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 interest in rupees will the customer have earned in total?

Introduction / Context:
This question involves compound interest with a high rate applied per half year, along with two separate deposits at different times within the year. The rate is 20 percent per half year, which is effectively 40 percent per annum nominal, but the key point is that compounding is done every half year. We must treat each deposit separately, depending on how many half years it stays invested, and then add the interests.

Given Data / Assumptions:
- Rate of compound interest: 20 percent per half year.
- First deposit: Rs 7600 on 1 January, remaining for two half year periods (January to June and July to December).
- Second deposit: Rs 7600 on 1 July, remaining for one half year period (July to December).
- Interest is compounded at the end of each half year.
- We must find the total interest earned by the end of the year.

Concept / Approach:
For compound interest, the amount after n periods at rate r per period is:
A = P * (1 + r)^nHere, r is the half yearly rate expressed as a decimal. For the first deposit, n = 2 half years; for the second deposit, n = 1 half year. We compute the compound amount for each deposit separately and subtract the original principal contributed by that deposit to get the interest. Adding both interests gives the total interest for the year.

Step-by-Step Solution:
Half yearly rate r = 20 % = 0.20.First deposit: P1 = 7600, n1 = 2 half years.Amount from first deposit: A1 = 7600 * (1.20)^2.(1.20)^2 = 1.44, so A1 = 7600 * 1.44 = 10944.Interest from first deposit: I1 = A1 − P1 = 10944 − 7600 = 3344.Second deposit: P2 = 7600, n2 = 1 half year.Amount from second deposit: A2 = 7600 * (1.20)^1 = 7600 * 1.20 = 9120.Interest from second deposit: I2 = A2 − P2 = 9120 − 7600 = 1520.Total interest for the year: I_total = I1 + I2 = 3344 + 1520 = 4864.Therefore, the customer earns Rs 4864 as interest by the end of the year.

Verification / Alternative check:
We can also check by total amount and total principal. The total amount at year end is A1 + A2 = 10944 + 9120 = 20064. The total principal deposited is P1 + P2 = 7600 + 7600 = 15200. The difference 20064 − 15200 = 4864 again gives the total interest. This matches the earlier calculation, confirming the result.

Why Other Options Are Wrong:
Rs 2432 and Rs 1216 are too small and would correspond to only one of the two deposits or to a much lower rate. Rs 9727 is larger than the total amount of interest plus principal for one deposit and is unrealistic for the given data. Rs 3648 might arise from miscalculating one of the deposits or forgetting the second half year of compounding for the first deposit. Only Rs 4864 agrees with correct compound interest calculations for both deposits.

Common Pitfalls:
Common mistakes include treating the rate of 20 percent per half year as 20 percent per year, ignoring the second deposit, or forgetting that the first deposit enjoys two compounding periods while the second deposit enjoys only one. Another error is to add simple interest instead of using compounding formulas. Clearly identifying the number of compounding periods for each deposit and applying A = P * (1 + r)^n correctly prevents these errors.

Final Answer:
The total interest earned by the customer by the end of the year is Rs 4864.