A bank offers compound interest at 15% per half year. A customer deposits Rs 8,800 on 1st January and another Rs 8,800 on 1st July of the same year. At the end of that year, how much total interest (in rupees) does the customer earn on these deposits?

Introduction:
This problem involves compound interest with two separate deposits made at different times during the year. It tests understanding of how time of investment affects the amount of interest earned, especially when interest is compounded every half year at a relatively high rate of 15% per half year.

Given Data / Assumptions:

• Rate of interest = 15% per half year
• First deposit = Rs 8,800 on 1st January
• Second deposit = Rs 8,800 on 1st July
• We consider the period up to the end of the same year
• Compounding is done half yearly

Concept / Approach:
Each deposit earns interest for a different number of half years:

• The January deposit earns interest for two half years.
• The July deposit earns interest for one half year.

We calculate the amount for each deposit separately using:
A = P * (1 + r)^nThen we subtract the total principal (both deposits) from the total final amount to get the interest earned.

Step-by-Step Solution:
Step 1: Rate per period.r = 15% per half year = 0.15Step 2: Amount from first deposit (two half years).A1 = 8,800 * (1 + 0.15)^2A1 = 8,800 * (1.15)^2(1.15)^2 = 1.3225A1 = 8,800 * 1.3225 = Rs 11,638 (approximately)Step 3: Amount from second deposit (one half year).A2 = 8,800 * (1 + 0.15)A2 = 8,800 * 1.15 = Rs 10,120Step 4: Total principal and final amount.Total principal = 8,800 + 8,800 = Rs 17,600Total amount = A1 + A2 = 11,638 + 10,120 = Rs 21,758 (approximately)Step 5: Total interest.Interest = 21,758 - 17,600 = Rs 4,158

Verification / Alternative check:
We can compute interest separately. For the first deposit, compound interest for two half years is:
CI1 = 8,800 * 1.3225 - 8,800 = Rs 2,838 (approximately)For the second deposit, simple compounding for one half year is:
CI2 = 8,800 * 0.15 = Rs 1,320Total interest = 2,838 + 1,320 = Rs 4,158, which matches our earlier result.

Why Other Options Are Wrong:

• Rs 8,315: Much too large for one year on these amounts.
• Rs 2,079 and Rs 1,039: These are too small; they correspond to partial calculations, not the total.
• Rs 3,120: Does not match the correct total interest for both deposits.

Common Pitfalls:
Students may mistakenly treat the entire Rs 17,600 as if deposited at the start of the year, or they may apply simple interest instead of compound interest. Another mistake is to ignore that the second deposit earns interest for only one half year, not the full year.

Final Answer:
The total interest earned at the end of the year is Rs 4,158.