Difficulty: Medium
Correct Answer: Rs 4096
Explanation:
Introduction:
This question asks you to reverse the compound interest process. Instead of finding the amount from a known principal, you are given the final amount, rate and time and must determine the original principal. The rate is a mixed fraction, 6 1/4%, which is convenient for expressing as a rational factor.
Given Data / Assumptions:
Concept / Approach:
For annual compounding, A = P * (1 + r/100)^t. We know A, r and t. The term (1 + r/100) becomes (1 + 6.25/100) = 1.0625. Notably, 1.0625 can be written as 17/16. Because A and P often come out as perfect cubes in such problems, recognizing (17/16)^3 is helpful. We solve P = A / (1.0625)^3.
Step-by-Step Solution:
Convert the rate: r = 6.25% = 6 1/4%.1 + r/100 = 1 + 6.25/100 = 1 + 0.0625 = 1.0625.Note that 1.0625 = 17/16.So the amount relation is A = P * (17/16)^3.Given A = 4913, so 4913 = P * (17/16)^3.Therefore P = 4913 / (17/16)^3 = 4913 * (16/17)^3.Notice that 4913 = 17^3, so:P = 17^3 * (16^3 / 17^3) = 16^3 = 4096.
Verification / Alternative Check:
Verify by forward calculation. Starting with P = 4096 at 6.25% per annum: factor per year = 1.0625 = 17/16. After 3 years, amount = 4096 * (17/16)^3. Since 4096 = 16^3, we get 16^3 * 17^3 / 16^3 = 17^3 = 4913, which matches the given amount.
Why Other Options Are Wrong:
Rs 3096, Rs 4076 and Rs 4085: None of these values, when grown at 6.25% compound interest for 3 years, will produce exactly Rs 4913.Rs 3500: Also does not match the given amount when the compound interest formula is applied.
Common Pitfalls:
Students often mishandle the mixed fraction rate or neglect to simplify 6 1/4% to 6.25%. Others attempt to compute (1.0625)^3 in decimal form and round too aggressively, leading to small discrepancies. Recognizing the rational form 17/16 and the cube structure helps achieve an exact and elegant solution.
Final Answer:
The required principal that amounts to Rs 4913 in 3 years at 6 1/4% compound interest is Rs 4096.
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