Difficulty: Easy
Correct Answer: Rs. 10,000
Explanation:
Introduction / Context:
This is a reverse compound interest problem where the interest amount over a given period is known, and the original principal must be determined. The interest is compounded annually at a rate of 15% per annum for 2 years, and the total compound interest earned over those 2 years is Rs. 3225.
Given Data / Assumptions:
Concept / Approach:
For annual compounding, the amount after 2 years is:
A = P * (1 + r/100)^2. The compound interest is:
CI = A − P = P * [(1 + r/100)^2 − 1]. Thus:
CI = P * [(1.15)^2 − 1]. We compute (1.15)^2, subtract 1 to get the interest factor, and then use CI = P * factor to solve for P.
Step-by-Step Solution:
Step 1: Compute the 2-year growth factor. (1.15)^2 = 1.15 * 1.15 = 1.3225. Step 2: Find the interest factor. (1.15)^2 − 1 = 1.3225 − 1 = 0.3225. Step 3: Relate CI to P. CI = P * 0.3225. 3225 = 0.3225 * P. Step 4: Solve for P. P = 3225 / 0.3225 = 10,000.
Verification / Alternative check:
Verify by calculating CI from P = 10,000. Amount after 2 years at 15% compounded annually:
A = 10,000 * 1.3225 = 13,225. CI = 13,225 − 10,000 = 3,225 rupees. This matches the given compound interest, confirming that the principal must be Rs. 10,000.
Why Other Options Are Wrong:
If P were 15,000, CI would be 15,000 * 0.3225 = 4,837.50. For 20,000, CI would be 6,450. For 32,250, CI would be even larger. None of these equal 3,225, so those options cannot be correct.
Common Pitfalls:
A common mistake is to use simple interest formulas or to treat 15% as a flat rate per year without squaring the factor for 2 years. Others may try to solve using yearly interest step-by-step without recognizing the compact formula, which can introduce arithmetic errors. Recognizing the clean factor 1.3225 in this problem simplifies the solution significantly.
Final Answer:
The original sum (principal) was Rs. 10,000.
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