earned by the end of the year. This is a practical type of problem that tests your ability to handle multiple deposits and different effective durations under the same compounding scheme.

Given Data / Assumptions:

• Rate per half year = 20 percent.
• First deposit: Rs. 7600 on 1 January.
• Second deposit: Rs. 7600 on 1 July of the same year.
• Compounding is half yearly, so compounding dates are effectively at the end of each half year.
• We consider the total interest earned by the end of the year.

Concept / Approach:
Under half yearly compounding, each half year the amount is multiplied by (1 + 20/100) = 1.2. The first deposit is invested for two half years (one full year), while the second deposit is invested for one half year only. We compute the amount from each deposit separately using the appropriate number of compounding periods, subtract the original principal for each, and then sum the two interest amounts. This gives the total interest earned by the end of the year.

Step-by-Step Solution:
Step 1: For the first deposit of Rs. 7600 on 1 January, there are two half yearly periods before year end. Step 2: Amount from first deposit after 2 half years: A₁ = 7600 * (1.2)^2. Step 3: Compute (1.2)^2 = 1.44, so A₁ = 7600 * 1.44 = 10944. Step 4: Interest from first deposit = A₁ − 7600 = 10944 − 7600 = 3344. Step 5: For the second deposit of Rs. 7600 on 1 July, there is only one half yearly period before year end. Step 6: Amount from second deposit after 1 half year: A₂ = 7600 * 1.2 = 9120. Step 7: Interest from second deposit = A₂ − 7600 = 9120 − 7600 = 1520. Step 8: Total interest by year end = 3344 + 1520 = 4864. Step 9: Therefore, the customer earns Rs. 4864 in total interest over the year.

Verification / Alternative check:
We can verify by computing the total amount and subtracting the total principal. Total principal = 7600 + 7600 = 15200. Total amount = A₁ + A₂ = 10944 + 9120 = 20064. Interest = Total amount − Total principal = 20064 − 15200 = 4864, which matches our earlier result. This confirms that the total interest has been computed correctly.

Why Other Options Are Wrong:
Rs. 2432 and Rs. 1216 are obtained if we mistakenly calculate interest for only one deposit or for only one half year period. Rs. 3648 corresponds roughly to taking interest for one and a half periods or mixing up the compounding effect. Rs. 9727 is far too high and would imply a much larger effective rate or longer investment duration. Only Rs. 4864 correctly represents the total interest on two deposits of Rs. 7600 each under the stated half yearly compounding scheme.

Common Pitfalls:
A common error is to treat the nominal rate of 20 percent per half year as 20 percent per annum and apply only one compounding factor for the full year, which underestimates the interest on the first deposit. Another mistake is to forget that the second deposit has only one half year to grow, not two. Some students simply double the interest from one deposit instead of accounting for different durations. Carefully tracking how long each deposit earns interest is crucial in such questions.

Final Answer:
By the end of the year, the total compound interest earned from the two Rs. 7600 deposits at 20 percent per half year is Rs. 4864.