In what time (in years) will a principal of Rs 8000 amount to Rs 9261 at 5% per annum compound interest, when interest is compounded annually?

Introduction / Context:
This problem checks understanding of the basic compound interest formula and how to find the time period when the principal, rate, and final amount are known. The interest is compounded annually at a rate of 5% per annum, and we are given the initial amount and the final amount. Such questions are very common in bank exams and aptitude tests and can often be solved using either the formula or observation of standard powers of common growth factors like 1.05, 1.10 and so on.

Given Data / Assumptions:

• Principal P = Rs 8000.
• Amount after t years, A = Rs 9261.
• Rate of interest r = 5% per annum.
• Interest is compounded annually.
• We need to find the time t in years.

Concept / Approach:
Under compound interest with annual compounding, the amount after t years is given by the formula A = P * (1 + r/100)^t. Here P, r and A are known and we need to solve for t. In many exam questions, the ratio A / P is chosen to match a known power of (1 + r/100), which allows us to recognize t quickly without using logarithms. Here, we will compute A / P and compare it with (1.05)^t for small integer values of t.

Step-by-Step Solution:
We know A = P * (1 + r/100)^t. So (1 + r/100)^t = A / P. Substitute the given values: P = 8000, A = 9261 and r = 5. Compute the ratio: A / P = 9261 / 8000. We have (1 + 5/100) = 1.05, so we need (1.05)^t = 9261 / 8000. Note that 9261 = 21^3 and 8000 = 20^3, so 9261 / 8000 = (21/20)^3. Also, 1.05 = 21/20, hence (1.05)^3 = (21/20)^3 = 9261 / 8000. Therefore, (1.05)^t = (1.05)^3, which implies t = 3 years.

Verification / Alternative check:
We can check by directly computing the amount after 3 years. Amount A = 8000 * (1.05)^3. First year: 8000 * 1.05 = 8400. Second year: 8400 * 1.05 = 8820. Third year: 8820 * 1.05 = 9261. This matches the given amount exactly, confirming that the required time is 3 years. For other times like 2 or 4 years, the amount would become either too small or too large compared to 9261, which further supports the correctness of our answer.

Why Other Options Are Wrong:
If t = 2 years, the amount would be 8000 * (1.05)^2 = 8000 * 1.1025 = 8820, which is less than 9261. If t = 3.5 or 4 years, the amount would exceed 9261 because we are compounding at a positive rate of 5%. Similarly, 5 years would produce an even higher amount. Therefore, none of the other time periods match the given final amount for the specified rate and principal.

Common Pitfalls:
Students sometimes incorrectly apply the simple interest formula instead of the compound interest formula, leading to linear rather than exponential growth. Another frequent mistake is rounding intermediate values too early, which can disturb the recognition of exact ratios like 9261/8000. It is often helpful to look for perfect cubes or squares in such problems, as exam setters intentionally choose convenient numbers to allow shortcut recognition without calculators.

Final Answer:
The principal of Rs 8000 will amount to Rs 9261 in 3 years at 5% per annum compounded annually.