The difference between the simple interest and the compound interest (compounded annually) on a certain sum of money for 2 years at 12% per annum is Rs 900. What is the value of this principal sum, in rupees?

Introduction / Context:
This question involves a known shortcut relating the difference between compound interest and simple interest for 2 years on the same principal and at the same rate. Instead of computing simple and compound interest separately, we can use a compact formula that directly connects the difference with the principal and the rate. This allows us to work backwards from the given difference to the principal. It is a common type of problem in competitive exams.

Given Data / Assumptions:

• Simple interest (SI) and compound interest (CI) are calculated on the same principal for 2 years.
• Rate of interest R = 12% per annum.
• Difference between CI and SI over 2 years = Rs 900.
• Interest is compounded annually for CI.
• We must find the principal sum P.

Concept / Approach:
For 2 years at the same rate R on principal P, with annual compounding, there is a direct relation:
Difference (CI - SI) for 2 years = P * (R / 100)^2This result comes from expanding the compound amount and comparing it with simple interest over 2 years. Once we know the difference and the rate, we can solve for P easily by rearranging the formula to:
P = Difference / (R / 100)^2

Step-by-Step Solution:
Step 1: Write the formula for the difference: CI - SI over 2 years = P * (R / 100)^2.Step 2: Substitute the given data. Here CI - SI = 900 and R = 12.Step 3: Compute (R / 100)^2 = (12 / 100)^2 = (0.12)^2 = 0.0144.Step 4: Set up the equation: 900 = P * 0.0144.Step 5: Solve for P by dividing both sides by 0.0144: P = 900 / 0.0144.Step 6: Convert 0.0144 to a fraction: 0.0144 = 144 / 10000.Step 7: Therefore P = 900 * 10000 / 144. Simplify 900 / 144 = 25 / 4, so P = (25 / 4) * 10000 = 62500.Step 8: Hence the principal sum is Rs 62,500.

Verification / Alternative check:
We can verify by computing both SI and CI using P = 62,500 and R = 12%. Simple interest for 2 years is SI = 62500 * 12 * 2 / 100 = 62500 * 0.24 = 15000. Compound interest is based on A = 62500 * (1.12)^2 = 62500 * 1.2544 = 78400. CI = 78400 - 62500 = 15900. The difference CI - SI = 15900 - 15000 = 900, which matches the given value. This confirms that the computed principal is correct.

Why Other Options Are Wrong:
Rs.125000 would give a difference four times as large, Rs.250000 even more, and Rs.187500 somewhere in between, none of which produce exactly Rs 900 as the difference. Rs.100000 also fails to satisfy the difference formula. Only Rs.62500 leads to a difference of Rs 900 between compound and simple interest at 12% per annum over 2 years.

Common Pitfalls:
Some students attempt to compute SI and CI separately without remembering the compact formula, which leads to more steps and potential arithmetic mistakes. Others misapply the formula, using P * R / 100 instead of P * (R / 100)^2, or forget that it applies specifically to 2 years. Carefully recalling that CI - SI for 2 years equals P * (R / 100)^2 makes the problem straightforward.

Final Answer:
The value of the principal sum that produces a Rs 900 difference between compound and simple interest in 2 years at 12% per annum is Rs.62500.