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

Introduction / Context:
Once again, this question uses the relationship between compound interest and simple interest over 2 years on the same principal at the same rate. Here the rate is 10 percent per annum, and the difference between compound and simple interest is Rs. 631. We must find the principal sum. Recognizing and applying the special 2 year formula simplifies the process significantly.

Given Data / Assumptions:

• Difference between compound interest and simple interest for 2 years is Rs. 631.
• Rate r = 10 percent per annum.
• Time n = 2 years.
• Interest is compounded annually for the compound interest part.
• We need to find the principal P.

Concept / Approach:
For a principal P at rate r percent per annum over 2 years, the difference between compound interest and simple interest is: Difference = P * (r/100)^2 This relation comes from expanding the compound interest formula for 2 years and subtracting the simple interest over the same period. We substitute r = 10 and the given difference to solve for P.

Step-by-Step Solution:
Given Difference = 631 Formula: Difference = P * (r/100)^2 So 631 = P * (10/100)^2 (10/100)^2 = (0.10)^2 = 0.01 Thus 631 = P * 0.01 P = 631 / 0.01 P = 63100 Therefore, the principal sum is Rs. 63100

Verification / Alternative check:
We can verify using explicit interest calculations. For P = 63100 and r = 10 percent per annum: Simple interest for 2 years = 63100 * 10 * 2 / 100 = 63100 * 0.20 = 12620 Compound amount after 2 years = 63100 * (1.10)^2 (1.10)^2 = 1.21, so amount = 63100 * 1.21 = 76351 Compound interest = 76351 - 63100 = 13251 Difference = 13251 - 12620 = 631 This matches the given difference exactly, confirming that P = 63100 is correct.

Why Other Options Are Wrong:
If we selected P = 60100, 61100, 62100, or 64100, then P * 0.01 would give differences of 601, 611, 621, or 641 respectively, not 631. Only P = 63100 satisfies the formula Difference = P * (0.10)^2 = P * 0.01 = 631. The quick check using explicit SI and CI computations also agrees with this value.

Common Pitfalls:
Some students mistakenly multiply by r/100 rather than (r/100)^2 and therefore get a completely different principal. Others may use the simple interest formula instead of the special difference formula and end up with more cumbersome equations. Misreading 10 percent as 0.010 instead of 0.10 when squaring can also introduce errors. Carefully applying the correct formula and paying attention to decimal placement are key to solving this accurately.

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