Sonika invested an amount of Rs. 5,800 for 2 years at compound interest compounded annually. If she earned a total compound interest of Rs. 594.50 at the end of 2 years, then at what annual rate of compound interest did she invest?

Introduction:
This is a reverse compound interest problem where the principal, time, and total compound interest are known, and we are asked to find the annual rate of interest. It checks understanding of how the compound interest formula can be rearranged to determine the rate when the amount of interest is given.

Given Data / Assumptions:
Principal P = Rs. 5,800. Time = 2 years. Total compound interest over 2 years = Rs. 594.50. Interest is compounded annually. We need the annual rate of interest r%.

Concept / Approach:
Compound interest for 2 years is given by: CI = P * [(1 + r/100)^2 − 1]. We know CI and P, so we can solve for (1 + r/100)^2, then take the square root to find (1 + r/100), and finally obtain r. Recognising perfect squares can make this process easier.

Step-by-Step Solution:
Step 1: Use the CI formula. 594.50 = 5800 * [(1 + r/100)^2 − 1]. (1 + r/100)^2 − 1 = 594.50 / 5800. 594.50 / 5800 = 0.1025. So (1 + r/100)^2 = 1 + 0.1025 = 1.1025. Step 2: Take square root. We notice 1.1025 = (1.05)^2. Therefore, 1 + r/100 = 1.05. r/100 = 0.05, so r = 5% per annum.

Verification / Alternative check:
Verify by recomputing CI with r = 5%: Amount A = 5800 * (1.05)^2 = 5800 * 1.1025 = Rs. 6,394.50. Compound interest = 6,394.50 − 5,800 = Rs. 594.50. This exactly matches the given interest, confirming that the rate is correct.

Why Other Options Are Wrong:
Rates like 6%, 6.5%, 4.5% or 7% would produce different values of (1 + r/100)^2 and would not reproduce Rs. 594.50 as the total compound interest. They come from guessing or incomplete handling of the algebraic steps.

Common Pitfalls:
Some students confuse simple and compound interest formulas and try to apply P * r * t / 100, which is incorrect here. Others may forget to add 1 to the interest fraction when forming (1 + r/100)^2 or make mistakes when taking the square root. Working carefully with percentages and recognising common decimal squares helps a lot.

Final Answer:
The required annual rate of compound interest is 5% per annum.