In how many years will a sum of ₹ 800 at 10% per annum compounded semi-annually become ₹ 926.10? (S.S.C., 2010)

### Concept & Conversion For semi-annual (half-yearly) compounding, the annual interest rate is divided by 2, and the compounding periods are in half-years. $$A = P \left(1 + \frac{R/2}{100}\right)^n$$ where $n$ is the number of half-years. ### Step-by-Step Solution * Given Principal $P = 800$ * Amount $A = 926.10$ * Annual Rate $= 10\% \implies$ Semi-annual Rate $r = 5\%$ * Substitute into the formula: $926.10 = 800 \left(1 + \frac{5}{100}\right)^n$ $\frac{926.10}{800} = \left(1 + \frac{1}{20}\right)^n$ $\frac{926.10}{800} = \left(\frac{21}{20}\right)^n$ * Remove the decimal by multiplying numerator and denominator by 10: $\frac{9261}{8000} = \left(\frac{21}{20}\right)^n$ * Observe the denominator: $8000 = 20^3$. Let us check if $9261$ is $21^3$. $21^2 = 441$. $441 \times 21 = 9261$. Yes. * Therefore, $\left(\frac{21}{20}\right)^3 = \left(\frac{21}{20}\right)^n$ $n = 3$ half-years. * Convert half-years to years: $3 \text{ half-years} = \frac{3}{2} \text{ years} = 1\frac{1}{2} \text{ years}$. ### Exam Strategy & Shortcut Always look for perfect squares or cubes when you have the fraction $\frac{A}{P}$. Once you see $8000$, recognize immediately that it is $20^3$. This instantly tells you $n = 3$ periods without calculating $21^3$. ### Common Pitfall A classic mistake is finding $n = 3$ and looking for "3 years" in the options, completely forgetting that $n$ represents the number of semi-annual periods, not years. ### Final Answer Therefore, the correct answer is **$1\frac{1}{2}$ years**.