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 g, he gains `20%`. In what proportion should the Desi Chai and Videshi Chai be mixed such that he can gain a profit of `5% `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 price of both types of tea, then calculate the average cost price needed for a desired profit margin, and finally use the method of allegations to find the required ratio for mixing the two types of tea.

### Step 1: Calculate the Cost Price of Desi Chai
Desi Chai is sold at Rs. 18 per kg with a loss of 10%.

Let the cost price (CP) of Desi Chai be \( x \).

Since there is a loss of 10%, we have:
\[ (100 - 10)\% \text{ of } x = 18 \]
\[ 90\% \text{ of } x = 18 \]
\[ x = \frac{18 \times 100}{90} = 20 \text{ Rs.} \]

### Step 2: Calculate the Cost Price of Videshi Chai
Videshi Chai is sold at Rs. 30 per kg with a gain of 20%.

Let the cost price (CP) of Videshi Chai be \( y \).

Since there is a gain of 20%, we have:
\[ (100 + 20)\% \text{ of } y = 30 \]
\[ 120\% \text{ of } y = 30 \]
\[ y = \frac{30 \times 100}{120} = 25 \text{ Rs.} \]

### Step 3: Determine the Required Average Cost Price for a 5% Profit
The selling price of the mixture is Rs. 27.5 per kg, and we want a profit of 5%.

Let the average cost price (CP_avg) of the mixture be \( z \).

To find \( z \):
\[ (100 + 5)\% \text{ of } z = 27.5 \]
\[ 105\% \text{ of } z = 27.5 \]
\[ z = \frac{27.5 \times 100}{105} = \frac{2750}{105} \approx 26.19 \text{ Rs.} \]

### Step 4: Use Allegation to Find the Ratio
Now we will use the method of allegation to find the ratio of Desi Chai and Videshi Chai.

- Cost price of Desi Chai (CPD) = Rs. 20
- Cost price of Videshi Chai (CPV) = Rs. 25
- Average cost price of the mixture (CP_avg) = Rs. 26.19

Now, we set up the allegation:

\[ \text{CPD} \quad \text{CP_avg} \quad \text{CPV} \]
\[ 20 \quad \quad 26.19 \quad \quad 25 \]

Calculate the differences:
- Difference between CP_avg and CPD:
\[ 26.19 - 20 = 6.19 \]
- Difference between CPV and CP_avg:
\[ 25 - 26.19 = -1.19 \]

### Step 5: Calculate the Ratio
The ratio of Desi Chai to Videshi Chai is given by the inverse of the differences:
\[ \text{Ratio} = \frac{6.19}{1.19} \]

Calculating the ratio:
\[ \text{Ratio} \approx 5.20 : 1 \text{ (approximately)} \]

### Final Answer
Thus, the proportion in which Desi Chai and Videshi Chai should be mixed is approximately **5.20:1**.

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