Simple interest on a certain amount is $\frac{9}{16}$ of the principal. If the numbers representing the rate of interest in percent and time in years be equal, then time, for which the principal is lent out, is

### Concept & Equal Rate and Time When the numerical value of Time ($T$) equals the numerical value of Rate ($R$), we can substitute $R$ with $T$ in the standard Simple Interest formula: $$SI = \frac{P \times T \times T}{100} = \frac{P \times T^2}{100}$$ ### Step-by-Step Solution * **Given:** Simple Interest ($SI$) = $\frac{9}{16}$ of Principal ($P$), and $R = T$. * **Calculation:** Substitute the known values into the modified formula: * $\frac{9}{16}P = \frac{P \times T^2}{100}$ * Divide both sides by $P$: * $\frac{9}{16} = \frac{T^2}{100}$ * Multiply by 100 to solve for $T^2$: * $T^2 = \frac{900}{16}$ * Take the square root of both sides: * $T = \sqrt{\frac{900}{16}} = \frac{30}{4} = \frac{15}{2}$ * Convert the improper fraction to a mixed fraction: * $T = 7\frac{1}{2}$ years. ### Exam Strategy & Shortcut Whenever $SI = \frac{a}{b} \times P$ and $R = T$, the formula simplifies to: $$T = \sqrt{\frac{a}{b} \times 100}$$ Here, $T = \sqrt{\frac{9}{16} \times 100} = \frac{30}{4} = 7.5 = 7\frac{1}{2}$ years. ### Common Pitfall Forgetting to take the square root of the denominator as well as the numerator, or calculating for Rate ($R$) but forgetting that it has the same numerical value as Time ($T$) and stopping too early. ### Final Answer Therefore, the correct answer is **7\frac{1}{2} years**.