Simple interest on a certain unknown sum for 3 years at 10% per annum is equal to half of the compound interest earned on Rs 6000 for 2 years at 10% per annum when interest is compounded annually. Using this relationship between simple interest and compound interest, find the sum of money that was placed on simple interest.

Introduction / Context:
This question combines both simple interest and compound interest concepts in a single scenario. The main idea is that the simple interest on an unknown principal over a given time equals half the compound interest on a known principal over a different time, both at the same rate of interest. Such mixed interest questions are important in competitive exams because they check a student’s ability to compute compound interest accurately and then relate it to simple interest through an algebraic equation.

Given Data / Assumptions:

• Rate of interest for both cases, R = 10% per annum.
• Simple interest case: time T1 = 3 years, principal P1 is unknown.
• Compound interest case: time T2 = 2 years, principal P2 = Rs 6000.
• Interest in the compound interest case is compounded annually.
• Simple interest for 3 years on P1 equals half of the compound interest for 2 years on Rs 6000.

Concept / Approach:
For simple interest, we use:
SI = (P1 * R * T1) / 100 For compound interest compounded annually, we use:
Amount A2 = P2 * (1 + R / 100) ^ T2 Compound interest CI = A2 − P2 The relation given is:
SI = (1 / 2) * CI First we compute the compound interest on Rs 6000 for 2 years at 10% and then take half of that value. That value is equal to the simple interest on P1 for 3 years, which can be expressed using the simple interest formula. Solving this equation gives P1.

Step-by-Step Solution:
P2 = Rs 6000, R = 10% per annum, T2 = 2 years. Amount after 2 years with compound interest: A2 = 6000 * (1 + 10 / 100) ^ 2. A2 = 6000 * (1.1) ^ 2 = 6000 * 1.21 = 7260. Compound interest CI = A2 − P2 = 7260 − 6000 = Rs 1260. Half of this compound interest = 1260 / 2 = Rs 630. Let P1 be the principal on simple interest for 3 years at 10%. Simple interest SI = (P1 * 10 * 3) / 100 = 0.3P1. Given that SI = 630, so 0.3P1 = 630. P1 = 630 / 0.3 = 2100. Therefore, the sum placed on simple interest is Rs 2100.

Verification / Alternative check:
Check the simple interest on Rs 2100 for 3 years at 10%. SI = (2100 * 10 * 3) / 100 = 630. This matches half of the compound interest computed earlier, which was 1260. Since both conditions are satisfied, P1 = Rs 2100 is consistent with the relationship stated in the question.

Why Other Options Are Wrong:
If P1 were Rs 4200, then simple interest for 3 years at 10% would be SI = 1260, which equals the full compound interest, not half of it.
If P1 were Rs 1680, then SI = (1680 * 10 * 3) / 100 = 504, which is less than 630.
If P1 were Rs 1050, then SI = 315, also not equal to 630.
If P1 were Rs 3000, then SI = 900, too large compared with 630. Hence only Rs 2100 satisfies the condition that simple interest equals half of the compound interest.

Common Pitfalls:
A typical mistake is to confuse simple interest and compound interest formulas or to treat compound interest as simple interest over 2 years. Some learners forget to subtract the principal when finding compound interest and accidentally use the amount directly. Another error is misinterpreting “half of the compound interest” as “compound interest at half the rate”, which is incorrect. Using the standard formulas carefully and reading the condition precisely avoids these misunderstandings.

Final Answer:
The sum of money that was placed on simple interest is Rs 2100.