Difficulty: Medium
Correct Answer: 21 percent
Explanation:
Introduction:
This question asks us to determine the unknown compound interest rate when we know the initial principal and the final amount after a specified time. It is a classic reverse compound interest problem that checks algebraic manipulation of the amount formula A = P * (1 + r/100)^n.
Given Data / Assumptions:
Principal P = Rs. 50,000. Amount after 2 years A = Rs. 73,205. Time t = 2 years. Interest is compounded annually at rate r% per annum.
Concept / Approach:
The amount formula for compound interest is: A = P * (1 + r/100)^n. We rearrange this to solve for r. For n = 2: (1 + r/100)^2 = A / P. Taking the square root will give 1 + r/100, from which we find r. Recognising familiar decimal squares can speed up the calculation.
Step-by-Step Solution:
Step 1: Form the ratio A / P. A / P = 73205 / 50000 = 1.4641. So (1 + r/100)^2 = 1.4641. Step 2: Take square root. We know (1.21)^2 = 1.4641 because 121^2 = 14641. Therefore 1 + r/100 = 1.21. Step 3: Solve for r. r/100 = 0.21, so r = 21 percent.
Verification / Alternative check:
Verify by recomputing the amount using r = 21 percent: A = 50000 * (1.21)^2. (1.21)^2 = 1.4641. A = 50000 * 1.4641 = Rs. 73,205. This matches the given amount exactly, confirming that the rate is correct.
Why Other Options Are Wrong:
At 19 percent, 17 percent, 15 percent or 23 percent per annum, the squared multipliers would be different from 1.4641 and therefore would not produce an amount of Rs. 73,205 from Rs. 50,000 in 2 years. These options correspond to incorrect guesses or miscalculated square roots.
Common Pitfalls:
Some learners place A / P directly equal to (1 + r/100) rather than its square, forgetting that the time is 2 years. Others take an approximate square root and do not verify it by squaring again. Always confirm that the square of your candidate multiplier returns the correct ratio A / P.
Final Answer:
The required annual compound interest rate is 21 percent per annum.
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