Rishi buys 30 kg of sugar at Rs. 9 per kg and 45 kg of sugar at Rs. 10 per kg. He mixes both types and sells the entire mixture at Rs. 9.75 per kg. What is his net gain or loss in rupees on the whole transaction?

Introduction / Context:
This mixture problem tests the ability to compute weighted average cost price and compare it with a single selling price. Rishi mixes two types of sugar bought at different rates and sells at one uniform rate. The question is whether he gains or loses on the mixture and by how much in rupee terms.

Given Data / Assumptions:

• First lot: 30 kg of sugar at Rs. 9 per kg.
• Second lot: 45 kg of sugar at Rs. 10 per kg.
• Selling price of the mixture = Rs. 9.75 per kg.
• All sugar bought is sold, and there is no wastage.

Concept / Approach:
First calculate the total cost of each lot and then add them to get the total cost of the entire mixture. Next, compute the total quantity of sugar and multiply by the selling price per kg to find the total revenue. The difference between total revenue and total cost gives the net gain or loss. This method is more reliable than trying to guess the average price directly.

Step-by-Step Solution:
Cost of first lot = 30 kg * Rs. 9 per kg = Rs. 270. Cost of second lot = 45 kg * Rs. 10 per kg = Rs. 450. Total cost = 270 + 450 = Rs. 720. Total quantity of sugar = 30 kg + 45 kg = 75 kg. Selling price per kg of mixture = Rs. 9.75. Total selling price = 75 kg * Rs. 9.75 per kg. Compute: 75 * 9.75 = 731.25 rupees. Net gain = total selling price - total cost = 731.25 - 720 = Rs. 11.25.

Verification / Alternative check:
We can also compute the average cost price per kg. Average CP per kg = total cost / total kg = 720 / 75 = Rs. 9.60 per kg. He sells at Rs. 9.75 per kg. Profit per kg = 9.75 - 9.60 = Rs. 0.15 per kg. On 75 kg, total profit = 75 * 0.15 = Rs. 11.25, which matches our earlier result.

Why Other Options Are Wrong:
Rs. 10.75, Rs. 12, and Rs. 13.50: These correspond to incorrect calculations of average cost or total selling price. Only Rs. 11.25 matches both the total cost comparison and the average cost approach.

Common Pitfalls:
A common mistake is to simply average the two prices (9 and 10) without weighting them by quantity. Another error is to forget to multiply by total kilograms when computing total selling price or to mis-handle decimal multiplication. Always compute total cost and total revenue carefully when working with mixtures.

Final Answer:
Rishi makes a net profit of Rs. 11.25 on the whole sugar mixture.