A bank offers 15% compound interest per half-year. A customer deposits Rs 7200 on 1st January and another Rs 7200 on 1st July of the same year. What total interest (in rupees) will the customer earn by the end of that year?

Introduction / Context:
This question involves compound interest with multiple deposits at different times within the same year. The bank offers 15% compound interest per half-year. One deposit is made at the beginning of the year, and the second deposit is made halfway through the year. We must calculate the interest earned on each deposit separately based on how long it stays in the account and then add the interests to get the total interest for the year. This tests the understanding of time-weighted compounding when there are multiple cash flows.

Given Data / Assumptions:

• Bank offers 15% compound interest per half-year (effectively 30% nominal per annum).
• First deposit: Rs 7,200 on 1st January.
• Second deposit: Rs 7,200 on 1st July.
• Interest is compounded every half-year.
• We want interest earned by 31st December of the same year.

Concept / Approach:
Since the compounding period is half-yearly, each half-year earns 15%. The first deposit remains invested for two half-year periods (from 1st January to 1st July, then to 31st December). The second deposit is invested for only one half-year (from 1st July to 31st December). For each deposit, we compute its amount using A = P * (1 + 0.15)^n, where n is the number of half-year periods. The interest for that deposit is A - P. Finally, we sum the interests from both deposits to get the total interest for the year.

Step-by-Step Solution:
For the first deposit: P1 = 7,200, n1 = 2 half-years. Amount on first deposit: A1 = 7,200 * (1.15)^2. Compute (1.15)^2 = 1.3225, so A1 = 7,200 * 1.3225 = 9,522. Interest on first deposit: I1 = A1 - P1 = 9,522 - 7,200 = 2,322. For the second deposit: P2 = 7,200, n2 = 1 half-year. Amount on second deposit: A2 = 7,200 * 1.15 = 8,280. Interest on second deposit: I2 = A2 - P2 = 8,280 - 7,200 = 1,080. Total interest earned in the year: I = I1 + I2 = 2,322 + 1,080 = 3,402.

Verification / Alternative check:
Another way to think about it is to treat interest separately: for the first deposit, total gain factor over the year is 1.3225, giving 32.25% overall on 7,200, which equals 2,322. For the second deposit, the money is in for only one half-year at 15%, giving 7,200 * 0.15 = 1,080. Adding these again results in 3,402. Since both methods give the same answer, the calculations are internally consistent.

Why Other Options Are Wrong:
Rs 850 and Rs 1,701 are much too small compared to a 15% half-yearly rate on two deposits of 7,200 each. Rs 6,803 is too large and would require substantially higher rates or more periods of compounding. Rs 2,552 is also smaller than the correctly computed combined interest. Only Rs 3,402 matches the detailed calculation from the bank's compounding rule.

Common Pitfalls:
A common mistake is to treat both deposits as if they were invested for the full year, which overstates the interest on the second deposit. Another error is to convert 15% per half-year into 30% and then apply it as a single annual simple interest rate, ignoring compounding. Always track how many compounding periods each deposit experiences and apply the compound interest formula to each cash flow separately.

Final Answer:
The total interest earned by the end of the year is Rs 3,402.