Simple interest on a certain sum of money for 3 years at 8% per annum is equal to half of the compound interest on Rs. 16,000 for 2 years at 10% per annum, compounded annually. What is the principal placed on simple interest?

Introduction / Context:
This question connects a simple interest calculation on an unknown principal with a compound interest calculation on a known principal. The key is to first find the compound interest on Rs. 16,000 at 10% for 2 years, then use half of that amount as the simple interest over 3 years to solve for the unknown principal.

Given Data / Assumptions:

Simple interest (SI) is charged on an unknown principal P for 3 years at 8% per annum.
Compound interest (CI) is calculated on Rs. 16,000 for 2 years at 10% per annum, compounded annually.
SI on P for 3 years at 8% equals half of this CI.
We must find the principal P placed on simple interest.

Concept / Approach:
First, compute the compound interest on Rs. 16,000 for 2 years at 10%: A = 16,000 * (1.10)^2, CI = A − 16,000. Then take half of this CI; that value equals the simple interest on P: SI = P * 8 * 3 / 100. Equate SI to half the CI and solve for P.

Step-by-Step Solution:
Step 1: Compute CI on Rs. 16,000 at 10% p.a. for 2 years. A = 16,000 * (1.10)^2 = 16,000 * 1.21 = Rs. 19,360. CI = 19,360 − 16,000 = Rs. 3,360. Step 2: Find half of this CI. Half of CI = 3,360 / 2 = Rs. 1,680. Step 3: Use simple interest formula for P. SI on P for 3 years at 8% = P * 8 * 3 / 100 = 0.24P. Given 0.24P = 1,680. P = 1,680 / 0.24 = 7,000.

Verification / Alternative check:
Check SI with P = 7,000. SI = 7,000 * 8 * 3 / 100 = 7,000 * 0.24 = 1,680. This matches half of the CI (3,360), so the condition in the question is satisfied exactly.

Why Other Options Are Wrong:
If P were 3,500, SI would be only 840. For P = 5,600, SI would be 1,344. For P = 14,000, SI would be 3,360, which equals the full CI, not half. Only P = 7,000 gives SI equal to half of the compound interest.

Common Pitfalls:
A frequent mistake is forgetting that the relation is with half the CI, not the full CI. Another is mixing up the rates or time periods between the SI and CI parts of the problem. Keep the two computations clearly separated, then connect them through the equality given in the problem statement.

Final Answer:
The sum placed on simple interest is Rs. 7,000.