Simple and Compound Interest on the Same Sum: The simple interest (SI) on a certain sum for 3 years is ₹6,000, and the compound interest (CI) on the same sum for 2 years at the same annual rate (compounded annually) is ₹4,160. What is the annual rate of interest (in %)?

Introduction / Context:
This question combines information about simple interest (SI) over 3 years and compound interest (CI) over 2 years on the same principal and at the same rate. You are required to find the common annual rate of interest. Such mixed SI and CI problems are popular in aptitude exams as they test your ability to use formulas and relate different time periods.

Given Data / Assumptions:

• Principal = P.
• Time for SI = 3 years.
• Simple interest for 3 years = ₹6,000.
• Time for CI = 2 years.
• Compound interest for 2 years (annual compounding) = ₹4,160.
• Rate of interest per annum = r% (same for both SI and CI).

Concept / Approach:
For SI: SI = (P * r * t) / 100. From the SI data, we can find the product P * r. For CI with annual compounding: CI = P * ((1 + r / 100)^2 - 1) over 2 years. Once P * r is known, we can test the given options for r, compute the corresponding principal P, and then verify which r makes the CI equal to ₹4,160.

Step-by-Step Solution:
Step 1: From SI, we have SI for 3 years = ₹6,000. Step 2: Use the SI formula: 6,000 = (P * r * 3) / 100. Step 3: Rearranging, P * r = (6,000 * 100) / 3 = 200,000. Step 4: For CI over 2 years, CI = P * ((1 + r / 100)^2 - 1) = ₹4,160. Step 5: Test r = 8% (a likely mid value). Then P = 200,000 / 8 = ₹25,000. Step 6: Compute CI with P = 25,000 and r = 8%: Amount A = 25,000 * (1.08)^2 = 25,000 * 1.1664 = ₹29,160. Step 7: CI = A - P = 29,160 - 25,000 = ₹4,160, which matches the given CI. Step 8: Therefore, the annual rate of interest is 8% per annum.

Verification / Alternative check:
To confirm, you can compute SI for 3 years at 8% per annum on ₹25,000: SI = (25,000 * 8 * 3) / 100 = 25,000 * 0.24 = ₹6,000, which matches the given SI. Both SI and CI conditions are satisfied simultaneously, so r = 8% is definitely correct.

Why Other Options Are Wrong:

• 9%: Leads to a different principal and CI around ₹4,180, not ₹4,160.
• 6%: Produces a CI of about ₹4,120 for 2 years, smaller than required.
• 10%: Gives a CI of around ₹4,200, larger than the given CI.
• 12%: Produces an even higher CI, far from ₹4,160.

Common Pitfalls:
Students sometimes directly equate SI and CI or confuse the time durations. Others forget that SI depends linearly on time (3 years) while CI has a squared factor for 2 years. The safest method is to first compute P * r from SI, then systematically test the provided rate options in the CI formula until you match the given CI.

Final Answer:
The required annual rate of interest is 8% per annum.