Some students planned a picnic. The budget for food was ₹ 500. But, 5 of them failed to go and thus the cost of food for each member increased by ₹ 5. How many students attended the picnic?

### Concept & Logic This problem uses the classic inverse relationship between the number of contributors and the cost per head when the total budget is fixed. $$\text{Cost per Head} = \frac{\text{Total Budget}}{\text{Number of People}}$$ ### Step-by-Step Solution * **Given:** Total food budget = ₹ 500. Dropouts = 5 students. Cost increase per head = ₹ 5. * **Calculation:** Let the originally planned number of students be $x$. * Original cost per head = $\frac{500}{x}$ * Number of students who actually attended = $x - 5$ * New cost per head = $\frac{500}{x - 5}$ * The difference between the new cost and the original cost is ₹ 5: $$\frac{500}{x - 5} - \frac{500}{x} = 5$$ * Divide by 5 to simplify: $$\frac{100}{x - 5} - \frac{100}{x} = 1$$ * Multiply through by $x(x - 5)$: $$100x - 100(x - 5) = x^2 - 5x$$ $$100x - 100x + 500 = x^2 - 5x$$ $$x^2 - 5x - 500 = 0$$ * Factorize (factors of -500 adding to -5 are -25 and +20): $$(x - 25)(x + 20) = 0$$ * Since the number of students must be positive, $x = 25$ (original plan). * The question asks for the number of students who **attended**: $$25 - 5 = 20$$ ### Exam Strategy & Shortcut **Option Testing:** Note that the question asks for the number who *attended*. If attended = 20 (Option b), then planned = 25. Check costs: $500 / 25 = 20$ per head originally. $500 / 20 = 25$ per head now. Difference = 5. Matches instantly! ### Common Pitfall The most common trap is solving the quadratic equation to get $x = 25$, and immediately marking 25 (Option c) without reading the final sentence. Always double-check what the question is asking—in this case, those who *attended* ($x - 5$). ### Final Answer **Therefore, the correct answer is 20.**