How many litres of oil costing Rs 40 per litre should be mixed with 240 litres of a second variety of oil costing Rs 60 per litre so as to obtain a mixture whose cost price is Rs 52 per litre?

Introduction:
This is an alligation and mixture problem on cost price. Two types of oil with different cost prices are mixed to obtain a mixture with an intermediate cost price. We are given the quantity of one type and must find how much of the other type is needed so that the final mixture has a specified cost per litre.

Given Data / Assumptions:

• First variety of oil costs Rs 40 per litre.
• Second variety of oil costs Rs 60 per litre.
• There are 240 litres of the second variety of oil.
• The required mixture must have a cost price of Rs 52 per litre.
• We must find how many litres of the first variety should be added.

Concept / Approach:
We can use the alligation rule or directly set up a weighted average equation. Let x be the litres of first oil (Rs 40 per litre). Then total cost of the mixture is 40x + 60 * 240. Total volume is x + 240 litres. The average cost per litre (which is the cost price of the mixture) equals Rs 52, so:
(40x + 60 * 240) / (x + 240) = 52. We solve this equation for x to find the required quantity of the first oil.

Step-by-Step Solution:
Step 1: Let x = litres of oil costing Rs 40 per litre. Step 2: Total cost of first oil = 40x rupees. Step 3: Total cost of second oil = 60 * 240 = 14400 rupees. Step 4: Total volume of mixture = x + 240 litres. Step 5: Required average cost per litre = 52 rupees. Step 6: Set up equation: (40x + 14400) / (x + 240) = 52. Step 7: Cross multiply: 40x + 14400 = 52 * (x + 240). Step 8: Expand right side: 40x + 14400 = 52x + 12480. Step 9: Rearrange: 14400 - 12480 = 52x - 40x. Step 10: 1920 = 12x, so x = 1920 / 12 = 160.

Verification / Alternative check:
If we mix 160 litres of Rs 40 oil with 240 litres of Rs 60 oil, total volume = 160 + 240 = 400 litres. Total cost = 160 * 40 + 240 * 60 = 6400 + 14400 = 20800 rupees. Average cost per litre = 20800 / 400 = 52 rupees, which matches the target price. This confirms that 160 litres is correct.

Why Other Options Are Wrong:
170, 150, 140 and 200 litres, when substituted as x, give mixture cost prices different from Rs 52. Each leads to a weighted average price that is either too low or too high compared to 52 rupees per litre.

Common Pitfalls:
A common mistake is to misapply the alligation rule by reversing the differences or mixing up which cost goes where. Another is to ignore the given fixed quantity of the second oil and treat both quantities as unknown. Some students also attempt to use percentage change concepts instead of the simple weighted average formula, which can complicate the problem unnecessarily.

Final Answer:
The quantity of Rs 40 per litre oil to be mixed is 160 litres.