A bank offers compound interest at the rate of 15% per half year. A customer deposits Rs. 2400 on 1st January and another Rs. 2400 on 1st July of the same year. What total interest will the customer earn by the end of that year?

Introduction / Context:
This question demonstrates how compound interest works when multiple deposits are made at different times during the year and interest is compounded at fixed intervals. The key is to treat each deposit separately, considering how long it earns interest before the final date, and then sum the interest from both deposits.

Given Data / Assumptions:

• Interest rate = 15% per half year
• First deposit = Rs. 2400 on 1st January
• Second deposit = Rs. 2400 on 1st July
• We are interested in the amount at the end of the year (31st December)
• Compounding occurs every half year

Concept / Approach:
Since interest is 15% per half year, we treat each half of the year as one compounding period. The deposit on 1st January earns interest for two half-year periods (January–June and July–December). The deposit on 1st July earns interest for one half-year period (July–December). We compute the future amount of each deposit and then deduct the total principal to get the total interest.

Step-by-Step Solution:
Step 1: Amount from first deposit after two half-years.Rate per period i = 15% = 0.15A1 = 2400 * (1 + 0.15)^2A1 = 2400 * (1.15)^2 = 2400 * 1.3225 = Rs. 3174Step 2: Amount from second deposit after one half-year.A2 = 2400 * (1 + 0.15) = 2400 * 1.15 = Rs. 2760Step 3: Total amount at year end.Total amount = A1 + A2 = 3174 + 2760 = Rs. 5934Step 4: Total interest.Total principal = 2400 + 2400 = Rs. 4800Interest = 5934 - 4800 = Rs. 1134

Verification / Alternative check:
Another way is to compute interest separately. For the first deposit: interest = 2400 * 1.3225 - 2400 = 774. For the second deposit: interest = 2400 * 1.15 - 2400 = 360. Sum = 774 + 360 = 1134, which matches the previous result.

Why Other Options Are Wrong:
Rs. 2268: This is exactly double the correct interest and would correspond to mistakenly thinking both deposits earn for two periods.Rs. 567 and Rs. 283: These are partial amounts and represent miscalculation or omitting one of the deposits or periods.Rs. 900: A rounded guess that ignores the exact compound growth factors.

Common Pitfalls:
Students may incorrectly treat the whole year as one compounding period or assume that both deposits earn for the same length of time. It is crucial to carefully track when each deposit is made and how many compounding periods it experiences before the final date.

Final Answer:
The customer will earn a total interest of Rs. 1134 by the end of the year.