The difference between the compound interest and the simple interest on a certain sum at 10% per annum for 2 years is Rs. 631. What is the value of this principal sum?

Introduction / Context:
This is a popular type of interest question where the difference between compound interest (C.I.) and simple interest (S.I.) on the same principal at the same rate and for the same time period is given. Using this difference, you must find the principal sum. The rate and time are such that a simple relationship emerges, especially when the time is 2 years.

Given Data / Assumptions:
- Let the principal be P rupees.
- Rate of interest r = 10% per annum.
- Time t = 2 years.
- Difference between compound interest and simple interest for this period = Rs. 631.
- Interest is compounded annually for the compound interest calculation.

Concept / Approach:
For 2 years at rate r, simple interest is S.I. = P * r * t / 100. Compound interest is C.I. = P * (1 + r/100)^2 - P. For r = 10%, simple interest for 2 years is 20% of P, and the total compound interest is 21% of P (because (1.10)^2 = 1.21). Thus the difference C.I. - S.I. is (21% - 20%) of P = 1% of P. This makes the principal easy to find from the given difference.

Step-by-Step Solution:
Step 1: For r = 10% and t = 2 years, simple interest S.I. = P * 10 * 2 / 100 = 0.20P.Step 2: Compound amount after 2 years: A = P * (1.10)^2 = P * 1.21.Step 3: Compound interest C.I. = A - P = 1.21P - P = 0.21P.Step 4: Difference between C.I. and S.I. = 0.21P - 0.20P = 0.01P = P/100.Step 5: Given that this difference equals Rs. 631, so P/100 = 631.Step 6: Solve for P: P = 631 * 100 = 63100.Step 7: Hence, the principal sum is Rs. 63,100.

Verification / Alternative check:
Check quickly by computing interests on Rs. 63,100. S.I. for 2 years at 10%: 63100 * 0.20 = 12620. C.I. for 2 years at 10%: 63100 * 0.21 = 13251. Difference = 13251 - 12620 = 631, exactly matching the given difference.

Why Other Options Are Wrong:
Rs. 60,100, Rs. 61,100 or Rs. 62,100 would produce differences of 601, 611 and 621 respectively, since the difference is always 1% of the principal for 10% over 2 years. Rs. 55,000 would give a difference of only 550. Only Rs. 63,100 gives a difference of Rs. 631.

Common Pitfalls:
Many students unnecessarily compute full compound interest and simple interest separately for each option instead of recognizing the simple pattern that the difference for 2 years at rate r is effectively P * (r/100)^2. For r = 10%, that becomes P * 0.01 = P/100. Recognizing such shortcuts can save time and reduce arithmetic mistakes in exams.

Final Answer:
The principal sum is Rs. 63,100.