The compound interest on ₹ 30,000 at 7% per annum is ₹ 4347. The period (in years) is (L.I.C.A.A.O., 2003)

### Concept & Formula The relationship between Amount ($A$), Principal ($P$), Rate ($R$), and Time ($n$) in compound interest is given by: $$A = P \left(1 + \frac{R}{100}\right)^n$$ The Amount is also equal to Principal + Compound Interest. ### Step-by-Step Solution * Given Principal $P = 30000$ * Compound Interest $C.I. = 4347$ * Rate $R = 7\%$ p.a. * Amount $A = P + C.I. = 30000 + 4347 = 34347$ * Using the formula: $34347 = 30000 \times \left(1 + \frac{7}{100}\right)^n$ $\frac{34347}{30000} = \left(\frac{107}{100}\right)^n$ $\frac{11449}{10000} = \left(\frac{107}{100}\right)^n$ *(Dividing numerator and denominator by 3)* * Notice that $10000 = 100^2$. Let us check if $11449 = 107^2$. $107^2 = (100 + 7)^2 = 10000 + 1400 + 49 = 11449$. * Therefore, $\left(\frac{107}{100}\right)^2 = \left(\frac{107}{100}\right)^n$ * Comparing the powers, we get $n = 2$. ### Exam Strategy & Shortcut Look at the denominator after isolating the ratio $A/P$. The fraction is $\frac{34347}{30000}$. Dividing by $3$ yields $\frac{11449}{10000}$. The denominator $10000$ is perfectly $100^2$. This strongly hints that the power $n$ is $2$, as the base rate fraction is $\frac{107}{100}$. ### Common Pitfall Getting stuck trying to solve for $n$ using logarithms when simple fraction simplification and recognizing perfect squares is much faster. ### Final Answer Therefore, the correct answer is **2 years**.