A certain sum of money at simple interest amounts to ₹ 1012 in $2\frac{1}{2}$ years and to ₹ 1067.20 in 4 years. The rate of interest per annum is

### Concept & Simple Interest The logic remains identical even with fractional years. The difference in the time period directly corresponds to the difference in the accrued amounts. $$Rate = \frac{100 \times S.I.}{P \times T}$$ ### Step-by-Step Solution * Amount in 4 years = ₹ 1067.20 * Amount in 2.5 years = ₹ 1012 * Time difference = 4 - 2.5 = 1.5 years. * S.I. for 1.5 years = ₹ 1067.20 - ₹ 1012 = ₹ 55.20 * S.I. for 1 year = $\frac{55.20}{1.5}$ = ₹ 36.80 * S.I. for 2.5 years = ₹ 36.80 × 2.5 = ₹ 92 * Principal = ₹ 1012 - ₹ 92 = ₹ 920 * Rate = $\frac{100 \times 36.80}{920 \times 1}$ * Rate = $\frac{3680}{920}$ = 4% ### Exam Strategy & Shortcut Avoid decimals by multiplying by 10 or working with fractions. $\frac{55.2}{1.5} = \frac{552}{15} = 36.8$. Find $P$: $1012 - 2.5(36.8) = 920$. Rate $= \frac{36.8}{920} \times 100 = 4\%$. ### Common Pitfall Mishandling the $2\frac{1}{2}$ years. Ensure it is converted to $2.5$ or $\frac{5}{2}$ properly when multiplying to find the interest to subtract. ### Final Answer Therefore, the correct answer is **4%**.