At compound interest, the amount on a certain principal at the end of the 2nd year is Rs 9,600 and at the end of the 3rd year is Rs 10,272. What is the annual rate of interest (in percent)?

Introduction / Context:
In this problem, we are given the compound interest amounts at the end of two successive years, specifically at the end of the 2nd and 3rd years, and we are asked to determine the annual rate of interest. This is a neat property of compound interest: the ratio of the amount at the end of year (n + 1) to the amount at the end of year n directly reveals the factor (1 + r/100). Therefore, once we know two consecutive year-end amounts, the rate can be found very quickly without even knowing the principal.

Given Data / Assumptions:

• Amount at the end of 2nd year, A2 = Rs 9,600.
• Amount at the end of 3rd year, A3 = Rs 10,272.
• Interest is compounded annually.
• Principal P is unknown, and we only have to find the rate r.

Concept / Approach:
Under compound interest with annual compounding, the amount after n years is A_n = P * (1 + r/100)^n. Therefore, A_(n+1) / A_n = (1 + r/100) for all n. Here, taking A3 / A2 immediately gives us (1 + r/100). We then subtract 1 and convert to percent to obtain r. This method avoids computing P and exploits the multiplicative structure of compound growth across consecutive years.

Step-by-Step Solution:
Given A2 = P * (1 + r/100)^2 and A3 = P * (1 + r/100)^3. Divide A3 by A2: A3 / A2 = (P * (1 + r/100)^3) / (P * (1 + r/100)^2) = 1 + r/100. Compute A3 / A2 = 10,272 / 9,600. 10,272 / 9,600 = 1.07. So, 1 + r/100 = 1.07. Therefore, r/100 = 0.07 and r = 7%. Thus, the annual compound interest rate is 7%.

Verification / Alternative check:
We can confirm by reconstructing the sequence of amounts. Suppose P is the amount at time zero. After 2 years: A2 = P * (1.07)^2 = 9,600. After 3 years: A3 = P * (1.07)^3 = A2 * 1.07 = 9,600 * 1.07 = 10,272, which matches the given value. This perfect consistency shows that 7% is indeed the correct annual rate. The exact principal is not required, but if needed, it could be obtained from P = 9,600 / (1.07)^2.

Why Other Options Are Wrong:
If the rate were 6%, then the ratio A3 / A2 would be 1.06, not 1.07. For 5%, the ratio would be 1.05, and for 8%, it would be 1.08. None matches the observed ratio of 10,272 / 9,600 = 1.07. A rate of 10% would give even larger growth. Hence, only 7% is consistent with the given pair of successive amounts.

Common Pitfalls:
Some students attempt to reintroduce the principal and form two separate equations in P and r, which makes the problem unnecessarily complicated. Others compute the difference in amounts (10,272 - 9,600) and try to interpret it directly as interest on principal, which is incorrect because the base amount for the 3rd year interest is A2, not P. The key is remembering that consecutive amounts under compound interest differ by a constant multiplicative factor equal to (1 + r/100).

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