A person purchases 1 kg of tomatoes from each of the 4 places at the rate of 1 kg, 2 kg, 3 kg, 4 kg per rupee respectively. On an average, he has purchased $x$ kg of tomatoes per rupee. Then the value of $x$ is

Concept & Logic Average quantity per unit of money is defined as the total quantity purchased divided by the total money spent. When identical physical quantities are purchased at varying rates, we calculate the individual monetary costs to find the total sum spent. $$Average = \frac{\text{Total Quantity}}{\text{Total Money Spent}}$$ Step-by-Step Solution * Total Quantity: The person purchases exactly 1 kg from 4 different places. Total weight = $1 + 1 + 1 + 1 = 4$ kg. * Money Spent at Place 1: Rate is 1 kg/rupee. Cost for 1 kg = $1 / 1 = 1$ rupee. * Money Spent at Place 2: Rate is 2 kg/rupee. Cost for 1 kg = $1 / 2 = 0.5$ rupees. * Money Spent at Place 3: Rate is 3 kg/rupee. Cost for 1 kg = $1 / 3$ rupees. * Money Spent at Place 4: Rate is 4 kg/rupee. Cost for 1 kg = $1 / 4 = 0.25$ rupees. * Find the total money spent: Total = $1 + 1/2 + 1/3 + 1/4$. * Taking the LCM of the denominators (1, 2, 3, 4) which is 12: Total Spent = $\frac{12 + 6 + 4 + 3}{12} = \frac{25}{12}$ rupees. * Calculate Average Quantity per Rupee = Total Quantity / Total Money Spent = $4 \div \frac{25}{12} = \frac{48}{25}$. * Convert the fraction to a decimal: $\frac{48}{25} = 1.92$ kg per rupee. Exam Strategy & Shortcut Recognize that since the quantity purchased is identical (1 kg) across all shops, the average rate (in kg/rupee) is essentially the Harmonic Mean of the individual rates (1, 2, 3, 4). Harmonic Mean for $n$ observations = $\frac{n}{1/a + 1/b + 1/c + 1/d}$ = $\frac{4}{1/1 + 1/2 + 1/3 + 1/4} = \frac{4}{25/12} = 1.92$. Common Pitfall A highly common mistake is calculating the simple arithmetic mean of the rates: $(1 + 2 + 3 + 4) / 4 = 2.5$ kg/rupee (Option C). This completely fails because the amount of money spent at each rate is not equal; rather, the physical quantity purchased is equal. Final Answer Therefore, the correct answer is 1.92.