What is the difference between the compound interests on ₹ 5000 for $1 \frac{1}{2}$ years at 4% per annum compounded yearly and half-yearly?

### Concept & Compounding Frequency When interest is compounded more frequently (e.g., half-yearly instead of yearly), the total interest earned is higher because you earn interest on the previously accrued interest sooner. $$ A = P(1 + \frac{R}{100})^n $$ ### Step-by-Step Solution * **Given:** Principal ($P$) = ₹ 5000, Rate = 4% p.a., Time = 1.5 years. * **Step 1:** Calculate Compound Interest (yearly compounding). For $1 \frac{1}{2}$ years, the amount is calculated for 1 full year at 4%, and the remaining half year at 2%. $A_{yearly} = 5000 \times (1 + \frac{4}{100}) \times (1 + \frac{2}{100})$ $A_{yearly} = 5000 \times 1.04 \times 1.02 = 5000 \times 1.0608 = 5304$ $CI_{yearly} = 5304 - 5000 = 304$ * **Step 2:** Calculate Compound Interest (half-yearly compounding). Rate becomes 2% per half-year, and time becomes $1.5 \times 2 = 3$ cycles. $A_{half} = 5000 \times (1 + \frac{2}{100})^3$ $A_{half} = 5000 \times (1.02)^3 = 5000 \times 1.061208 = 5306.04$ $CI_{half} = 5306.04 - 5000 = 306.04$ * **Step 3:** Find the difference. Difference = $306.04 - 304 = 2.04$ ### Exam Strategy & Shortcut We can focus purely on effective percentage rates. Yearly effective rate for 1.5 years = $4\% + 2\% + \frac{4 \times 2}{100} = 6.08\%$. Half-yearly effective rate for 3 cycles of 2% = $3(2) + 3(2^2)/100 + (2^3)/10000 = 6.1208\%$. Difference in rates = $6.1208\% - 6.08\% = 0.0408\%$. Difference in Amount = $0.0408\%$ of $5000 = 2.04$. ### Common Pitfall Calculating simple interest for the fractional part incorrectly or using the full 4% rate for a half-year period. ### Final Answer Therefore, the correct answer is **₹ 2.04**.