Rs. 1,000 is invested at a compound interest rate of 50% per annum, compounded annually. What is the interest earned during the 3rd year alone (that is, in the third year) under compound interest?

Introduction / Context:
This question asks specifically for the interest earned in the third year under compound interest, not the total interest over three years. The sum grows rapidly because the rate is very high (50% per annum). The key is to find the amount at the beginning and at the end of the third year and take the difference, which represents the interest credited in that particular year.

Given Data / Assumptions:
- Principal P = Rs. 1,000.
- Rate r = 50% per annum, compounded annually.
- Time in question: the third year only.
- We must find the interest gained during year 3 under compound interest.

Concept / Approach:
Under compound interest, interest for each year is calculated on the amount (principal plus accumulated interest) at the beginning of that year. The interest in the nth year can be found as the amount at the end of that year minus the amount at the end of the previous year. For this problem, we use the growth factor (1.5) for each year, as 50% per annum implies a multiplication by 1.5 each year.

Step-by-Step Solution:
Step 1: Compute amount after Year 1: A1 = 1000 * 1.5 = 1500.Step 2: Compute amount after Year 2: A2 = A1 * 1.5 = 1500 * 1.5 = 2250.Step 3: Compute amount after Year 3: A3 = A2 * 1.5 = 2250 * 1.5 = 3375.Step 4: Total interest over 3 years would be A3 - P = 3375 - 1000 = 2375, but we only need the third year's interest.Step 5: Interest earned in the 3rd year = A3 - A2 = 3375 - 2250.Step 6: Interest in 3rd year = Rs. 1,125.

Verification / Alternative check:
Third year interest can also be calculated directly as 50% of the amount at the end of the second year, because interest is applied on the beginning-of-year amount. Amount at the end of year 2 is Rs. 2250, so interest in year 3 = 2250 * 50/100 = 1125. This matches the difference A3 - A2 and confirms the answer.

Why Other Options Are Wrong:
Rs. 625 might be mistakenly taken as 50% of the original principal, forgetting that by year 3 the interest is on a larger amount. Rs. 1,025 and Rs. 1,122 are near but incorrect values, arising from miscalculating the intermediate amounts. Rs. 750 would be 50% of 1500, which corresponds to year 2 interest, not year 3. Only Rs. 1,125 correctly reflects 50% of 2250, the amount at the start of year 3.

Common Pitfalls:
Students sometimes confuse total interest over the whole period with the interest in a particular year. Others keep applying simple interest logic and simply multiply the initial principal by 150% across years. Remember that under compound interest, each year's interest is based on the amount accumulated up to that year, not just the original principal.

Final Answer:
The interest earned during the 3rd year is Rs. 1,125.