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. Find the sum of money that was placed on simple interest.

Introduction / Context:
This question compares simple interest on one sum with compound interest on another sum at the same percentage rate. Compound interest grows faster than simple interest because interest is earned on both principal and previous interest, while simple interest grows only on principal. Here, simple interest on some unknown amount for 3 years at 10 percent per annum is half of the compound interest on Rs. 6000 for 2 years at 10 percent per annum. The learner must correctly compute the compound interest first and then use the relationship to find the unknown principal under simple interest.

Given Data / Assumptions:
Rate of interest for both cases is 10 percent per annum.
Time for simple interest case is 3 years.
Time for compound interest case is 2 years, compounded annually.
Principal for compound interest case is Rs. 6000.
Simple interest on the unknown sum is half of the compound interest on Rs. 6000.
Interest is annually compounded for the compound interest case.

Concept / Approach:
Compound interest uses the formula A = P * (1 + r / 100)^n, where A is the amount after n years. Compound interest CI is CI = A - P. For the first part, we compute compound interest on Rs. 6000 at 10 percent for 2 years. Then we know that simple interest on the unknown principal, at the same rate for 3 years, equals half of that compound interest. For simple interest, the formula is I = P * r * t / 100. Using these relationships, we set up an equation and solve for the unknown principal.

Step-by-Step Solution:
First, calculate the amount on Rs. 6000 at compound interest for 2 years at 10 percent per annum.Amount A = 6000 * (1 + 10 / 100)^2 = 6000 * (1.1)^2.Compute (1.1)^2 = 1.21, so A = 6000 * 1.21 = 7260 rupees.Compound interest CI on Rs. 6000 is CI = A - P = 7260 - 6000 = 1260 rupees.Half of this compound interest is 1260 / 2 = 630 rupees.Let the unknown principal on simple interest be P rupees.Simple interest for 3 years at 10 percent is I = P * 10 * 3 / 100 = 0.3P.Set 0.3P equal to 630: 0.3P = 630.Solve for P: P = 630 / 0.3 = 2100.Therefore the sum placed on simple interest is Rs. 2100.

Verification / Alternative check:
Check the simple interest on Rs. 2100 at 10 percent for 3 years. Simple interest equals 2100 * 10 * 3 / 100 = 2100 * 30 / 100 = 630 rupees. Earlier we found that half of the compound interest on Rs. 6000 for 2 years at 10 percent is 630 rupees. Since the simple interest computed matches this value, the relationship given in the question is satisfied and the answer is confirmed.

Why Other Options Are Wrong:
If the principal were Rs. 4200, simple interest would be 4200 * 10 * 3 / 100 = 1260 rupees, which equals the full compound interest, not half of it.
For Rs. 1680, simple interest is 504 rupees, less than half of the compound interest found earlier.
For Rs. 1050, simple interest is 315 rupees, which is only one quarter of the compound interest.
Rs. 3000 would give simple interest of 900 rupees, which is greater than half of the compound interest. None of these satisfy the half relationship.

Common Pitfalls:
Students sometimes confuse compound interest with simple interest and use the simple interest formula for both parts. Another mistake is to forget subtracting the principal from the amount when computing compound interest, leading to A instead of CI. There can also be arithmetic errors when squaring 1.1 or when dividing 1260 by 2. Writing the formulas clearly and checking each arithmetic step greatly reduces errors.

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