If the difference between the compound interest and the simple interest on a certain sum at the rate of 5% per annum for 2 years is Rs 20, then what is the value of that sum (principal), in rupees?

Introduction / Context:
This is another question that uses the relationship between simple interest and compound interest on the same principal over a period of 2 years. The difference between compound interest and simple interest is given, and we must find the principal. The rate is moderate (5% per annum) and the time period is 2 years, which allows use of a direct formula. This type of question is very common in competitive exams because it tests whether a student knows this formula and can manipulate it correctly.

Given Data / Assumptions:

• Rate of interest R = 5% per annum.
• Time period T = 2 years.
• Compound interest is computed annually.
• Difference between compound interest (CI) and simple interest (SI) for 2 years is Rs 20.
• Principal P is unknown and must be determined.

Concept / Approach:
For 2 years at the same rate R on a principal P, the difference between CI and SI is given by:
CI - SI = P * (R / 100)^2This formula is derived from expanding the 2 year compound amount and comparing it with the simple interest expression for 2 years. Given the difference and the rate, we can solve directly for P. No separate calculation of SI and CI is required.

Step-by-Step Solution:
Step 1: Write the formula for the difference between CI and SI over 2 years: CI - SI = P * (R / 100)^2.Step 2: Substitute the known values: CI - SI = 20 and R = 5.Step 3: Compute (R / 100)^2 = (5 / 100)^2 = (0.05)^2 = 0.0025.Step 4: So the equation becomes 20 = P * 0.0025.Step 5: Solve for P by dividing both sides by 0.0025: P = 20 / 0.0025.Step 6: Note that 0.0025 = 25 / 10000. So P = 20 * 10000 / 25.Step 7: Simplify 20 / 25 = 4 / 5, hence P = (4 / 5) * 10000 = 8000.Step 8: Therefore, the principal sum is Rs 8,000.

Verification / Alternative check:
We can verify by directly computing SI and CI on Rs 8,000 at 5% per annum for 2 years. Simple interest SI = 8000 * 5 * 2 / 100 = 800. For compound interest: amount A = 8000 * (1.05)^2. Compute 1.05^2 = 1.1025, so A = 8000 * 1.1025 = 8820. CI = A - P = 8820 - 8000 = 820. The difference CI - SI = 820 - 800 = 20, exactly as given, which confirms that the principal is Rs 8,000.

Why Other Options Are Wrong:
Rs.2000, Rs.4000, and Rs.6000 would produce smaller differences between CI and SI at 5% over 2 years, because the difference scales directly with P. Rs.10000 would yield a larger difference of 25, not 20. Only Rs.8000 gives a difference of exactly Rs 20 between compound and simple interest at 5% for 2 years.

Common Pitfalls:
Students sometimes forget the specific formula for CI - SI in 2 years and instead attempt a full SI and CI computation, which is longer but still feasible. Others mistakenly use P * R / 100 instead of P * (R / 100)^2, confusing simple interest for one year with the extra interest in compounding. Correctly recalling that CI - SI = P * (R / 100)^2 for 2 years makes the calculation quick and reliable.

Final Answer:
The principal sum that produces a difference of Rs 20 between compound interest and simple interest at 5% per annum for 2 years is Rs.8000.