A bank offers 15% compound interest per half year. A customer deposits Rs. 8,800 on 1st January and another Rs. 8,800 on 1st July of the same year. What total interest does the customer earn by the end of that year?

Introduction / Context:
This problem deals with multiple deposits made at different times within the same year under half yearly compounding. It illustrates how the time remaining until the end of the year affects the interest earned by each deposit, since earlier deposits earn interest for more periods than later ones.

Given Data / Assumptions:

• Interest rate: 15% per half year.
• First deposit: Rs. 8,800 on 1st January.
• Second deposit: Rs. 8,800 on 1st July.
• We need the total interest earned by the end of the year.
• We assume the financial year ends on 31st December.

Concept / Approach:
Since interest is 15% per half year, there are two compounding periods within the year. The deposit on 1st January remains invested for a full year and therefore experiences two half year compounding periods. The deposit on 1st July remains invested only for half a year and experiences one compounding period. We compute the amount for each deposit separately and subtract the total principal to obtain total interest.

Step-by-Step Solution:
Step 1: For the first deposit, P1 = 8,800 and rate per half year i = 0.15.Step 2: It remains for 2 periods, so amount A1 = P1 * (1 + i)^2.Step 3: Compute A1 = 8,800 * (1.15)^2 = 8,800 * 1.3225 = 11,626 (approximately).Step 4: Interest from the first deposit is I1 = A1 - P1 = 11,626 - 8,800 = 2,826 (approximately). Using exact multiplication gives 2,838 but the given answer uses rounded arithmetic.Step 5: For the second deposit, P2 = 8,800, which is invested for one half year, so amount A2 = P2 * (1 + i) = 8,800 * 1.15 = 10,120.Step 6: Interest from the second deposit is I2 = A2 - P2 = 10,120 - 8,800 = 1,320.Step 7: Total interest earned is approximately I1 + I2 = 2,838 + 1,320 = 4,158 rupees.

Verification / Alternative check:
We can also compute the combined amount directly. The total principal is 8,800 + 8,800 = 17,600. The combined amount at the end of the year is A1 + A2 = 11,638 (rounded) plus 10,120, which is about 21,758. The difference between the combined amount and the total principal is about 4,158 rupees, which matches the total interest obtained by adding I1 and I2 separately.

Why Other Options Are Wrong:

• Rs. 8315: This is far too large to be interest on only 17,600 rupees over parts of one year at the stated rate.
• Rs. 2079: This is too small, as the first deposit alone earns more interest than this amount.
• Rs. 1039: Much too low and inconsistent with a 15% rate applied over several periods.

Common Pitfalls:
Students sometimes treat the full year rate as 15% and forget that it is per half year, or they incorrectly assume both deposits earn interest for the same duration. Another mistake is adding 15% twice linearly for the first deposit instead of compounding it, which underestimates the interest. Careful tracking of time periods and correct use of the compound interest formula for each deposit separately are essential.

Final Answer:
The customer earns a total interest of Rs. 4158 by the end of the year.