For a certain principal invested at compound interest, the amount at the end of the second year is Rs 30250 and the amount at the end of the third year is Rs 33275. What is the annual rate of compound interest?

Introduction / Context:
This question focuses on using successive compound amounts to determine the annual rate of interest. Instead of giving principal and rate directly, we are told what the investment is worth after two and three years. The key idea is that the ratio of consecutive year end amounts under compound interest is simply 1 plus the annual rate. Such problems test your ability to recognize patterns in the growth of money under compounding and to extract the interest rate from those patterns.

Given Data / Assumptions:

• Amount after 2 years A2 = Rs 30250
• Amount after 3 years A3 = Rs 33275
• Principal P is unknown but stays constant
• Interest is compounded once per year at a constant rate r

Concept / Approach:
Under annual compounding, the amount after each year is multiplied by the same factor (1 + r). Therefore, A2 = P * (1 + r)^2 and A3 = P * (1 + r)^3. Dividing A3 by A2 cancels both P and one power of (1 + r), giving A3 / A2 = 1 + r. This ratio directly equals the growth factor per year, so the annual rate can be found as r = (A3 / A2) - 1. This method avoids having to find the principal first and gives a neat and quick solution.

Step-by-Step Solution:
A2 = P * (1 + r)^2 A3 = P * (1 + r)^3 Compute A3 / A2: A3 / A2 = [P * (1 + r)^3] / [P * (1 + r)^2] = 1 + r Given A3 = 33275 and A2 = 30250 1 + r = 33275 / 30250 33275 / 30250 = 1.10 So 1 + r = 1.10 and r = 0.10 = 10% per annum

Verification / Alternative check:
We can verify this by checking whether A2 grows correctly to A3 at 10%. Applying 10% to 30250 gives 30250 * 0.10 = 3025. Adding this interest to 30250 yields 30250 + 3025 = 33275, which matches the given third year amount. This confirms that the annual rate of 10% is consistent with the data, and therefore correct.

Why Other Options Are Wrong:
A 5% rate would give a factor of 1.05, so A3 would be 30250 * 1.05 which is smaller than 33275. A 15% rate corresponds to factor 1.15, which would produce an amount larger than 33275. A 20% rate would increase the amount even more sharply. Only a 10% rate produces exactly the given progression from 30250 to 33275 in one year.

Common Pitfalls:
A frequent mistake is to try to calculate the principal first and perform unnecessary algebra. Some learners also incorrectly take the difference A3 - A2 and treat it as simple interest, forgetting the effect of compounding. Remembering that the ratio of successive compound amounts directly equals 1 plus the annual rate makes these problems much easier.

Final Answer:
The annual rate of compound interest is 10% per annum.