Simple interest on a certain sum of money for 3 years at 8% per annum is equal to half the compound interest on Rs 4,000 for 2 years at 10% per annum. What is the value of the sum placed on simple interest, in rupees?

Introduction / Context:
This question combines both simple interest and compound interest in a comparative statement. We are told that simple interest on an unknown sum for a certain time and rate equals half the compound interest on a known sum for a different time and rate. The task is to compute the compound interest on the known principal first and then use the given relationship to find the unknown principal that would produce the given simple interest amount. Such cross linked interest problems are typical in aptitude exams and test a student's ability to coordinate multiple formulas.

Given Data / Assumptions:

• Unknown principal for simple interest = P (to be found).
• Simple interest rate for P = 8% per annum.
• Time for simple interest = 3 years.
• Known principal for compound interest = Rs 4,000.
• Compound interest rate on 4,000 = 10% per annum.
• Time for compound interest = 2 years, compounded annually.
• Simple interest on P for 3 years at 8% equals half the compound interest on Rs 4,000 for 2 years at 10%.

Concept / Approach:
First compute the compound interest on Rs 4,000 at 10% per annum for 2 years. Then take half of that amount. This half is equal to the simple interest on P for 3 years at 8%. Using the simple interest formula, we can then solve for P. The main formulas are:
A = 4000 * (1 + 10 / 100)^2 for the compound amountCI = A - 4000SI = P * R * T / 100 for simple interest

Step-by-Step Solution:
Step 1: Compute the compound amount on 4,000 for 2 years at 10% per annum. A = 4000 * (1 + 10 / 100)^2 = 4000 * (1.10)^2.Step 2: Calculate (1.10)^2 = 1.21.Step 3: So A = 4000 * 1.21 = 4840.Step 4: Compound interest on 4,000 is CI = A - principal = 4840 - 4000 = 840.Step 5: According to the problem, simple interest on P equals half of this CI, so SI on P = 840 / 2 = 420.Step 6: Use the simple interest formula SI = P * R * T / 100. Here SI = 420, R = 8, and T = 3.Step 7: Substitute the values: 420 = P * 8 * 3 / 100 = P * 24 / 100.Step 8: Rearrange: 420 = 0.24 * P, so P = 420 / 0.24.Step 9: Rewrite 0.24 as 24 / 100. So P = 420 * 100 / 24.Step 10: Simplify 420 / 24 = 17.5. Thus P = 17.5 * 100 = 1750.Step 11: Therefore, the sum placed on simple interest is Rs 1,750.

Verification / Alternative check:
We can verify by going back to the definitions. Simple interest on 1750 at 8% for 3 years is SI = 1750 * 8 * 3 / 100 = 1750 * 24 / 100 = 1750 * 0.24 = 420. Compound interest on 4000 at 10% for 2 years is 840 as computed earlier. Half of 840 is 420, which matches the simple interest on 1750. This confirms the relationship given in the problem and validates our answer.

Why Other Options Are Wrong:
At 1550, simple interest would be 1550 * 24 / 100 = 372, which is less than half the compound interest of 840. At 1650, simple interest would be 396, still not equal to 420. At 1850, simple interest becomes 444, greater than 420. At 2000, simple interest would be 480, again too high. None of these match the required 420. Only P = 1750 produces the exact equality specified in the question.

Common Pitfalls:
Some students misinterpret the statement and mistakenly equate the compound interest directly with simple interest, ignoring the factor of one half. Others compute simple interest and compound interest on the wrong principals or mix the rates and times. Misplacing the factor of 2 in the relationship can lead to completely wrong results. Carefully translating the English sentence into an algebraic equation is crucial in such word problems.

Final Answer:
The sum placed on simple interest, whose 3 year simple interest at 8% equals half the compound interest on Rs 4,000 at 10% for 2 years, is Rs.1750.