The difference between the simple interest and the compound interest (compounded annually) on a certain sum of money for 2 years at 5% per annum is Rs. 45. What is the value of that sum (in rupees)?

Introduction:
This problem is similar to an earlier one but uses a different rate and difference amount. We are given the difference between compound interest and simple interest over 2 years at 5% per annum and asked to find the principal sum that produces this difference.

Given Data / Assumptions:

• Difference between compound interest and simple interest for 2 years = Rs. 45.
• Rate r = 5% per annum.
• Time t = 2 years.
• Interest is compounded annually when calculating compound interest.
• Principal P is unknown.

Concept / Approach:
For 2 years, the difference between compound interest (CI) and simple interest (SI) on the same principal, at the same rate, is given by the formula:
CI - SI = P * (r^2) / 100^2This formula is derived by expanding the compound interest formula and subtracting simple interest for 2 years.

Step-by-Step Solution:
Step 1: Use the formula CI - SI = P * r^2 / 100^2.Step 2: Substitute CI - SI = 45 and r = 5.Step 3: Compute r^2: 5^2 = 25.Step 4: Hence, 45 = P * 25 / 10,000.Step 5: Simplify the fraction 25 / 10,000 = 0.0025.Step 6: So 45 = 0.0025 * P.Step 7: Solve for P: P = 45 / 0.0025.Step 8: Perform the division: P = 18,000.Therefore the principal sum is Rs. 18,000.

Verification / Alternative check:
To verify, compute SI and CI for P = 18,000 at 5% for 2 years. SI = (P * r * t) / 100 = (18,000 * 5 * 2) / 100 = 1,800. Compound amount A = 18,000 * (1.05)^2 = 18,000 * 1.1025 = 19,845. CI = A - P = 19,845 - 18,000 = 1,845. Difference CI - SI = 1,845 - 1,800 = 45, matching the given value.

Why Other Options Are Wrong:
For P = 36,000, the difference would double to 90. For P = 54,000 or 72,000, the difference would be proportionally larger, not equal to 45. A sum of 9,000 would give half the correct difference, only 22.5. Therefore only Rs. 18,000 satisfies the condition that CI - SI equals Rs. 45 at 5% over 2 years.

Common Pitfalls:
Students sometimes forget that this formula involves r squared. Using r instead of r^2 will produce an incorrect value for P. Others make mistakes in the division when solving for the principal, especially when dealing with decimals. It is best to carefully multiply and divide step by step or rewrite 0.0025 as 25/10,000 to keep track of fractions.

Final Answer:
The sum of money for which the difference between simple and compound interest is Rs. 45 at 5% per annum for 2 years is Rs. 18,000.