Simple interest on a certain sum of money for 3 years at 8 percent per annum is half the compound interest on Rs. 4000 for 2 years at 10 percent per annum. What is the sum invested at simple interest?

Introduction / Context:
This question links simple interest on an unknown principal with compound interest on a known principal. The simple interest for 3 years is given to be half of the compound interest on Rs. 4000 for 2 years at 10 percent per annum. We need to find the principal that was placed at simple interest. This problem checks understanding of both simple and compound interest formulas and the ability to connect two different investments through a given condition.

Given Data / Assumptions:

• Simple interest (SI) on an unknown principal P for 3 years at 8 percent per annum.
• Compound interest (CI) on Rs. 4000 for 2 years at 10 percent per annum.
• The simple interest is equal to half of this compound interest.
• Rates are annual and remain constant; compounding is annual.

Concept / Approach:
First we compute the compound interest on Rs. 4000 for 2 years at 10 percent. Then we take half of that amount as the given simple interest. After that, we use the simple interest formula SI = P * r * t / 100 and solve for P. The key steps are computing CI correctly and then equating it to the expression for SI.

Step-by-Step Solution:
Step 1: Compute CI on Rs. 4000 at 10 percent for 2 years. Amount A = 4000 * (1 + 10/100)^2 (1 + 10/100) = 1.10, so (1.10)^2 = 1.21 A = 4000 * 1.21 = 4840 CI = A - principal = 4840 - 4000 = 840 Step 2: Simple interest equals half of this CI. Therefore, SI = 840 / 2 = 420 Step 3: Use SI formula with P, rate 8 percent, time 3 years. SI = P * 8 * 3 / 100 = 0.24 * P So 0.24 * P = 420 P = 420 / 0.24 = 1750 Therefore, the principal placed at simple interest is Rs. 1750

Verification / Alternative check:
We can verify by computing the simple interest on Rs. 1750 at 8 percent for 3 years. SI = 1750 * 8 * 3 / 100 = 1750 * 24 / 100 = 1750 * 0.24 = 420 We already know half of the compound interest on Rs. 4000 for 2 years at 10 percent is 420, since the full CI is 840. Thus the condition in the question is satisfied, confirming that P = 1750 is correct.

Why Other Options Are Wrong:
Principals like Rs. 1550, Rs. 1650, or Rs. 1850 would give different simple interest values when substituted into the SI formula and would not yield 420. For example, using 1850 would give SI = 1850 * 0.24 = 444, which is not half of 840. Rs. 2000 would give SI = 480, which is greater than half of the computed compound interest. Only Rs. 1750 leads to the correct simple interest of Rs. 420.

Common Pitfalls:
A common mistake is to confuse half of the compound amount with half of the compound interest. Students may accidentally take half of 4840 instead of half of 840. Another pitfall is treating CI like SI and incorrectly applying a linear formula over 2 years. Some also mis apply the percentage when solving for P. Writing each step clearly and keeping a separate symbol for the CI amount helps avoid these issues.

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