In what ratio should sugar costing Rs 30 per kg be mixed with sugar costing Rs 45 per kg so that, when the mixture is sold at Rs 42 per kg, the shopkeeper earns a profit of 20% on the cost price?

Introduction / Context:
This aptitude question on profit and loss and mixtures tests whether you can connect selling price, profit percentage, and weighted average cost. Two varieties of sugar are mixed and then sold at a common rate. You must determine in what ratio the cheaper and costlier sugar should be mixed so that the trader earns a 20% profit overall when selling the mixture at Rs 42 per kg.

Given Data / Assumptions:

• Cheaper sugar costs Rs 30 per kg.
• Costlier sugar costs Rs 45 per kg.
• The mixture is sold at Rs 42 per kg.
• Required overall profit on cost price is 20%.
• We assume no wastage or extra cost of mixing.

Concept / Approach:
The key idea is that selling price (SP) and profit percentage determine the effective cost price (CP) of the mixture. If profit is 20%, then SP = 1.2 * CP. Knowing SP of the mixture, we first find CP per kg of the mixture. Then we treat the mixture as a weighted average of the two components with prices Rs 30 and Rs 45. The ratio in which they are mixed is obtained using the alligation or weighted average method.

Step-by-Step Solution:
Let CP of the mixture per kg be M. Given profit = 20%, SP = 42, so 42 = 1.2 * M. Therefore M = 42 / 1.2 = 35 rupees per kg. We now have cheaper = 30, mixture CP = 35, costlier = 45. By alligation, ratio of cheaper : costlier = (45 - 35) : (35 - 30) = 10 : 5 = 2 : 1.

Verification / Alternative check:
Suppose we take 2 kg of sugar at Rs 30 and 1 kg of sugar at Rs 45. The total cost is 2 * 30 + 1 * 45 = 60 + 45 = 105 rupees for 3 kg, so CP per kg = 105 / 3 = 35 rupees. If the mixture is sold at Rs 42 per kg, the profit per kg is 42 - 35 = 7 rupees. Profit percentage = (7 / 35) * 100 = 20%, which matches the requirement, confirming the ratio is correct.

Why Other Options Are Wrong:
The ratio 2 : 3 gives a weighted cost closer to the higher price and would not result in CP = 35. The ratio 5 : 2 gives a different weighted average cost, not equal to Rs 35. The ratio 3 : 7 heavily favors the costlier sugar and leads to a much higher CP, which cannot yield exactly 20% profit at Rs 42 per kg.

Common Pitfalls:
A common error is to apply alligation directly between 30 and 45 with 42 in the middle, without adjusting for the given profit percentage. Another mistake is to treat Rs 42 as the cost price instead of the selling price. Some learners also confuse the roles of cheaper and costlier components when forming the ratio and accidentally invert the ratio values.

Final Answer:
The required mixing ratio of sugar costing Rs 30 per kg to sugar costing Rs 45 per kg is 2 : 1.