In how many years, ₹ 150 will produce the same interest @ 8% as ₹ 800 produce in 3 years @ $4 \frac{1}{2}$% ?

### Concept & Equating Two Interest Scenarios When the problem states that the interest produced in two different scenarios is the same, set up an equation where $SI_1 = SI_2$. $$\frac{P_1 \times R_1 \times T_1}{100} = \frac{P_2 \times R_2 \times T_2}{100}$$ ### Step-by-Step Solution * **Calculate Target Interest ($SI_2$):** $P_2 = \text{₹ } 800, T_2 = 3 \text{ years}, R_2 = 4 \frac{1}{2}\% = 4.5\%$ $SI_2 = \frac{800 \times 4.5 \times 3}{100} = 8 \times 4.5 \times 3 = 8 \times 13.5 = \text{₹ } 108$ * **Set up Scenario 1 to match:** $P_1 = \text{₹ } 150, R_1 = 8\%, T_1 = \text{unknown}$ $SI_1 = \frac{150 \times 8 \times T_1}{100} = \frac{1200 \times T_1}{100} = 12 \times T_1$ * **Equate and Solve:** $12 \times T_1 = 108$ $T_1 = \frac{108}{12} = 9 \text{ years}$ ### Exam Strategy & Shortcut Equate them without the $100$ denominators: $150 \times 8 \times T = 800 \times 3 \times 4.5$. Simplify: $1200 \times T = 800 \times 13.5$. Divide both sides by $400$: $3 \times T = 2 \times 13.5$. $3 \times T = 27 \implies T = 9$. ### Common Pitfall Miscalculating $4 \frac{1}{2}$ as a decimal (e.g., writing it incorrectly) or struggling to multiply $8 \times 13.5$. Keeping terms as fractions ($9/2$) often prevents arithmetic mistakes: $800 \times 3 \times \frac{9}{2} = 400 \times 27 = 10800$. ### Final Answer Therefore, the correct answer is **9**.