In what ratio sugar at Rs. 30 per kg should be mixed with sugar at Rs. 45 per kg so that on selling the mixture at Rs. 42 per kg there is a profit of 20%?

Options

A. `2:1`
B. `2:3`
C. `5:2`
D. `3:7`

Answer

To solve the problem of mixing sugar at different prices to achieve a desired selling price with a profit, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Cost Price (CP) and Selling Price (SP):**
- The cost price of the first sugar is Rs. 30 per kg.
- The cost price of the second sugar is Rs. 45 per kg.
- The selling price of the mixture is Rs. 42 per kg.
- We need to achieve a profit of 20%.

2. **Calculate the Cost Price (CP) of the Mixture:**
- Since there is a profit of 20%, we can express the selling price in terms of cost price:
\[ SP = CP + \text{Profit} \]
- Profit can be expressed as:
\[ \text{Profit} = 20\% \text{ of } CP = \frac{20}{100} \times CP = \frac{1}{5} \times CP \]
- Therefore, we can write:
\[ SP = CP + \frac{1}{5} \times CP = \frac{6}{5} \times CP \]
- Rearranging gives:
\[ CP = \frac{5}{6} \times SP \]
- Substituting the selling price:
\[ CP = \frac{5}{6} \times 42 = 35 \text{ Rs. per kg} \]

3. **Set Up the Allegation:**
- Now we will use the allegation method to find the ratio.
- We have:
- Cost Price of first sugar (CP1) = Rs. 30
- Cost Price of second sugar (CP2) = Rs. 45
- Cost Price of the mixture (CPm) = Rs. 35
- Set up the allegation:
\[ \text{Difference between CP1 and CPm} = 35 - 30 = 5 \]
\[ \text{Difference between CP2 and CPm} = 45 - 35 = 10 \]
- The ratio of the quantities of the two sugars is given by the differences:
\[ \text{Ratio} = \frac{10}{5} = 2:1 \]

4. **Conclusion:**
- Therefore, the sugar at Rs. 30 per kg should be mixed with sugar at Rs. 45 per kg in the ratio of **2:1**.