In what ratio should sugar costing ₹34/kg be mixed with sugar costing ₹58/kg so that by selling the mixture at ₹52.80/kg, there is a profit of 32%?

The correct option is 3 : 1.

Step 1: Find the Cost Price (CP) of the mixture
We are given:
Selling Price (SP) of the mixture = ₹52.80/kg
Profit percentage = 32%

The formula to find the Cost Price from the Selling Price and Profit percentage is:

$\mathrm{CP}=\frac{\mathrm{SP}}{1+\frac{\mathrm{Profit}\%}{100}}$

Substituting the given values into the formula:

$\mathrm{CP}=\frac{52.80}{1+\frac{32}{100}}$

$\mathrm{CP}=\frac{52.80}{1.32}$

$\mathrm{CP}=40$

So, the Cost Price of the mixture is ₹40/kg.

Step 2: Apply the Rule of Alligation
We have two varieties of sugar:
- Cost Price of cheaper sugar = ₹34/kg
- Cost Price of dearer sugar = ₹58/kg
- Mean Cost Price (CP of the mixture) = ₹40/kg

According to the rule of alligation:

$\frac{\mathrm{Quantity}\mathrm{of}\mathrm{Cheaper}}{\mathrm{Quantity}\mathrm{of}\mathrm{Dearer}}=\frac{\mathrm{CP}\mathrm{of}\mathrm{Dearer}-\mathrm{Mean}\mathrm{CP}}{\mathrm{Mean}\mathrm{CP}-\mathrm{CP}\mathrm{of}\mathrm{Cheaper}}$

Substituting the values:

$\frac{\mathrm{Quantity}\mathrm{of}\mathrm{Cheaper}}{\mathrm{Quantity}\mathrm{of}\mathrm{Dearer}}=\frac{58-40}{40-34}$

$\frac{\mathrm{Quantity}\mathrm{of}\mathrm{Cheaper}}{\mathrm{Quantity}\mathrm{of}\mathrm{Dearer}}=\frac{18}{6}$

$\frac{\mathrm{Quantity}\mathrm{of}\mathrm{Cheaper}}{\mathrm{Quantity}\mathrm{of}\mathrm{Dearer}}=\frac{3}{1}$

Thus, the two varieties of sugar must be mixed in the ratio 3 : 1.