Simple interest on a certain unknown sum of money for 3 years at 8% per annum is equal to half of the compound interest on Rs 1200 for 2 years at 10% per annum, assuming the compound interest is calculated annually. Using this relationship, find the principal that was placed on simple interest.

Introduction / Context:
This question is another mixed-interest problem that links simple interest to compound interest. The simple interest on an unknown principal is given to be half of the compound interest on a known principal. Students must compute the compound interest on Rs 1200 for 2 years at 10% per annum, take half of that value, and then relate it to the simple interest formula over 3 years at 8% per annum to find the unknown principal. Such questions are excellent practice for combining different interest concepts in a single problem.

Given Data / Assumptions:

• Rate for simple interest case = 8% per annum.
• Time for simple interest case = 3 years.
• Principal for simple interest case = P rupees (unknown).
• Rate for compound interest case = 10% per annum.
• Time for compound interest case = 2 years, compounded annually.
• Principal for compound interest case = Rs 1200.
• Simple interest for 3 years equals half of the compound interest for 2 years on Rs 1200.

Concept / Approach:
For compound interest with annual compounding:
Amount A = P * (1 + R / 100) ^ T Compound interest CI = A − P For simple interest:
SI = (P * R * T) / 100 The problem gives:
SI (on unknown principal) = 0.5 * CI (on Rs 1200) So we first compute CI on Rs 1200 at 10% for 2 years, then take half, and finally set that equal to the simple interest expression for P at 8% for 3 years. Solving this equation yields P.

Step-by-Step Solution:
Compute compound interest on Rs 1200 at 10% for 2 years. Amount A = 1200 * (1 + 10 / 100) ^ 2 = 1200 * (1.1) ^ 2 = 1200 * 1.21. A = 1452. Compound interest CI = A − Principal = 1452 − 1200 = Rs 252. Half of this compound interest = 252 / 2 = Rs 126. Now consider simple interest case: Rate = 8% per annum, time = 3 years, principal P rupees. Simple interest SI = (P * 8 * 3) / 100 = 0.24P. Given that SI = 126, so 0.24P = 126. P = 126 / 0.24 = 525. Therefore, the principal placed on simple interest is Rs 525.

Verification / Alternative check:
Check the simple interest on Rs 525 at 8% for 3 years. SI = (525 * 8 * 3) / 100. First compute 8 * 3 = 24, so SI = (525 * 24) / 100 = 12600 / 100 = Rs 126. This matches half of the compound interest on Rs 1200, which was 252. Hence, the relationship is satisfied and P = Rs 525 is correct.

Why Other Options Are Wrong:
If P were Rs 1050, simple interest would be 0.24 * 1050 = 252, which equals the full compound interest, not half of it.
If P were Rs 260, SI would be 0.24 * 260 = 62.4, which is less than 126.
If P were Rs 420, SI would be 100.8, still not equal to 126.
If P were Rs 630, SI would be 151.2, larger than 126. Only Rs 525 satisfies the required equality condition.

Common Pitfalls:
Students sometimes confuse half of compound interest with compound interest at half the rate, which is incorrect. Another mistake is to miscalculate the compound amount by using simple interest instead of compounding, or by forgetting to subtract the principal when finding CI. Rounding errors may appear if intermediate steps are approximated. Using exact values and carefully following the formula helps to avoid these common issues.

Final Answer:
The principal that was placed on simple interest is Rs 525.