A bank offers 10% compound interest per half year. A customer deposits Rs 3600 on 1st January and another Rs 3600 on 1st July of the same year. What total amount does the customer gain by way of interest by the end of that year?

Introduction / Context:
This question involves multiple deposits within the same year under a scheme that pays compound interest per half year. You must calculate the interest earned separately on each deposit because they are invested for different lengths of time, and then add the results to obtain the total interest gained at the end of the year.

Given Data / Assumptions:

• Rate of compound interest = 10% per half year.
• First deposit = Rs 3600 on 1st January.
• Second deposit = Rs 3600 on 1st July.
• Interest is compounded half-yearly.
• We are interested in the interest amount at the end of the same calendar year.

Concept / Approach:
For the first deposit, invested on 1st January, there will be two half-yearly compounding periods (January to June and July to December). For the second deposit, placed on 1st July, there will be only one half-yearly compounding period (July to December). We compute the final amount for each deposit using A = P * (1 + r/100)^n and subtract the principal to get the interest. Finally, we add the two interest amounts.

Step-by-Step Solution:
Step 1: For the 1st deposit of Rs 3600: Number of half-yearly periods n1 = 2, rate per period r = 10%. Step 2: Amount after 2 periods: A1 = 3600 * (1.10)^2 = 3600 * 1.21 = Rs 4356. Step 3: Interest from 1st deposit: I1 = A1 - 3600 = 4356 - 3600 = Rs 756. Step 4: For the 2nd deposit of Rs 3600: Number of periods n2 = 1, same rate 10%. Step 5: Amount after 1 period: A2 = 3600 * 1.10 = Rs 3960. Step 6: Interest from 2nd deposit: I2 = 3960 - 3600 = Rs 360. Step 7: Total interest earned: I_total = I1 + I2 = 756 + 360 = Rs 1116.
Verification / Alternative check:
You can also compute the combined final amount: 4356 + 3960 = 8316. Total principal invested is 3600 + 3600 = 7200, and 8316 - 7200 = 1116, confirming the total interest.
Why Other Options Are Wrong:
Rs 2232 is double the correct interest and would correspond to miscounting the periods. Rs 558 or Rs 279 correspond roughly to the interest from just one of the deposits, not both.
Common Pitfalls:
A typical mistake is to treat the entire Rs 7200 as if it were deposited at the same time or to use a single interest period for both deposits. Another error is to convert the half-yearly rate into an annual rate incorrectly and then apply annual compounding instead of half-yearly compounding.
Final Answer:
The total interest earned by the customer at the end of the year is Rs 1116.