Difficulty: Easy
Correct Answer: 11 percent
Explanation:
Introduction / Context:
This question asks you to determine the annual compound interest rate when the principal, final amount and time are given. Such problems are very common in banking, investment and aptitude tests, and they test your understanding of the reverse use of the compound interest formula.
Given Data / Assumptions:
Concept / Approach:
For annual compounding, the basic relation is:
A = P * (1 + r/100)^t. Rearranging, we have:
A / P = (1 + r/100)^t. Since we know A, P and t, we first compute the ratio A / P, then take the square root (because t = 2) to find (1 + r/100), and hence r.
Step-by-Step Solution:
Step 1: Compute the ratio A / P. A / P = 12,321 / 10,000 = 1.2321. Step 2: Relate this to the rate. (1 + r/100)^2 = 1.2321. Step 3: Take the square root. 1 + r/100 = √1.2321 = 1.11 (since 1.11 * 1.11 = 1.2321). Step 4: Solve for r. r/100 = 1.11 − 1 = 0.11, so r = 11%.
Verification / Alternative check:
Check the result by recomputing the amount at 11%. After 2 years at 11% compound interest:
A = 10,000 * (1.11)^2 = 10,000 * 1.2321 = 12,321. This matches the given final amount exactly, confirming that the rate must be 11% per annum compounded annually.
Why Other Options Are Wrong:
A rate of 7% or 15% would lead to much smaller or larger final amounts than 12,321, while 22% would grow the principal far too quickly. Only 11% results in the exact given amount, so all other options are mathematically inconsistent with the data.
Common Pitfalls:
Students sometimes forget that the exponent t must be handled properly and attempt to treat the growth as simple interest. Another mistake is to not recognize convenient squares like 1.2321, which is 1.11 squared, and instead approximate poorly. When the numbers look neat, always check whether they match a known power such as (1.1)^2 or (1.11)^2.
Final Answer:
The annual compound interest rate is 11 percent.
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