Simple interest on a certain sum of money for 3 years at 18% per annum is equal to half of the compound interest on Rs. 9000 for 2 years at 10% per annum. What is the principal sum (in rupees) that is placed on simple interest?

Introduction / Context:
This problem mixes both compound interest and simple interest in a single question. It states that the simple interest on an unknown principal at 18% per annum for 3 years equals half the compound interest on a known principal of Rs. 9000 for 2 years at 10% per annum. Such questions test your ability to handle both interest types correctly and then relate the two to find the missing principal on which simple interest is calculated.

Given Data / Assumptions:

• Unknown principal for simple interest = P rupees.
• Simple interest rate = 18% per annum.
• Time for simple interest = 3 years.
• Known principal for compound interest = Rs. 9000.
• Compound interest rate = 10% per annum.
• Time for compound interest = 2 years (annual compounding).
• Simple interest on P equals half the compound interest on Rs. 9000.

Concept / Approach:
First, compute the compound interest on Rs. 9000 at 10% per annum for 2 years using the formula Amount = P * (1 + R/100) ^ T, then CI = Amount - P. Half of this compound interest is then equated to the simple interest on the unknown principal P at 18% for 3 years. For simple interest, we use SI = P * R * T / 100. Equating SI to half of CI allows us to solve for P. This approach ensures that the relationship between the two types of interest is properly captured.

Step-by-Step Solution:
Step 1: Compute compound amount on Rs. 9000 at 10% for 2 years: Amount = 9000 * (1 + 10/100) ^ 2. Step 2: 1 + 10/100 = 1.10, so Amount = 9000 * (1.10) ^ 2 = 9000 * 1.21 = 10890. Step 3: Compound interest CI = Amount - Principal = 10890 - 9000 = 1890. Step 4: Half of this compound interest is 1890 / 2 = 945. Step 5: Let the unknown principal on simple interest be P. Simple interest formula: SI = P * 18 * 3 / 100. Step 6: Simplify: SI = P * 54 / 100 = 0.54 * P. Step 7: Given that SI = 945, we have 0.54 * P = 945. Step 8: Therefore P = 945 / 0.54 = 1750. Step 9: Hence, the principal on simple interest is Rs. 1750.

Verification / Alternative check:
Verify by recomputing SI on Rs. 1750 at 18% for 3 years: SI = 1750 * 18 * 3 / 100 = 1750 * 54 / 100 = 1750 * 0.54 = 945, which matches half the compound interest. Also, we already found CI on Rs. 9000 is 1890, so half is indeed 945. Since the simple interest computed from P = 1750 equals 945, the condition in the question is satisfied, confirming that our solution is correct.

Why Other Options Are Wrong:
If P = Rs. 3500, SI would be 3500 * 18 * 3 / 100 = 1890, which equals the full compound interest, not half. If P = Rs. 875, SI would be 875 * 18 * 3 / 100 = 472.5, which is far too small. If P = Rs. 1400, SI would be 1400 * 18 * 3 / 100 = 756. All these values differ from half the compound interest (945), so these options are not correct.

Common Pitfalls:
A common mistake is to confuse the condition and equate simple interest directly to the full compound interest instead of half. Some students also forget to subtract principal from the compound amount, using the amount itself as the compound interest. Another frequent error is miscalculating (1.10) ^ 2 or incorrectly simplifying 54/100. Being careful with basic arithmetic and clearly distinguishing between amount and interest helps avoid these errors.

Final Answer:
The sum placed on simple interest is Rs. 1750.