In what ratio should sugar costing ₹38 per kg be mixed with sugar costing ₹55 per kg so that by selling the mixture at ₹57.20 per kg, there is a profit of 10%?

The correct option is 3 : 14.

Step 1: Calculate the cost price (CP) of the mixture per kg
We are given that the selling price (SP) of the mixture is ₹57.20 per kg and the profit percentage is 10%.
The relationship between Selling Price (SP), Cost Price (CP), and Profit percentage is given by:

$\text{SP}=\text{CP}\times (1+\frac{\text{Profit \%}}{100})$

Substituting the given values into the equation:

$57.20=\text{CP}\times (1+\frac{10}{100})$

$57.20=\text{CP}\times 1.10$

$\text{CP}=\frac{57.20}{1.10}=52$

So, the mean cost price of the sugar mixture is ₹52 per kg.

Step 2: Apply the Rule of Alligation to find the required ratio
Let Sugar 1 cost ₹38 per kg (cheaper quantity) and Sugar 2 cost ₹55 per kg (dearer quantity).
The mean price (cost price of mixture) is ₹52 per kg.

By the rule of alligation:
(Quantity of Cheaper Sugar) : (Quantity of Dearer Sugar) = (Cost price of Dearer Sugar - Mean Price) : (Mean Price - Cost price of Cheaper Sugar)

Difference 1 (Dearer cost - Mean cost):

$55-52=3$

Difference 2 (Mean cost - Cheaper cost):

$52-38=14$

Therefore, the required ratio in which the two types of sugar should be mixed is:

$\text{Ratio}=3:14$