In what ratio must a fruit vendor mix two varieties of mangoes costing ₹60/kg and ₹68.40/kg, so that by selling the mixed variety of mangoes at ₹79.80/kg, he may gain 18.75%?

The correct option is 1 : 6.

To find the ratio in which the two varieties of mangoes must be mixed, we first need to determine the cost price (CP) of the mixture. We are given the selling price (SP) of the mixture and the percentage gain.

Step 1: Calculate the Cost Price (CP) of the mixture
The selling price (SP) of the mixture is ₹79.80/kg.
The gain percentage is 18.75%.

The relationship between Cost Price and Selling Price given a gain percentage is:
$$\text{SP}=\text{CP}\times (1+\frac{\text{Gain\%}}{100})$$
Substituting the given values into the formula:
$$79.80=\text{CP}\times (1+\frac{18.75}{100})$$
Convert the percentage fraction into a simplified form:
$$\frac{18.75}{100}=\frac{1875}{10000}=\frac{3}{16}$$
So, the equation becomes:
$$79.80=\text{CP}\times (1+\frac{3}{16})$$
$$79.80=\text{CP}\times \frac{19}{16}$$
Now, solve for CP:
$$\text{CP}=\frac{79.80\times 16}{19}$$
Dividing 79.80 by 19:
$$\frac{79.80}{19}=4.20$$
Therefore:
$$\text{CP}=4.20\times 16=67.20$$
So, the cost price of the mixture is ₹67.20/kg.

Step 2: Apply the Rule of Alligation
Let Variety 1 have a cost of ₹60/kg (cheaper quantity) and Variety 2 have a cost of ₹68.40/kg (dearer quantity). The mean price (cost price of the mixture) is ₹67.20/kg.

Using the alligation formula:
$$\frac{\text{Quantity of Cheaper (Variety 1)}}{\text{Quantity of Dearer (Variety 2)}}=\frac{\text{Price of Dearer}-\text{Mean Price}}{\text{Mean Price}-\text{Price of Cheaper}}$$
Substitute the respective values:
$$\frac{\text{Quantity of Variety 1}}{\text{Quantity of Variety 2}}=\frac{68.40-67.20}{67.20-60}$$
$$\frac{\text{Quantity of Variety 1}}{\text{Quantity of Variety 2}}=\frac{1.20}{7.20}$$
Simplify this fraction:
$$\frac{1.20}{7.20}=\frac{12}{72}=\frac{1}{6}$$
Thus, the vendor must mix the two varieties of mangoes in the ratio of 1 : 6.