Difficulty: Medium
Correct Answer: Rs. 8,800
Explanation:
Introduction:
In this question we are asked to compute the compound interest earned in 1 year when a high annual rate of 40% is applied, but compounding occurs half yearly. This problem highlights the impact of compounding frequency on the total interest earned.
Given Data / Assumptions:
Principal P = Rs. 20,000. Nominal annual rate r = 40% per annum. Compounding frequency = half yearly. Total time = 1 year.
Concept / Approach:
For half yearly compounding: Rate per half year = r / 2 = 40 / 2 = 20%. Number of half years in 1 year = 2. We then use the compound interest formula: A = P * (1 + rate per half year / 100)^2. Compound interest CI is then A − P.
Step-by-Step Solution:
Step 1: Compute half yearly rate and periods. Rate per half year = 20%. Number of periods n = 2. Step 2: Compute the amount. A = 20000 * (1.20)^2. (1.20)^2 = 1.44. A = 20000 * 1.44 = Rs. 28,800. Step 3: Compute compound interest. CI = A − P = 28,800 − 20,000 = Rs. 8,800.
Verification / Alternative check:
We can reason that if interest were not compounded half yearly, simple interest at 40% for 1 year would be 0.40 * 20,000 = Rs. 8,000. Because of half yearly compounding, the interest is slightly greater than Rs. 8,000, and Rs. 8,800 aligns with the exact calculation using the 1.44 multiplier.
Why Other Options Are Wrong:
Rs. 8,000 corresponds to simple interest or annual compounding, not half yearly compounding. Rs. 8,650 and Rs. 8,750 are intermediate guesses that do not match the exact squared multiplier 1.44. Rs. 7,200 is far too small and inconsistent with a 40% nominal rate.
Common Pitfalls:
A common mistake is to forget that interest is compounded half yearly and to use 40% once. Others divide the rate incorrectly or square 1.4 instead of 1.2. Always remember that half yearly compounding at 40% annual rate means two periods with 20% each, leading to the factor (1.20)^2.
Final Answer:
The compound interest for 1 year is Rs. 8,800.
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