What will be the ratio of simple interest earned by certain amount at the same rate of interest for 6 years and that for 9 years ?

### Concept & Formula Simple interest is directly proportional to the time period when the principal and rate of interest remain constant. $$SI = \frac{Principal \times Rate \times Time}{100}$$ $$\frac{SI_1}{SI_2} = \frac{Time_1}{Time_2}$$ ### Step-by-Step Solution 1. Let the principal be $P$ and the rate of interest be $R$. 2. Simple Interest for 6 years ($SI_1$) = $\frac{P \times R \times 6}{100}$ 3. Simple Interest for 9 years ($SI_2$) = $\frac{P \times R \times 9}{100}$ 4. Required Ratio = $\frac{SI_1}{SI_2} = \frac{\frac{P \times R \times 6}{100}}{\frac{P \times R \times 9}{100}}$ 5. Simplify the fraction: $\frac{6}{9} = \frac{2}{3}$ 6. The ratio is $2 : 3$. Since this is not present in options (a), (b), or (c), the answer is "None of these". ### Exam Strategy & Shortcut Since principal and rate are constant, skip writing the full formula. The ratio of the simple interests will simply be the ratio of the time periods: $6:9$, which simplifies to $2:3$. ### Common Pitfall A common mistake is selecting "Data inadequate" because the principal and rate are not given. However, since we are only asked for a ratio, the unknown constants cancel out. ### Final Answer Therefore, the correct answer is **None of these**.