On a certain principal, the simple interest for 2 years is Rs. 2,800 and the compound interest for the same principal, rate and time (2 years, compounded annually) is Rs. 2,996. What is the annual rate of interest?

Introduction / Context: This question asks you to use both simple interest and compound interest information to determine the interest rate. Knowing the simple interest for 2 years and the compound interest for 2 years on the same principal offers enough information to solve for the rate without explicitly knowing the principal.

Given Data / Assumptions:

Simple interest (SI) for 2 years = Rs. 2,800. Compound interest (CI) for 2 years, compounded annually = Rs. 2,996. Principal is P and rate is r% per annum. Time t = 2 years in both cases.

Concept / Approach: For simple interest: SI = P * r * t / 100. For 2 years, SI = 2P * r / 100 = 2,800. For compound interest over 2 years: CI = P * (1 + r/100)^2 − P. Also, for 2-year CI, the difference CI − SI equals the interest on the first year's interest: CI − SI = P * r^2 / 100^2. We can use SI to relate P and r, then use CI − SI to solve for r directly.

Step-by-Step Solution: Step 1: Write SI equation. SI = 2P * r / 100 = 2,800. So P * r = 2,800 * 100 / 2 = 140,000. Step 2: Write CI − SI equation. CI − SI = 2,996 − 2,800 = 196. Also, CI − SI = P * r^2 / 100^2. So 196 = P * r^2 / 10,000. Step 3: Use P * r = 140,000 to eliminate P. P * r^2 / 10,000 divided by P * r gives (r / 10,000). Hence, r / 10,000 = 196 / 140,000 → r = 196 * 10,000 / 140,000 = 14.

Verification / Alternative check: Check with r = 14%. SI for 2 years: SI = 2P * 14 / 100 = 0.28P = 2,800 → P = 10,000. Now CI: Amount A = 10,000 * (1.14)^2 = 10,000 * 1.2996 = 12,996. CI = A − P = 12,996 − 10,000 = 2,996, matching the given CI.

Why Other Options Are Wrong: Rates of 7%, 21% or 28% give SI and CI values that do not produce a difference of exactly 196. For example, at 7% the difference CI − SI would be far smaller; at higher rates it would be much larger.

Common Pitfalls: A common error is to try to solve for P and r independently without using the simplifying relation between CI − SI and r^2. Another mistake is miscomputing the difference CI − SI or forgetting to square r when using the formula for CI − SI over 2 years.

Final Answer: The annual rate of interest is 14 percent.