Rice costing Rs. 126 per kg and Rs. 135 per kg are mixed with a third variety of rice in the ratio 1 : 1 : 2. If the resulting mixture is worth Rs. 153 per kg, what is the price per kg of the third variety of rice?

Introduction / Context:
This is a standard mixture and alligation problem involving three varieties of rice. Two known varieties with given prices per kilogram are mixed with a third unknown variety in a specific ratio, and the price of the resulting mixture is given. We are required to find the price per kg of the third variety. Such questions test weighted average calculations and understanding of how ratios distribute cost contributions in mixtures.

Given Data / Assumptions:
• Price of first variety = Rs. 126 per kg. • Price of second variety = Rs. 135 per kg. • Price of third variety = Rs. x per kg (unknown). • Mixing ratio of the three varieties = 1 : 1 : 2. • Price of final mixture = Rs. 153 per kg.

Concept / Approach:
The price of the mixture is the weighted average of the individual prices, where the weights are the quantities mixed in the given ratio. With a ratio of 1 : 1 : 2, the total of ratio parts is 4. We set up an equation: (126*1 + 135*1 + x*2) / 4 = 153. Solving this linear equation gives the value of x. This is a direct application of the weighted mean formula for mixtures.

Step-by-Step Solution:
Step 1: Let the quantities of the three rice types be 1 kg, 1 kg, and 2 kg, respecting the ratio 1 : 1 : 2. Step 2: Total weight of the mixture = 1 + 1 + 2 = 4 kg. Step 3: Total cost of the mixture = 1 * 126 + 1 * 135 + 2 * x. Step 4: So total cost = 126 + 135 + 2x = 261 + 2x. Step 5: Average cost per kg of mixture = (total cost) / 4 = 153. Step 6: Therefore (261 + 2x) / 4 = 153. Step 7: Multiply both sides by 4: 261 + 2x = 153 * 4 = 612. Step 8: Subtract 261 from both sides: 2x = 612 − 261 = 351. Step 9: Divide by 2: x = 351 / 2 = 175.5.

Verification / Alternative check:
Check by substituting back: total cost = 126 + 135 + 2 * 175.5 = 261 + 351 = 612. Dividing 612 by the total quantity 4 kg gives 612 / 4 = 153, which matches the mixture price given in the question. Thus, the computed price of the third variety is consistent with the information.

Why Other Options Are Wrong:
• Rs. 571.25 and Rs. 751.75 are unrealistically high for a small change in mixture price from 126–135 to 153. • Rs. 850 would make the mixture price far higher than Rs. 153 for the given ratio.

Common Pitfalls:
Some candidates mistakenly average the three prices directly without weighting by the mixing ratio, which is incorrect when quantities differ. Another common error is to miscalculate the total cost or to divide by the wrong total quantity. Always ensure that the weighted sum is divided by the sum of ratio parts, not just the count of varieties.

Final Answer:
The price per kg of the third variety of rice is Rs. 175.5.