A sum of money lent at compound interest for 2 years at 20% per annum would fetch ₹ 482 more, if the interest was payable half-yearly than if it was payable annually. The sum is :

### Concept & Compound Interest Frequency The effective interest earned changes based on compounding frequency. Compounding more frequently yields higher total interest. $$Amount = P\left(1 + \frac{R}{n \times 100}\right)^{n \times T}$$ where $n$ is the number of compounding periods per year. ### Step-by-Step Solution 1. Let the principal sum be $P$. 2. **Case 1: Annually** Amount after 2 years = $P\left(1 + \frac{20}{100}\right)^2 = P(1.2)^2 = 1.44P$ Compound Interest (CIA) = $1.44P - P = 0.44P$ 3. **Case 2: Half-Yearly** Rate per half-year = $\frac{20\%}{2} = 10\%$. Number of periods = $2 \times 2 = 4$. Amount after 2 years = $P\left(1 + \frac{10}{100}\right)^4 = P(1.1)^4 = 1.4641P$ Compound Interest (CIH) = $1.4641P - P = 0.4641P$ 4. **Difference** Given difference = ₹ 482 $0.4641P - 0.44P = 482$ $0.0241P = 482$ $P = \frac{482}{0.0241} = 20,000$ ### Exam Strategy & Shortcut For 20% over 2 years: Annual effective CI is 44%. For 10% over 4 periods: Effective CI is 46.41% (memorizing Pascal's triangle or successive percentage changes helps: 10% twice is 21%, 21% twice is $21 + 21 + 4.41 = 46.41\%$). Difference = 2.41%. $2.41\% \text{ of } P = 482 \implies 1\% \text{ of } P = 200 \implies 100\% = 20,000$. ### Common Pitfall A common mistake is forgetting to adjust both the rate and the time period when switching from annual to half-yearly compounding. ### Final Answer Therefore, the correct answer is **₹ 20,000**.