Simple interest on ₹ 500 for 4 years at 6.25% per annum is equal to the simple interest on ₹ 400 at 5% per annum for a certain period of time. The period of time is

### Concept & Equating Simple Interests When two simple interest scenarios are equal, their respective $\frac{P \times R \times T}{100}$ calculations can be set equal to one another to solve for the missing variable. $$SI_1 = SI_2 \implies \frac{P_1 \times R_1 \times T_1}{100} = \frac{P_2 \times R_2 \times T_2}{100}$$ ### Step-by-Step Solution * **Scenario 1:** $P_1 = \text{₹ } 500, T_1 = 4 \text{ years}, R_1 = 6.25\%$ $SI_1 = \frac{500 \times 6.25 \times 4}{100} = 5 \times 25 = \text{₹ } 125$ * **Scenario 2:** $P_2 = \text{₹ } 400, R_2 = 5\%, T_2 = \text{unknown}$ $SI_2 = \frac{400 \times 5 \times T_2}{100} = 20 \times T_2$ * **Equate them:** $20 \times T_2 = 125$ $T_2 = \frac{125}{20} = \frac{25}{4} = 6.25 \text{ years}$ * Convert to fraction: $6.25 = 6 \frac{1}{4} \text{ years}$ ### Exam Strategy & Shortcut Equate directly and cancel terms without full multiplication: $500 \times 4 \times 6.25 = 400 \times 5 \times T_2$ $2000 \times 6.25 = 2000 \times T_2$ Divide both sides by $2000$: $T_2 = 6.25$. No complex arithmetic is required! ### Common Pitfall A student might perform unnecessary, long multiplication on the left side before realizing the right side also contains factors ($400 \times 5 = 2000$) that easily cancel out the left side ($500 \times 4 = 2000$). ### Final Answer Therefore, the correct answer is **$6 \frac{1}{4}$ years**.