The difference between the simple interest and the compound interest (compounded annually) on a certain sum of money at 12% per annum for 2 years is Rs. 72. What is the value of the original principal sum (in rupees)?

Introduction:
This question compares simple interest and compound interest over the same period and rate and gives you the difference between them. Using a known formula for the difference between compound interest and simple interest over two years, you can directly compute the principal without having to find the interests separately.

Given Data / Assumptions:

• Difference between compound interest and simple interest for 2 years = Rs. 72.
• Rate of interest r = 12% per annum.
• Time t = 2 years.
• Interest is compounded annually for CI.
• Principal P is unknown.

Concept / Approach:
For the same principal, rate, and time, the difference between compound interest and simple interest for 2 years has a neat formula:
CI - SI = P * (r^2) / 100^2This arises from expanding the compound interest formula for 2 years and comparing it with the simple interest expression. Here r is in percent and P is principal.

Step-by-Step Solution:
Step 1: Write the formula for the difference over 2 years: CI - SI = P * r^2 / 100^2.Step 2: Substitute known values: CI - SI = 72 and r = 12%.Step 3: Compute r^2: 12^2 = 144.Step 4: Therefore 72 = P * 144 / 10,000.Step 5: Simplify the fraction: 144 / 10,000 = 0.0144.Step 6: So 72 = 0.0144 * P.Step 7: Solve for P by dividing: P = 72 / 0.0144.Step 8: Compute: P = 5,000.Hence the principal sum is Rs. 5,000.

Verification / Alternative check:
We can verify by calculating SI and CI directly. Simple interest for 2 years: SI = (P * r * t) / 100 = (5,000 * 12 * 2) / 100 = 1,200. Compound amount after 2 years: A = 5,000 * (1.12)^2 = 5,000 * 1.2544 = 6,272. So CI = 6,272 - 5,000 = 1,272. The difference CI - SI = 1,272 - 1,200 = 72, which matches the given value.

Why Other Options Are Wrong:
If P were 10,000, the difference would be double 72, that is 144. For 20,000, the difference would be four times 72, and so on. The options 8,000, 10,000, 15,000, and 20,000 do not yield a difference of exactly Rs. 72 when using the formula, so they cannot be correct.

Common Pitfalls:
Many students attempt to compute SI and CI separately without using the shortcut formula, which is more error prone. Others forget the square on the rate in the formula and use r instead of r^2. Always remember that for 2 years, CI - SI depends on r squared and not just r. Carefully substituting values and simplifying avoids mistakes.

Final Answer:
The original principal sum for which the difference between simple and compound interest is Rs. 72 at 12% per annum for 2 years is Rs. 5,000.