The simple interest on a sum of money for 2 years at 5% per annum is Rs 50. What will be the compound interest on the same principal at the same rate for the same period of 2 years, when interest is compounded annually?

Introduction / Context:
This question connects simple interest and compound interest for the same principal, rate, and time. Instead of giving the principal directly, the problem tells you the simple interest and asks you to determine the compound interest. This tests your ability to first reverse the simple interest formula to find the principal, and then apply the compound interest formula correctly. Such conversions are common in exam questions and help deepen understanding of the differences between simple and compound interest.

Given Data / Assumptions:

• Simple interest SI for 2 years at 5% per annum is Rs 50
• Annual rate of interest r = 5% per annum
• Time period t = 2 years
• Interest for the compound case is compounded annually
• We must find compound interest CI for the same P, r, and t

Concept / Approach:
First, we use the simple interest formula SI = P * r * t / 100 to find the principal P. After that, we treat this principal as the starting amount for the compound interest calculation. For annual compounding, the amount after n years is A = P * (1 + r)^n. The compound interest CI is then A - P. It is important to remember that simple interest grows linearly, while compound interest grows on a growing amount, causing a slightly higher interest for the same period and rate.

Step-by-Step Solution:
Given SI = 50, r = 5% per annum, t = 2 years Simple interest formula: SI = P * r * t / 100 50 = P * 5 * 2 / 100 50 = P * 10 / 100 = P * 0.10 So P = 50 / 0.10 = 500 Now apply compound interest with P = 500, r = 5%, n = 2 A = 500 * (1 + 0.05)^2 = 500 * 1.05^2 1.05^2 = 1.1025, so A = 500 * 1.1025 = 551.25 CI = A - P = 551.25 - 500 = Rs 51.25

Verification / Alternative check:
We can verify by computing year wise. At the end of year 1, interest is 500 * 0.05 = 25 and amount is 525. At the end of year 2, interest is 525 * 0.05 = 26.25. The total compound interest over 2 years is 25 + 26.25 = 51.25, which matches our previous result. This confirms that the compound interest is slightly greater than the simple interest of Rs 50 for the same period.

Why Other Options Are Wrong:
Rs 52, Rs 53, and Rs 54 are all larger than the correctly computed compound interest. None of these values match the precise calculation from either the formula or the year by year approach. The difference between simple and compound interest for small rates and short periods is modest, so large departures from Rs 51.25 are unlikely.

Common Pitfalls:
One common error is to assume that compound interest must be much larger than simple interest, and to pick a bigger number without calculation. Another mistake is to forget to find the principal from the simple interest data and instead guess or assume an arbitrary principal. Correctly following the two stage process avoids these problems.

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