There were 35 students in a hostel. If the number of the students is increased by 7, then the expenses of the mess increase by ₹ 42 per day, while the average expenditure per head diminishes by ₹ 1. The original expenditure of the mess per day was

Concept & Logic When a group expands, total expenditure changes by a fixed addition, while the new average becomes the old average minus the reduction amount. Use linear equations tracking the original average expenditure per head. Step-by-Step Solution * Let the original average expenditure per student per day be ₹ $x$. * Original daily total expenditure for 35 students = $35x$. * New number of students = $35 + 7 = 42$. * New average expenditure per head = ₹ $(x - 1)$. * New daily total expenditure for 42 students = $42(x - 1)$. * Given that the total daily expenditure increased by ₹ 42: $$42(x - 1) - 35x = 42$$ * Solve the linear equation: $$42x - 42 - 35x = 42$$ $$7x = 84$$ $$x = 12$$ * Find the original daily expenditure of the mess: $$\text{Original Expenditure} = 35x = 35 \times 12 = \text{₹ } 420$$ Exam Strategy & Shortcut **Option Elimination Method:** The original daily expenditure must be a multiple of 35 because $\text{Total} = 35 \times x$. * 400 is not divisible by 35. * 420 divided by 35 gives exactly 12 (an integer value for initial head cost). Let's check: New total = $420 + 42 = 462$. New per head cost = $462 / 42 = 11$, which is exactly 1 less than 12. Option (b) matches perfectly! Common Pitfall Be cautious when structuring the difference equation. Ensure you subtract the original cost from the new cost ($42(x-1) - 35x = 42$) rather than accidentally swapping them, which results in negative numbers. Final Answer Therefore, the correct answer is ₹ 420.