For a certain principal sum of money, the difference between the simple interest and the compound interest (compounded annually) for 2 years at 5% per annum is Rs 45. What is the value of the principal (the original sum)?

Introduction / Context:
This question tests the relationship between simple interest (SI) and compound interest (CI) for a fixed period and rate, and how to use the known difference between them to find the original principal. The rate of interest is 5% per annum and the time period is 2 years, with compound interest calculated annually. The key idea is that for small time periods, there is a simple formula for the difference between CI and SI which makes such problems quick to solve in competitive exams.

Given Data / Assumptions:

• The principal (original sum) is P rupees.
• Rate of interest, r = 5% per annum.
• Time period, t = 2 years.
• Interest is compounded annually for CI.
• Difference between CI for 2 years and SI for 2 years is Rs 45.

Concept / Approach:
For a given principal P, rate r% per annum and time 2 years, the simple interest is P * r * 2 / 100. The compound interest for 2 years is P * [(1 + r/100)^2 - 1]. For 2 years, the difference between CI and SI can be directly written as P * r^2 / 100^2. Using this shortcut, we equate the difference to the given Rs 45 and solve for P. This avoids computing full SI and CI separately and saves time.

Step-by-Step Solution:
Let the principal be P rupees. For 2 years at r% per annum, difference between CI and SI is given by: Difference = P * r^2 / 100^2. Here, r = 5, so r^2 = 25. Therefore, Difference = P * 25 / 10000 = P * 0.0025. We are told that this difference is Rs 45, so: P * 0.0025 = 45. Rearranging, P = 45 / 0.0025. Compute: 0.0025 = 25 / 10000, so 45 / 0.0025 = 45 * 400 = 18,000. Thus, the required principal is Rs 18,000.

Verification / Alternative check:
We can verify by calculating SI and CI directly. Simple interest for 2 years at 5%: SI = P * r * t / 100 = 18,000 * 5 * 2 / 100 = 18,000 * 0.10 = Rs 1,800. Amount with CI after 2 years: A = P * (1 + r/100)^2 = 18,000 * (1.05)^2 = 18,000 * 1.1025 = Rs 19,845. So CI = 19,845 - 18,000 = Rs 1,845. Difference CI - SI = 1,845 - 1,800 = Rs 45, which matches the given value. This confirms our answer is consistent.

Why Other Options Are Wrong:
Rs 36,000 would double the difference to Rs 90, not Rs 45. Rs 9,000 would give half the correct difference (Rs 22.50). Rs 27,000 would produce a difference of Rs 67.50, not Rs 45. Rs 22,500 also gives a larger difference than Rs 45. Therefore, none of these values satisfy the given condition exactly.

Common Pitfalls:
A frequent mistake is to try to compute CI and SI fully without using the shortcut formula, which can lead to arithmetic errors. Another error is to confuse the formula for difference between SI and CI for 2 years with that for 3 years. Some students also forget that r must be squared as a whole number of percent, not as a decimal, before dividing by 100^2. Careful substitution and correct handling of percentages are essential to avoid such mistakes.

Final Answer:
Thus, the principal sum of money is Rs 18,000.