If Rs. 10000 amounts to Rs. 11664 in 2 years at compound interest, compounded annually, what is the annual rate of compound interest?

Introduction / Context:
This question requires finding the compound interest rate when the principal, final amount, and time are known. It is an inverse compound interest problem where the rate is the unknown. Such problems test the learner's familiarity with growth factors and their ability to recognise common squares or use simple calculations to isolate the rate per annum.

Given Data / Assumptions:

• Principal P = Rs. 10000.
• Amount after 2 years A = Rs. 11664.
• Time period n = 2 years.
• Interest is compounded annually.
• We need to find the annual rate of compound interest r in percent.

Concept / Approach:
The compound amount formula for annual compounding is A = P * (1 + r) ^ n, where r is in decimal form. Here n = 2, so A = P * (1 + r) ^ 2. We can solve for the growth factor (1 + r) by dividing A by P and taking the square root, because n = 2. That is, (1 + r) ^ 2 = A / P, so 1 + r = square root of (A / P). Then r = (square root of (A / P)) - 1. Finally, we convert r into a percentage by multiplying by 100.

Step-by-Step Solution:
Step 1: Compute the ratio A / P = 11664 / 10000 = 1.1664. Step 2: Since A = P * (1 + r) ^ 2, we have (1 + r) ^ 2 = 1.1664. Step 3: Take the square root of 1.1664 to find 1 + r. Step 4: The square root of 1.1664 is 1.08. Step 5: Therefore, 1 + r = 1.08, which implies r = 1.08 - 1 = 0.08. Step 6: Convert r to a percentage: r = 0.08 * 100% = 8%. Step 7: Hence, the annual rate of compound interest is 8%.

Verification / Alternative check:
We can check this rate by forward calculation. At 8% per annum on 10000, the amount after one year is 10000 * 1.08 = 10800. After two years, the amount becomes 10800 * 1.08 = 11664. This matches the given amount exactly, confirming that 8% is the correct rate. Since the arithmetic uses a neat square number, the result is clean and typical for exam style questions that avoid messy decimals for the rate.

Why Other Options Are Wrong:
If the rate were 10%, the amount after two years would be 10000 * (1.10) ^ 2 = 12100, which is greater than 11664. For 9%, the amount would be 10000 * (1.09) ^ 2 = 11881, also too large. A rate of 6% would give 10000 * (1.06) ^ 2 = 11236, which is less than 11664. A rate of 7% would give 10000 * (1.07) ^ 2 = 11449, still less than the target amount. Only 8% produces the exact figure of 11664.

Common Pitfalls:
Learners sometimes mistakenly apply the simple interest formula and conclude that the effective two year rate is 16.64%, then divide by 2 to get about 8.32%, which is not correct for compound interest. Others may incorrectly take half of 1.1664 instead of its square root, confusing the relationship between squares and doubles. Accurate understanding of the exponent in the compound amount formula and careful handling of square roots are essential in this kind of problem.

Final Answer:
The annual rate of compound interest that turns Rs. 10000 into Rs. 11664 in 2 years is 8%, which corresponds to option C.