For a certain principal invested at compound interest, the amounts received at the end of the 2nd year and at the end of the 3rd year are Rs. 2100 and Rs. 2268 respectively. What is the annual rate of interest (in percent), assuming yearly compounding?

Introduction / Context:
Here we work with successive amounts under compound interest. Instead of being given principal and rate directly, we are told the total amounts at the end of the 2nd and 3rd years. By using the relationship between these amounts, we can find the annual compound interest rate without knowing the principal explicitly.

Given Data / Assumptions:

Amount at end of 2nd year, A2 = Rs. 2100
Amount at end of 3rd year, A3 = Rs. 2268
Interest is compounded annually at a constant rate r%
Principal P is the same throughout but is not directly needed
We must find r in percent

Concept / Approach:
Under annual compounding at rate r%, the amount each year is multiplied by a factor (1 + r/100). Therefore, successive amounts are related as:
A3 = A2 * (1 + r/100). Taking the ratio A3 / A2 gives us the single-year growth factor directly. From that factor we can find r. This is often easier than working back to the principal first.

Step-by-Step Solution:
Step 1: Compute the ratio of the two amounts. A3 / A2 = 2268 / 2100. This simplifies to 1.08. Step 2: Relate this to the annual growth factor. A3 = A2 * (1 + r/100) ⇒ 1 + r/100 = 1.08. Step 3: Solve for r. r/100 = 1.08 − 1 = 0.08. Therefore, r = 8% per annum.

Verification / Alternative check:
We can check the consistency. If the amount at the end of year 2 is Rs. 2100 and grows at 8%, then:
A3 = 2100 * 1.08 = 2268. This matches the given third-year amount exactly, confirming the rate. The value of the principal P does not need to be calculated for this question, which is one of the advantages of using ratios in compound interest problems.

Why Other Options Are Wrong:
A rate of 7% would give A3 = 2100 * 1.07 = 2247, not 2268. At 9%, A3 would be 2100 * 1.09 = 2289. At 10%, the amount would be even higher. None of these match 2268, so they are incorrect.

Common Pitfalls:
A common error is to mistakenly treat 2100 and 2268 as interests instead of total amounts, which complicates the analysis unnecessarily. Another pitfall is to attempt to compute the principal first when it is not required, leading to algebraic mistakes. Always look for simple ratios between successive amounts when dealing with compound interest over consecutive years.

Final Answer:
The rate of interest is 8% per annum.