In aptitude (Simple Interest vs Compound Interest), the difference between compound interest (CI) and simple interest (SI) on a certain sum for 2 years at 8% per annum is ₹40. What is the principal sum (in ₹)?

Introduction / Context:
Here you are given the difference between compound interest and simple interest on the same principal, at the same rate, over the same period of 2 years. This is a standard type of aptitude question that relies on a compact formula relating that difference directly to the principal and rate, making the solution quick once you know the relationship.

Given Data / Assumptions:

• The rate of interest r = 8% per annum.
• Time period t = 2 years.
• Difference between CI and SI for 2 years = ₹40.
• Principal P is the same for both CI and SI.
• Interest is compounded annually for the CI calculation.
• We must find P in rupees.

Concept / Approach:
For 2 years at rate r%, the difference between compound interest and simple interest on principal P is: CI - SI = P * (r / 100)^2. This happens because compound interest for 2 years can be written as: CI = SI + extra interest, and that extra interest equals P * (r / 100)^2. Since r is known and the difference CI - SI is given as ₹40, we can directly solve for P using this formula without computing the full CI and SI separately.

Step-by-Step Solution:
Step 1: Use the difference formula: CI - SI = P * (r / 100)^2. Step 2: Substitute CI - SI = 40 and r = 8. Step 3: Compute (r / 100)^2 = (8 / 100)^2 = 0.08^2 = 0.0064. Step 4: So the equation becomes 40 = P * 0.0064. Step 5: Rearrange for P: P = 40 / 0.0064. Step 6: Compute 40 / 0.0064 = 6,250. Step 7: Therefore, the principal sum P is ₹6,250.

Verification / Alternative check:
Verify by computing SI and CI for P = ₹6,250 at 8% for 2 years. Simple interest: SI = (6,250 * 8 * 2) / 100 = (6,250 * 16) / 100 = 100,000 / 100 = 1,000. Compound interest: Amount A = 6,250 * (1 + 8 / 100)^2 = 6,250 * 1.08^2. Compute 1.08^2 = 1.1664, so A = 6,250 * 1.1664 = 7,290. CI = A - P = 7,290 - 6,250 = 1,040. Difference CI - SI = 1,040 - 1,000 = 40, which matches the given difference.

Why Other Options Are Wrong:
₹12,500, ₹18,750, ₹25,000, and ₹10,000 all produce differences between CI and SI that are either larger or smaller than ₹40 when the rate is 8% and the time is 2 years. Only the principal of ₹6,250 yields exactly a ₹40 difference, so the other options are not valid.

Common Pitfalls:
Many learners ignore the direct formula for CI - SI in 2 years and instead try to compute SI and CI for several trial principals, which is slow and error prone. Another mistake is to use the formula for longer periods incorrectly or to miscalculate (r / 100)^2. Always remember that for exactly 2 years, the difference between CI and SI depends only on P and the square of the rate fraction, making these problems fast to solve once you know the relationship.

Final Answer:
The principal sum for which the difference between compound interest and simple interest at 8% per annum over 2 years is ₹40 is ₹6,250.