Simple Interest vs Compound Interest: On a certain principal, the simple interest (SI) for 2 years is ₹4,000 and the compound interest (CI) for the same 2 years at the same annual rate is ₹4,180. What is the rate of interest per annum (in %)?

Introduction / Context:
This question compares simple interest (SI) and compound interest (CI) on the same principal over the same time period and asks you to determine the annual rate of interest. Such questions are common in bank exams and aptitude tests and help you understand how compounding increases the interest earned compared with simple interest.

Given Data / Assumptions:

• Principal = P (same for SI and CI).
• Time = 2 years.
• Simple interest for 2 years = ₹4,000.
• Compound interest for 2 years = ₹4,180.
• Rate of interest per annum = r% (same for SI and CI).
• Interest is compounded annually for CI.

Concept / Approach:
For simple interest, SI = (P * r * t) / 100. For compound interest with annual compounding for 2 years, CI = P * ((1 + r / 100)^2 - 1). Because SI and CI are both given, we can first find the product P * r from the SI expression and then substitute into the CI expression to test which rate from the options satisfies both conditions.

Step-by-Step Solution:
Step 1: Using simple interest: SI for 2 years = (P * r * 2) / 100 = ₹4,000. Step 2: Rearranging, P * r = (4,000 * 100) / 2 = 200,000. Step 3: For compound interest for 2 years: CI = P * ((1 + r / 100)^2 - 1) = ₹4,180. Step 4: Test the given options for r (in %): 6, 12, 9, 7.5, 4.5. Step 5: Try r = 9%. Then P = 200,000 / 9 ≈ 22,222.22. Step 6: Compute CI for r = 9%: CI = 22,222.22 * ((1.09^2) - 1) = 22,222.22 * (1.1881 - 1) = 22,222.22 * 0.1881 ≈ ₹4,180. Step 7: The result exactly matches the given CI = ₹4,180, so r = 9% satisfies both conditions. Step 8: Therefore, the required rate of interest is 9% per annum.

Verification / Alternative check:
Once you have r = 9%, you can quickly verify SI: SI = (P * r * 2) / 100 = (22,222.22 * 9 * 2) / 100 ≈ ₹4,000. This confirms that both the SI and CI values are consistent with r = 9% per annum, so no other option is needed.

Why Other Options Are Wrong:

• 6%: At this rate, the CI for 2 years is too small (around ₹4,120) and does not match ₹4,180.
• 12%: At 12%, the 2 year CI is about ₹4,240, which is greater than ₹4,180.
• 7.5%: At 7.5%, CI is about ₹4,150, still not equal to ₹4,180.
• 4.5%: At 4.5%, CI is about ₹4,090, clearly less than ₹4,180.

Common Pitfalls:
Students often try to apply a shortcut based only on the difference between CI and SI without writing the equations. Another mistake is to forget that SI for 2 years is proportional to 2r, whereas CI depends on (1 + r / 100)^2. Always form the correct equations for SI and CI and then test the options logically.

Final Answer:
The correct annual rate of interest is 9% per annum.