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

Introduction / Context:
This problem illustrates how compound interest works when there are multiple deposits at different times in the same year. The bank offers a relatively high rate of 15% per half year, so each six month period generates a significant amount of interest. The candidate must handle two separate cash flows, one invested for the full year and the other for only half the year, then sum the resulting interests to obtain the total interest by year end.

Given Data / Assumptions:

• Rate of interest = 15% per half year.
• First deposit = Rs. 2400 on 1st January.
• Second deposit = Rs. 2400 on 1st July of the same year.
• Interest is compounded every half year at 15%.
• We want the total interest earned by 31st December of that year.
• There are two half year periods in the year: January to June and July to December.

Concept / Approach:
Since the rate is per half year, each period of six months multiplies the current amount by the factor (1 + 0.15). The first deposit remains invested for two half years, so it earns interest twice. The second deposit remains invested for only one half year, so it earns interest once. The overall interest is the sum of the interest on each deposit. We will compute the amount for each deposit separately and then subtract the total principal from the combined amount to get the total compound interest earned in the year.

Step-by-Step Solution:
Step 1: For the first deposit, the principal P1 is 2400 and the number of half year periods is 2. Step 2: The amount from the first deposit after one year is A1 = 2400 * (1 + 0.15) ^ 2. Step 3: Compute the factor: (1 + 0.15) = 1.15, so (1.15) ^ 2 = 1.3225. Step 4: Therefore A1 = 2400 * 1.3225 = 3174. Step 5: The interest on the first deposit is I1 = A1 - 2400 = 3174 - 2400 = 774. Step 6: For the second deposit, the principal P2 is 2400 and it is invested for one half year only. Step 7: The amount from the second deposit is A2 = 2400 * (1 + 0.15) = 2400 * 1.15 = 2760. Step 8: The interest on the second deposit is I2 = A2 - 2400 = 2760 - 2400 = 360. Step 9: Total interest earned is I = I1 + I2 = 774 + 360 = 1134.

Verification / Alternative check:
A quick check can be done by considering the total amount at the end of the year. The total principal is 2400 + 2400 = 4800. The amount obtained from both deposits is 3174 + 2760 = 5934. The difference 5934 - 4800 equals 1134, which matches our interest computation. Also, if the rate per half year were 15%, a rough simple interest estimate for 2400 for one year and another 2400 for half a year gives about 2400 * 0.30 + 2400 * 0.15 = 720 + 360 = 1080. The compound interest should be slightly higher than this rough figure, and 1134 fits that expectation well.

Why Other Options Are Wrong:
Rs. 2268 is exactly double the correct interest and would imply counting the same deposits twice or treating each half year as a full year. Rs. 567 is only half of the correct interest, which can arise if a learner incorrectly treats only one of the two deposits or divides instead of adding. Rs. 283 is far too small and comes from a serious misapplication of the rate or a single simple interest step. Rs. 1340 is somewhat higher than the correct value and suggests that the interest factor has been exaggerated or applied for too many half year periods.

Common Pitfalls:
Students often misread the phrase "15% per half year" and treat it as 15% per annum instead of each six month period. Another error is to combine the two deposits and assume they have both been invested for the same length of time, which is not true since the second deposit is made later. Some candidates ignore compounding and calculate interest as if it were simple interest, which leads to an underestimate of the total interest. It is also easy to forget that there are exactly two half year periods in a year and that the first deposit enjoys both while the second deposit enjoys only one.

Final Answer:
By treating each deposit separately and applying the 15% half yearly rate correctly, the total interest earned by the end of the year is Rs. 1134, which corresponds to option B.