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 two types of sugar at different prices to achieve a desired selling price with a profit, we can follow these steps:

### Step 1: Determine the Cost Price
We know the selling price (SP) and the profit percentage (P). The formula to find the cost price (CP) is:

\[ CP = \frac{100}{100 + P} \times SP \]

Given:
- Selling Price (SP) = Rs. 42
- Profit Percentage (P) = 20%

Substituting the values into the formula:

\[ CP = \frac{100}{100 + 20} \times 42 = \frac{100}{120} \times 42 \]

Calculating it:

\[ CP = \frac{5}{6} \times 42 = 35 \]

### Step 2: Identify the Prices of Sugars
We have:
- Price of Sugar 1 (S1) = Rs. 30 per kg
- Price of Sugar 2 (S2) = Rs. 45 per kg

### Step 3: Calculate the Differences
Now, we calculate the differences between the cost price of the mixture and the prices of the two types of sugar:

1. Difference between S2 and CP:
\[ S2 - CP = 45 - 35 = 10 \]

2. Difference between CP and S1:
\[ CP - S1 = 35 - 30 = 5 \]

### Step 4: Set Up the Ratio
The ratio of the quantities of Sugar 1 and Sugar 2 can be set up using the differences calculated:

\[ \text{Ratio} = \frac{S2 - CP}{CP - S1} = \frac{10}{5} = 2:1 \]

### Step 5: Conclusion
Thus, the ratio in which sugar at Rs. 30 per kg should be mixed with sugar at Rs. 45 per kg to achieve a selling price of Rs. 42 per kg with a profit of 20% is:

\[ \text{Ratio} = 2:1 \]