If the simple interest on a certain sum of money for 2 years at 5% per annum is Rs 50, then what is the compound interest on the same sum at the same rate and for the same time, when interest is compounded annually?

Introduction / Context:
This problem links simple interest (SI) and compound interest (CI) on the same sum, rate, and time. We are told the simple interest for 2 years at 5% per annum and must find the compound interest for the same period, with annual compounding. Such questions are useful for practicing the conversion between SI and CI and for reinforcing the basic formulas for both types of interest.

Given Data / Assumptions:

• Simple interest SI for 2 years = Rs 50.
• Rate r = 5% per annum.
• Time t = 2 years.
• Interest is to be compounded annually for the CI calculation.
• Principal P is unknown but can be derived from the SI information.

Concept / Approach:
For simple interest, SI = P * r * t / 100. We use this equation to find the principal P. Once we know P, we apply the compound interest formula A = P * (1 + r/100)^t to find the amount A after 2 years. The compound interest is then CI = A - P. Because the time and rate are small, the calculations are straightforward, and we can find the exact value of CI without complex tools.

Step-by-Step Solution:
We know SI = P * r * t / 100. Substitute SI = 50, r = 5, t = 2. So 50 = P * 5 * 2 / 100 = P * 10 / 100 = 0.10P. Therefore, P = 50 / 0.10 = 500. Now compute the amount under compound interest for 2 years at 5% per annum: A = 500 * (1.05)^2. Compute (1.05)^2 = 1.1025. So A = 500 * 1.1025 = 551.25. Hence, compound interest CI = A - P = 551.25 - 500 = 51.25. Therefore, the required compound interest is Rs 51.25.

Verification / Alternative check:
We can compare SI and CI directly. For P = 500, SI at 5% for 2 years is 500 * 5 * 2 / 100 = 50, which matches the given value. The first year's CI is 500 * 0.05 = 25, making the amount 525. The second year's CI is 525 * 0.05 = 26.25. Total CI = 25 + 26.25 = 51.25. This stepwise reasoning confirms the compound interest value obtained from the formula method.

Why Other Options Are Wrong:
Rs 61.25 is too high and would require a larger rate or longer time period. Rs 51 is close but slightly less than the exact CI and typically results from rounding errors or treating both years as if they had the same interest of 25.5. Rs 42 is too low and Rs 50 is exactly the SI, not the CI. Only Rs 51.25 fits the detailed compound interest computation.

Common Pitfalls:
Some students incorrectly assume that CI will always exceed SI by a large margin and choose the option that appears significantly larger. Others forget that the principal for the second year of CI includes the first year's interest. Using simple interest logic in the compound interest phase leads to wrong answers. Careful application of the formula or a year-by-year breakdown is essential to obtain the correct result.

Final Answer:
The compound interest on the sum at 5% per annum for 2 years is Rs 51.25.