Simple Interest vs Compound Interest on Different Sums: The simple interest on a certain sum for 3 years at 14% per annum is equal to half of the compound interest on ₹10,000 for 2 years at 10% per annum (compounded annually). What is the sum (principal) placed at simple interest?

Introduction / Context:
This question compares simple interest (SI) on one sum with compound interest (CI) on another sum. You are told that the SI for 3 years at 14% per annum equals half of the CI on ₹10,000 for 2 years at 10% per annum. The aim is to find the principal on which the simple interest was calculated. This is a standard aptitude problem on the relationship between SI and CI.

Given Data / Assumptions:

• Principal for SI = P (unknown).
• Time for SI = 3 years.
• SI rate = 14% per annum.
• Principal for CI = ₹10,000.
• CI rate = 10% per annum, compounded annually.
• Time for CI = 2 years.
• SI for 3 years = half of CI on ₹10,000 for 2 years.

Concept / Approach:
First, compute the compound interest on ₹10,000 at 10% per annum for 2 years. Then divide that CI value by 2 to obtain the simple interest earned on the unknown principal P. Next, use the SI formula SI = (P * r * t) / 100 with r = 14% and t = 3 years to solve for P. This algebraic approach is straightforward and avoids unnecessary trial and error.

Step-by-Step Solution:
Step 1: Compute the CI on ₹10,000 at 10% per annum for 2 years. Step 2: Amount A = 10,000 * (1 + 10 / 100)^2 = 10,000 * (1.1)^2 = 10,000 * 1.21 = ₹12,100. Step 3: CI = A - P = 12,100 - 10,000 = ₹2,100. Step 4: Given that the SI on P for 3 years at 14% equals half of this CI, SI = 2,100 / 2 = ₹1,050. Step 5: Use SI formula: 1,050 = (P * 14 * 3) / 100. Step 6: Simplify: 1,050 = (42P) / 100, so 1,050 * 100 = 42P. Step 7: P = (1,050 * 100) / 42 = 105,000 / 42 = ₹2,500. Step 8: Therefore, the required principal is ₹2,500.

Verification / Alternative check:
Check SI on ₹2,500 at 14% per annum for 3 years: SI = (2,500 * 14 * 3) / 100 = 2,500 * 0.42 = ₹1,050. Half of the CI on ₹10,000 for 2 years at 10% was found to be ₹1,050, so the condition is satisfied exactly. This confirms that the principal is correct.

Why Other Options Are Wrong:

• ₹5,000: SI for 3 years at 14% would be ₹2,100, which equals the full CI, not half of it.
• ₹2,000: SI would be ₹840, which is less than half of the given CI.
• ₹1,250: SI would be ₹525, far too small.
• ₹3,000: SI would be ₹1,260, larger than the required ₹1,050.

Common Pitfalls:
A common mistake is to misinterpret the phrase “half of the compound interest” and incorrectly relate SI directly to the amount instead of the CI. Another error is to mix up the principals or rates for SI and CI. Always compute CI on the given sum first, then apply the given relationship to find SI and, finally, use the SI formula to determine the unknown principal.

Final Answer:
The sum placed at simple interest is ₹2,500.