A shopkeeper has two types of rice. The first one is at Rs. 1.80 and the second one is at Rs. 1.20 respectively. What is the proportion in which they should be mixed to sell the mixture at Rs. 1.75 and at a gain of 25%?

To solve this problem, we need to use the rule of alligation which is a rule that allows us to find the ratio in which two or more ingredients at a given price should be mixed to produce a mixture of a desired price.

Step 1: Calculate the cost price of the mixture

The shopkeeper wants to gain 25% profit by selling the mixture at Rs. 1.75. So, we need to calculate the cost price of the mixture first.

Let's denote the cost price as CP and the selling price as SP. We know that:

SP = CP + Gain

or

CP = SP - Gain

We also know that Gain is 25% of CP. So we can write:

CP = SP - 0.25*CP

Solving this equation for CP gives:

CP = SP / 1.25

Substituting SP = 1.75 (the selling price of the mixture) gives:

CP = 1.75 / 1.25 = Rs. 1.40

So, the cost price of the mixture is Rs. 1.40.

Step 2: Use the rule of alligation

Now we can use the rule of alligation. The rule of alligation states that:

(CP of dearer - Mean price) : (Mean price - CP of cheaper) = Ratio of quantities

Here, the CP of dearer is Rs. 1.80 (the cost price of the first type of rice), the CP of cheaper is Rs. 1.20 (the cost price of the second type of rice), and the mean price is Rs. 1.40 (the cost price of the mixture).

Substituting these values into the formula gives:

(1.80 - 1.40) : (1.40 - 1.20) = 0.40 : 0.20 = 2 : 1

So, the shopkeeper should mix the two types of rice in the ratio 2:1 to sell the mixture at Rs. 1.75 and at a gain of 25%.