For a certain principal invested at compound interest, the accumulated amount at the end of the 2nd year is Rs 2100 and at the end of the 3rd year is Rs 2268. What is the annual rate of interest (in %)?

Introduction / Context:
This question tests your understanding of how compound interest grows from year to year and how you can use successive yearly amounts to determine the annual rate of interest. Instead of giving you the principal directly, the problem provides the accumulated amounts at the end of the second and third years. From this information you must infer the rate of interest per year in percentage form.

Given Data / Assumptions:

• Principal is some unknown amount P invested at compound interest.
• Amount at the end of the 2nd year, A2 = Rs 2100.
• Amount at the end of the 3rd year, A3 = Rs 2268.
• Interest is compounded annually at a constant rate r%.
• We assume there are no deposits or withdrawals between these years.

Concept / Approach:
At compound interest, the amount after each year is multiplied by the same factor (1 + r/100). Therefore, the ratio of the amount at the end of the 3rd year to the amount at the end of the 2nd year is exactly equal to 1 + r/100. By taking this ratio and simplifying, we can solve directly for the annual rate r.

Step-by-Step Solution:
Step 1: Write the compound interest relations: A2 = P * (1 + r/100)^2 and A3 = P * (1 + r/100)^3. Step 2: Form the ratio A3 / A2: A3 / A2 = [P * (1 + r/100)^3] / [P * (1 + r/100)^2] = 1 + r/100. Step 3: Substitute the given amounts: 1 + r/100 = 2268 / 2100. Step 4: Simplify the fraction: 2268 / 2100 = 1.08. Step 5: Therefore, 1 + r/100 = 1.08, so r/100 = 0.08 and r = 8.
Verification / Alternative check:
Assume r = 8%. Then 1 + r/100 = 1.08. If A2 is 2100, the next year amount should be A3 = 2100 * 1.08 = 2268, which matches the given data, confirming that 8% is correct.
Why Other Options Are Wrong:
7% would give a growth factor of 1.07, and 2100 * 1.07 is 2247, not 2268. 9% would give a factor of 1.09, and 2100 * 1.09 is 2289, which is too large. 10% would give 2100 * 1.10 = 2310, which also does not match 2268.
Common Pitfalls:
A common mistake is to try to find the principal first using both amounts, which is unnecessary and can lead to algebraic errors. Another frequent error is to apply simple interest logic instead of compound interest, forgetting that each year builds on the previous year's amount.
Final Answer:
The correct annual rate of compound interest is 8%.