Baniya sells two types of tea viz. Desi Chai and Videshi Chai. He sells Desi Chai at Rs. 18 per kg and incurs a loss of 10% whereas on selling the Videshi Chai at Rs. 30 per kg, he gains 20%. In what proportion should the Desi Chai and Videshi Chai be mixed such that he can gain a profit of 25% by selling the mixture at Rs. 27.5 per kg?

Options

A. `3:2`
B. `2:3`
C. `2:5`
D. `3:5`

Answer

To solve the problem step by step, we will first determine the cost prices (CP) of both types of tea, then find the cost price of the mixture, and finally use the method of allegations to find the required proportion of Desi Chai and Videshi Chai.

### Step 1: Calculate the Cost Price of Desi Chai
Given:
- Selling Price (SP) of Desi Chai = Rs. 18 per kg
- Loss = 10%

Using the formula for selling price in case of loss:
\[ SP = CP \times \left(1 - \frac{Loss\%}{100}\right) \]
Substituting the values:
\[ 18 = CP \times \left(1 - \frac{10}{100}\right) \]
\[ 18 = CP \times \left(\frac{90}{100}\right) \]
\[ 18 = CP \times \frac{9}{10} \]
Now, solving for CP:
\[ CP = 18 \times \frac{10}{9} = 20 \text{ Rs. per kg} \]

### Step 2: Calculate the Cost Price of Videshi Chai
Given:
- Selling Price (SP) of Videshi Chai = Rs. 30 per kg
- Profit = 20%

Using the formula for selling price in case of profit:
\[ SP = CP \times \left(1 + \frac{Profit\%}{100}\right) \]
Substituting the values:
\[ 30 = CP \times \left(1 + \frac{20}{100}\right) \]
\[ 30 = CP \times \left(\frac{120}{100}\right) \]
\[ 30 = CP \times \frac{6}{5} \]
Now, solving for CP:
\[ CP = 30 \times \frac{5}{6} = 25 \text{ Rs. per kg} \]

### Step 3: Calculate the Cost Price of the Mixture
Given:
- Selling Price of the mixture = Rs. 27.5 per kg
- Profit = 25%

Using the formula for selling price in case of profit:
\[ SP = CP \times \left(1 + \frac{Profit\%}{100}\right) \]
Substituting the values:
\[ 27.5 = CP \times \left(1 + \frac{25}{100}\right) \]
\[ 27.5 = CP \times \left(\frac{125}{100}\right) \]
\[ 27.5 = CP \times \frac{5}{4} \]
Now, solving for CP:
\[ CP = 27.5 \times \frac{4}{5} = 22 \text{ Rs. per kg} \]

### Step 4: Apply Allegation to Find the Ratio
Now we have:
- CP of Desi Chai = Rs. 20 per kg
- CP of Videshi Chai = Rs. 25 per kg
- CP of the mixture = Rs. 22 per kg

Using the method of allegations:
\[ \text{Difference between CP of Videshi Chai and CP of mixture} = 25 - 22 = 3 \]
\[ \text{Difference between CP of mixture and CP of Desi Chai} = 22 - 20 = 2 \]

Thus, the ratio of Desi Chai to Videshi Chai is:
\[ \text{Ratio} = \frac{3}{2} \]

### Final Answer
The proportion in which Desi Chai and Videshi Chai should be mixed is **3:2**.

---