A sum of money becomes \frac{7}{6} of itself in 3 years at a certain rate of simple interest. The rate per annum is

### Concept & Amount in Fractions When a sum "becomes" a fraction of itself, that fraction represents the Amount ($A$). Simple Interest ($SI$) is the difference between Amount and Principal: $$SI = A - P$$ ### Step-by-Step Solution * **Given:** Amount ($A$) = $\frac{7}{6}P$, Time ($T$) = 3 years. * **Calculation:** First, calculate the Simple Interest ($SI$): * $SI = \frac{7}{6}P - P = \left(\frac{7}{6} - 1\right)P = \frac{1}{6}P$ * Now, plug $SI$ into the simple interest formula: * $SI = \frac{P \times R \times T}{100}$ * $\frac{1}{6}P = \frac{P \times R \times 3}{100}$ * Cancel $P$ from both sides: * $\frac{1}{6} = \frac{3R}{100}$ * Solve for $R$: * $100 = 18R$ * $R = \frac{100}{18} = \frac{50}{9}$ * Convert to a mixed fraction: * $\frac{50}{9} = 5\frac{5}{9}\%$ ### Exam Strategy & Shortcut Let Principal ($P$) be 6 units. Then Amount ($A$) is 7 units. Interest earned = $7 - 6 = 1$ unit. This 1 unit of interest is earned in 3 years. Interest per year = $\frac{1}{3}$. Rate % = $\left(\frac{\text{Interest per year}}{P}\right) \times 100 = \left(\frac{\frac{1}{3}}{6}\right) \times 100 = \frac{1}{18} \times 100 = \frac{50}{9} = 5\frac{5}{9}\%$. ### Common Pitfall A common mistake is using $\frac{7}{6}P$ as the Simple Interest rather than the Amount, which results in heavily inflated and incorrect rate calculations. ### Final Answer Therefore, the correct answer is **5\frac{5}{9}\%**.