What is the difference between the compound interests on ₹ 5000 for 1 year at 4% per annum compounded yearly and half yearly? [SSC—Junior Associates (Pre.) Exam, 2016]

### Concept & Effect of Compounding Frequency When interest is compounded half-yearly for one year, the principal effectively earns "interest on the interest" from the first half-year during the second half-year. This "interest on interest" is exactly the difference between annual and half-yearly compounding for a single year. $$Difference = P \left(\frac{R_{half}}{100}\right)^2$$ ### Step-by-Step Solution * **Given:** Principal $P = 5000$, Annual Rate $R = 4\%$, Time = 1 year. * **Yearly Compounding:** Rate = 4%, Time = 1 year. Interest = $\frac{5000 \times 4 \times 1}{100} = 200$. * **Half-Yearly Compounding:** Rate per half-year = $\frac{4\%}{2} = 2\%$. Number of periods = $1 \times 2 = 2$. Interest for 1st half-year = $\frac{5000 \times 2}{100} = 100$. Interest for 2nd half-year = Interest on principal + Interest on previous interest $2nd\ half\ interest = 100 + \left(\frac{100 \times 2}{100}\right) = 100 + 2 = 102$. Total Half-Yearly CI = $100 + 102 = 202$. * **Difference:** $Difference = 202 - 200 = 2$. ### Exam Strategy & Shortcut For exactly one year, the difference between yearly and half-yearly compounding is simply the interest calculated on the first half-year's interest. First half-year interest = $2\%$ of $5000 = 100$. Interest on that 100 at $2\%$ = $2\%$ of $100 = 2$. The answer is just 2. ### Common Pitfall A time-consuming mistake is manually applying the full compound interest formula $A = P(1+R/100)^N$ for both scenarios, which introduces unnecessary arithmetic steps and risk of calculation errors. ### Final Answer Therefore, the correct answer is **2**.