Simple vs Compound Interest — The simple interest (SI) on a certain sum for 3 years at 8% per annum is half the compound interest (CI) on ₹ 8,000 for 2 years at 10% per annum. Find the sum on which SI is calculated.

Introduction / Context:
We are asked to compare a given simple-interest amount to half of a known compound-interest amount. The trick is to compute the CI on ₹ 8,000 over 2 years at 10% first, then halve it, and finally back-solve the principal that generates that simple interest at 8% for 3 years.

Given Data / Assumptions:

• CI base: ₹ 8,000, rate = 10% per annum, time = 2 years, compounding annually.
• SI case: unknown principal P at 8% per annum for 3 years.
• SI (3 years, 8%) = (1/2) × CI(₹ 8,000, 2 years, 10%).

Concept / Approach:
For 2 years, compound-interest factor is (1 + 0.10)^2 = 1.21. CI = P_ci * (1.21 − 1) = 8,000 * 0.21 = ₹ 1,680. Half of that is ₹ 840, which equals the SI on the unknown principal for 3 years at 8%.

Step-by-Step Solution:
Half of CI = 1,680 / 2 = ₹ 840SI formula: SI = P * r * t840 = P * 0.08 * 3P = 840 / 0.24 = ₹ 3,500

Verification / Alternative check:
Check SI: 3,500 * 0.08 * 3 = 840, exactly half of 1,680. Hence consistent.

Why Other Options Are Wrong:
₹ 3,800, ₹ 3,600, and ₹ 3,200 result from algebra or arithmetic slips; ₹ 4,000 overestimates the principal and would give SI = 960, not 840.

Common Pitfalls:
Confusing CI with SI formulas; using 10% for 3 years instead of 2; or forgetting to halve the computed CI amount.

Final Answer:
₹ 3,500