₹ 2000 amounts to ₹ 2226.05 in 2 years at compound interest. What will be the rate of interest? (E.S.I.C., 2006)

### Concept & Formula The fundamental formula for calculating the amount under compound interest is: $$A = P \left(1 + \frac{R}{100}\right)^n$$ To find the rate, isolate the $(1 + R/100)$ term and take the $n$-th root of the $A/P$ ratio. ### Step-by-Step Solution * Given Principal $P = 2000$ * Amount $A = 2226.05$ * Time $n = 2$ years * Plugging these into the formula: $2226.05 = 2000 \times \left(1 + \frac{R}{100}\right)^2$ $\left(1 + \frac{R}{100}\right)^2 = \frac{2226.05}{2000}$ * Multiply numerator and denominator by $100$ to remove the decimal: $\left(1 + \frac{R}{100}\right)^2 = \frac{222605}{200000}$ * Simplify the fraction by dividing by $5$: $\left(1 + \frac{R}{100}\right)^2 = \frac{44521}{40000}$ * The denominator $40000$ is a perfect square: $200^2$. We need to find the square root of $44521$. Since it ends in $1$, the root ends in $1$ or $9$. It is larger than $200^2$ ($40000$) and smaller than $210^2$ ($44100$). Wait, $210^2 = 44100$. So it must be $211^2$. Let us check $211^2$: $211 \times 211 = 44521$. * So, $\left(1 + \frac{R}{100}\right)^2 = \left(\frac{211}{200}\right)^2$ $1 + \frac{R}{100} = \frac{211}{200} = 1.055$ $\frac{R}{100} = 0.055 \implies R = 5.5\%$ ### Exam Strategy & Shortcut Once you have the ratio $\frac{A}{P} = \frac{2226.05}{2000} = 1.113025$, you are looking for $\sqrt{1.113025}$. Check the options: (a) $5\% \implies 1.05^2 = 1.1025$ (d) $6\% \implies 1.06^2 = 1.1236$ Our value $1.113025$ is squarely between them. Let us test $5.5\% \implies 1.055^2$. Using the rule for squaring numbers ending in $5$: $105 \times 106 = 11130$, append $25 \implies 1113025$. Perfect match. ### Common Pitfall Not eliminating the decimal points before trying to simplify the fraction $\frac{2226.05}{2000}$. Multiplying by 100 first makes the perfect squares much easier to identify. ### Final Answer Therefore, the correct answer is **5.5%**.