A sum of ₹ 1600 gives a simple interest of ₹ 252 in 2 years and 4 months. The rate of interest per annum is

### Concept & Conversion of Time Units In simple interest problems, the time ($T$) must always be expressed in years when calculating the per annum rate. Months must be converted by dividing by 12. $$R = \frac{SI \times 100}{P \times T}$$ ### Step-by-Step Solution * Given Principal, $P = \text{₹ } 1600$ * Given Simple Interest, $SI = \text{₹ } 252$ * Convert Time to years: $T = 2 \text{ years and } 4 \text{ months} = 2 + \frac{4}{12} = 2 + \frac{1}{3} = \frac{7}{3} \text{ years}$. * Apply the Rate formula: $R = \frac{252 \times 100}{1600 \times (\frac{7}{3})}$ $R = \frac{252 \times 100 \times 3}{1600 \times 7}$ $R = \frac{252 \times 3}{16 \times 7} = \frac{36 \times 3}{16}$ $R = \frac{108}{16} = \frac{27}{4} = 6.75\%$ or $6 \frac{3}{4}\%$ ### Exam Strategy & Shortcut Keep fractions intact rather than using decimals (like $2.333...$ years) to avoid rounding errors and make cancellation with $100$ and $P$ much easier. ### Common Pitfall The most common mistake is entering time as $2.4$ years instead of $2 \frac{1}{3}$ years. 4 months is $\frac{1}{3}$ of a year, not $0.4$ of a year. ### Final Answer Therefore, the correct answer is **$6 \frac{3}{4}$%**.