If the simple interest on a certain sum for 15 months at $7\frac{1}{2}\%$ per annum exceeds the simple interest on the same sum for 8 months at $12\frac{1}{2}\%$ per annum by ₹ 32.50, then the sum (in ₹) is :

### Concept & Difference of Two Simple Interests When comparing two different investment scenarios on the same principal, we calculate the simple interest for each and find their difference. Remember to always convert months into years when the rate is per annum. $$SI = \frac{P \times R \times T}{100}$$ ### Step-by-Step Solution * **Given:** * Scenario 1: $T_1 = \frac{15}{12}$ years, $R_1 = 7.5\% = \frac{15}{2}\%$. * Scenario 2: $T_2 = \frac{8}{12}$ years, $R_2 = 12.5\% = \frac{25}{2}\%$. * Difference in Interest = ₹ 32.50. * **Calculation:** Set up the equation for $SI_1 - SI_2 = 32.50$. * $\left(\frac{P \times \frac{15}{2} \times \frac{15}{12}}{100}\right) - \left(\frac{P \times \frac{25}{2} \times \frac{8}{12}}{100}\right) = 32.50$ * Simplify the terms inside the parentheses: * $\left(\frac{P \times 225}{2400}\right) - \left(\frac{P \times 200}{2400}\right) = 32.50$ * Combine the fractions since they have a common denominator: * $\frac{225P - 200P}{2400} = 32.50$ * $\frac{25P}{2400} = 32.50$ * Solve for $P$: * $25P = 32.50 \times 2400$ * $P = \frac{78000}{25} = 3120$. ### Exam Strategy & Shortcut Instead of large denominators, you can simplify the "Rate $\times$ Time" product mentally first. Product 1 = $\frac{15}{12} \times 7.5 = 1.25 \times 7.5 = 9.375\%$ Product 2 = $\frac{8}{12} \times 12.5 = \frac{2}{3} \times 12.5 = 8.333\%$ Difference = $9.375\% - 8.333\% = 1.0416\% \approx \frac{25}{24}\%$ So, $\frac{25}{24}\%$ of $P = 32.50 \implies P = \frac{32.50 \times 24 \times 100}{25} = 3120$. ### Common Pitfall Forgetting to convert the time periods from months to years. Using 15 and 8 directly will yield massively incorrect principal values because the rates are stated "per annum" (yearly). ### Final Answer Therefore, the correct answer is **₹ 3120**.