Find the compound interest on Rs. 8000 at 15% per annum for 2 years 4 months, when interest is compounded annually and the extra 4 months are treated using simple interest on the amount after 2 years.

Introduction / Context:
This question focuses on calculating compound interest for a non-integer time period where the interest is compounded annually, and an extra fraction of a year is handled using simple interest on the accumulated amount. Such problems combine compound interest over whole years with simple interest over remaining months, which is a common pattern in competitive exams. Here, the principal is Rs. 8000, the rate is 15% per annum, and the time is 2 years 4 months.

Given Data / Assumptions:

• Principal (P) = Rs. 8000.

• Rate of interest (R) = 15% per annum.

• Total time = 2 years 4 months.

• Interest is compounded annually.

• The extra 4 months are treated using simple interest on the amount after 2 full years.

Concept / Approach:
For the first two full years, we calculate compound interest using the standard formula for annual compounding: A = P * (1 + R/100)^n, where n is the number of years. After obtaining the amount at the end of 2 years, we treat the remaining 4 months as 4/12 of a year and compute simple interest on this amount at the same annual rate. Adding this simple interest to the 2-year amount gives the final amount, and subtracting the original principal yields the total interest earned over 2 years 4 months.

Step-by-Step Solution:
Step 1: Compute amount after 2 years with annual compounding. Step 2: A2 = P * (1 + R/100)^2 = 8000 * (1 + 15/100)^2. Step 3: 1 + 15/100 = 1.15, so A2 = 8000 * (1.15)^2. Step 4: (1.15)^2 = 1.3225, so A2 = 8000 * 1.3225 = Rs. 10,580. Step 5: Time remaining = 4 months = 4/12 year = 1/3 year. Step 6: Simple interest for 4 months on A2 at 15% per annum: SI = A2 * (R/100) * (1/3). Step 7: SI = 10,580 * (15/100) * (1/3) = 10,580 * 0.15 * (1/3). Step 8: 0.15 * (1/3) = 0.05, so SI = 10,580 * 0.05 = Rs. 529. Step 9: Final amount after 2 years 4 months = A2 + SI = 10,580 + 529 = Rs. 11,109. Step 10: Total interest = final amount − principal = 11,109 − 8000 = Rs. 3,109.

Verification / Alternative check:
We can verify intermediate values. First, check that the 2-year compounding is correct: 8000 at 15% for 1 year gives 8000 * 1.15 = 9200. One more year at 15% gives 9200 * 1.15 = 10,580, which matches A2. Then 4 months is one-third of a year, and 15% of 10,580 for a full year is 1,587. For one-third year, it is approximately 1,587 / 3 = 529, which matches the computed SI. Thus, the final interest of 3,109 is consistent.

Why Other Options Are Wrong:
• Rs. 2109: This ignores or underestimates the extra simple interest for 4 months after compounding for 2 years.
• Rs. 4109: This overstates the interest, likely by mistakenly compounding for the entire 2 years 4 months without splitting into compound and simple parts correctly.
• Rs. 6109: This is far too high and does not align with the given rate and time period for the principal amount.

Common Pitfalls:
A common mistake is to treat the entire 2 years 4 months as if compounding occurs continuously or quarterly without reading the phrase "compounded annually" carefully. Another error is to forget that the extra months should be converted to a fraction of a year and used with simple interest on the amount after full years of compounding. Carefully separating the compound interest and simple interest portions ensures an accurate solution.

Final Answer:
The compound interest earned on Rs. 8000 at 15% per annum for 2 years 4 months, with annual compounding and simple interest for the extra months, is Rs. 3109.