The amounts received at the end of the 2nd year and the 3rd year on a certain principal invested at compound interest are Rs. 1800 and Rs. 1926 respectively. Assuming interest is compounded annually, what is the annual rate of interest?

Introduction / Context:
As in similar compound interest questions, we are given two consecutive year-end amounts and asked to find the annual interest rate. Because the investment grows by the same percentage each year, the ratio of the amount at the end of the 3rd year to the amount at the end of the 2nd year gives us the yearly growth factor directly.

Given Data / Assumptions:

Amount at end of 2nd year, A2 = Rs. 1800
Amount at end of 3rd year, A3 = Rs. 1926
Interest is compounded annually at a constant rate r%
Principal P is fixed but not required explicitly
We must find the annual rate r

Concept / Approach:
Under annual compounding:
A3 = A2 * (1 + r/100). Thus, the ratio A3 / A2 equals the factor (1 + r/100). Once we compute this ratio, we subtract 1 and convert to a percentage to obtain r. This method avoids dealing with the original principal.

Step-by-Step Solution:
Step 1: Compute the ratio of amounts. A3 / A2 = 1926 / 1800. 1926 / 1800 = 1.07. Step 2: Relate this to the growth factor. 1 + r/100 = 1.07. Step 3: Solve for r. r/100 = 1.07 − 1 = 0.07. Therefore, r = 7% per annum.

Verification / Alternative check:
To verify, assume some principal P. After 2 years, amount A2 = P * (1.07)^2 = 1800. After 3 years, A3 = P * (1.07)^3 = A2 * 1.07 = 1800 * 1.07 = 1926. This matches the given 3rd-year amount, confirming that 7% is correct. We never actually need to compute P for the question asked.

Why Other Options Are Wrong:
At 6%, the factor would be 1.06, giving A3 = 1800 * 1.06 = 1908. At 6.5%, the factor is 1.065, so A3 would be 1917. For 7.5%, the factor is 1.075, giving A3 = 1935. None of these equal 1926, so those rates cannot be correct.

Common Pitfalls:
A typical mistake is to treat the difference (1926 − 1800) as simple interest for one year on the original principal instead of on the second-year amount. Another error is to try to compute the principal first, which is unnecessary and can lead to algebraic confusion. Always use the direct ratio between consecutive amounts when dealing with compound interest over successive years.

Final Answer:
The annual rate of interest is 7%.