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

Introduction / Context:
This problem links a simple interest scenario with a compound interest scenario. The simple interest earned on an unknown principal for a certain time and rate is given to be half the compound interest earned on a known principal under different conditions. The goal is to determine the unknown principal in the simple interest case. Such mixed interest comparisons are popular in competitive exams because they test understanding of both types of interest and the ability to combine them in an equation.

Given Data / Assumptions:

Simple interest is on an unknown principal S at 8 percent per annum for 3 years
Compound interest is on Rs 16,000 at 10 percent per annum for 2 years, compounded yearly
Simple interest amount equals half of the compound interest amount
We need to find the principal S used in the simple interest case

Concept / Approach:
First, compute the compound interest on Rs 16,000 at 10 percent per annum for 2 years using the formula A = P * (1 + r)^2 and then subtract the principal to get compound interest. Then take half of that compound interest. Next, express the simple interest on S at 8 percent per annum for 3 years as S * 0.08 * 3. Set this equal to half of the compound interest and solve for S. This gives a direct equation that can be solved algebraically.

Step-by-Step Solution:
Step 1: Compute compound amount on Rs 16,000 for 2 years at 10 percent. P1 = 16,000 and r1 = 10 percent = 0.10. Amount A1 = 16,000 * (1 + 0.10)^2 = 16,000 * (1.10)^2. (1.10)^2 = 1.21, so A1 = 16,000 * 1.21 = 19,360 rupees. Step 2: Compute compound interest on Rs 16,000. C.I. = A1 - P1 = 19,360 - 16,000 = 3,360 rupees. Step 3: Find half of this compound interest. Half of C.I. = 3,360 / 2 = 1,680 rupees. Step 4: Express simple interest on S at 8 percent for 3 years. Rate r2 = 8 percent = 0.08, time T2 = 3 years. Simple interest = S * 0.08 * 3 = S * 0.24. Step 5: Set simple interest equal to half the compound interest. S * 0.24 = 1,680. Step 6: Solve for S. S = 1,680 / 0.24 = 7,000 rupees.

Verification / Alternative check:
Verify by reapplying the formulas. Simple interest on Rs 7,000 at 8 percent for 3 years is 7,000 * 0.08 * 3 = 7,000 * 0.24 = 1,680 rupees. Compound interest on 16,000 at 10 percent for 2 years is 3,360 rupees, as calculated earlier, and half of that is 1,680 rupees. Because both sides equal 1,680 rupees, the equality required by the question is satisfied, confirming that S = 7,000 rupees is correct.

Why Other Options Are Wrong:
If S were Rs 14,000, the simple interest at 8 percent for 3 years would be 14,000 * 0.24 = 3,360 rupees, which equals the full compound interest, not half of it. If S were Rs 3,500, the simple interest would be only 840 rupees, which is one quarter of the compound interest. For S = 5,600, the simple interest would be 1,344 rupees, which does not match half of 3,360. Only S = 7,000 rupees produces simple interest equal to half the compound interest as required.

Common Pitfalls:
A typical mistake is to equate simple interest directly to the full compound interest instead of to half of it, which changes the resulting principal. Others may miscalculate the compound amount or incorrectly square 1.10. It is also easy to forget to convert percentages to decimal form when applying formulas. Careful reading of the phrase half the compound interest and precise arithmetic at each step are crucial.

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