Simple interest (S.I.) and compound interest (C.I.) are both calculated at the same rate of 10% per annum on the same principal. If compound interest is calculated yearly for 2 years, then for how many years must simple interest be calculated so that the total simple interest equals the 2 year compound interest?

Introduction / Context:
This question compares simple interest and compound interest for the same principal and annual rate. It asks for the time period under simple interest that will generate the same total interest as compound interest generates in 2 years. This highlights the difference between linear growth (simple interest) and exponential growth (compound interest), and it is a common conceptual test in quantitative aptitude and banking exams.

Given Data / Assumptions:

Principal P is the same for both cases (exact amount is not required)
Annual interest rate r = 10 percent per annum
Compound interest is calculated yearly for 2 years
Simple interest is to be calculated at the same rate for an unknown number of years n
We want simple interest over n years to equal compound interest over 2 years

Concept / Approach:
First, compute compound interest on principal P for 2 years at 10 percent per year compounded annually using A = P * (1 + r)^2. The compound interest C.I. is A minus P. Then express simple interest S.I. for n years as S.I. = P * r * n (with r in decimal). Equate S.I. to C.I. and solve for n. Note that because P and r are common factors, we can cancel them and solve directly for n without needing the actual principal.

Step-by-Step Solution:
Step 1: Convert rate to decimal: r = 10 percent = 0.10. Step 2: Compute the compound amount for 2 years. A = P * (1 + r)^2 = P * (1 + 0.10)^2 = P * (1.10)^2. Step 3: Evaluate (1.10)^2 = 1.21. So A = P * 1.21. Step 4: Compound interest C.I. for 2 years is A - P. C.I. = P * 1.21 - P = P * (1.21 - 1) = P * 0.21. Step 5: Express simple interest S.I. for n years. S.I. = P * r * n = P * 0.10 * n = 0.10 * P * n. Step 6: Set simple interest equal to compound interest. 0.10 * P * n = 0.21 * P. Step 7: Cancel P from both sides (P is non zero). 0.10 * n = 0.21. Step 8: Solve for n: n = 0.21 / 0.10 = 2.1 years.

Verification / Alternative check:
As a check, assume a convenient principal such as P = 1,000 rupees. Compound interest at 10 percent for 2 years is P * 0.21 = 210 rupees. Simple interest at 10 percent per year for 2.1 years would be S.I. = 1,000 * 0.10 * 2.1 = 210 rupees, which matches the compound interest exactly. This confirms that 2.1 years of simple interest at 10 percent per annum produces the same interest as 2 years of compound interest at the same rate.

Why Other Options Are Wrong:
Option 4.2 years would produce simple interest S.I. = P * 0.10 * 4.2 = 0.42 * P, which is double the required interest of 0.21 * P. Option 1.1 years would yield S.I. = 0.11 * P, which is too small. Option 1.4 years gives S.I. = 0.14 * P, also less than 0.21 * P. Only 2.1 years results in simple interest that matches the 2 year compound interest.

Common Pitfalls:
Some students mistakenly equate the amounts instead of the interest components or forget to subtract principal when computing compound interest. Others treat 10 percent for 2 years under compound interest as the same as 20 percent under simple interest, ignoring the extra interest on interest that arises in the compound case. Careful attention to the definitions of simple and compound interest and a clear distinction between amount and interest help avoid these errors.

Final Answer:
The simple interest must be calculated for 2.1 years in order to equal the 2 year compound interest at 10 percent per annum.