The difference between the compound interest and the simple interest on an amount of Rs. 18,000 for 2 years is Rs. 405. Assuming the same rate of interest per annum for both cases and that compound interest is compounded annually, what is the annual rate of interest in percent?

Introduction:
This question uses a special relationship between compound interest and simple interest for a period of 2 years. Instead of computing both interests separately, there is a direct formula for the difference between compound and simple interest over 2 years, which greatly simplifies the process of finding the interest rate.

Given Data / Assumptions:
Principal P = Rs. 18,000. Time t = 2 years. Difference between CI and SI over 2 years = Rs. 405. Rate r% per annum is the same for both CI and SI, with CI compounded annually.

Concept / Approach:
For 2 years, the difference between compound interest and simple interest on a principal P at rate r% is: CI − SI = P * (r/100)^2. This result comes from expanding the compound interest formula for 2 years. We use this relation to form an equation in r and then solve for r.

Step-by-Step Solution:
Formula: CI − SI = P * (r/100)^2. 405 = 18000 * (r/100)^2. (r/100)^2 = 405 / 18000. 18000 / 10000 = 1.8, so 405 / 18000 = 405 / (1.8 * 10000) = 0.0225. Thus (r/100)^2 = 0.0225. r/100 = sqrt(0.0225) = 0.15. Therefore r = 0.15 * 100 = 15% per annum.

Verification / Alternative check:
Check by computing CI and SI explicitly with r = 15%. For 2 years, SI = 18000 * 15 * 2 / 100 = Rs. 5,400. CI amount = 18000 * (1.15)^2 = 18000 * 1.3225 = Rs. 23,805. CI = 23,805 − 18,000 = Rs. 5,805. Difference = 5,805 − 5,400 = Rs. 405, which matches the given value.

Why Other Options Are Wrong:
Rates like 12%, 13% or 14% would give much smaller differences between CI and SI, while 10% would give an even smaller difference. None of these satisfy the required difference of Rs. 405 when substituted into the formula.

Common Pitfalls:
A common mistake is to attempt full calculation of CI and SI separately for each option, which is time consuming and error prone. Others forget the formula for CI − SI for 2 years and try to use an incorrect relation. Remembering the compact formula P * (r/100)^2 is very helpful in such questions.

Final Answer:
The annual rate of interest is 15% per annum.