Two varieties of sugar are mixed together in a certain ratio. The cost of the mixture per kg is Rs. 0.50 less than that of the superior and Rs. 0.75 more than the inferior variety. The ratio in which the superior and inferior varieties of sugar have been mixed is:

Options

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

Answer

To solve the problem using the allegation method, we will follow these steps:

### Step 1: Define Variables
Let:
- The price of the superior variety of sugar be \( S \) (in Rs.)
- The price of the inferior variety of sugar be \( I \) (in Rs.)
- The price of the mixture be \( M \) (in Rs.)

### Step 2: Set Up the Equations
According to the problem:
- The price of the mixture is Rs. 0.50 less than that of the superior variety:
\[ M = S - 0.50 \]
- The price of the mixture is Rs. 0.75 more than that of the inferior variety:
\[ M = I + 0.75 \]

### Step 3: Equate the Two Expressions for M
Since both expressions equal \( M \), we can set them equal to each other:
\[ S - 0.50 = I + 0.75 \]

### Step 4: Rearrange the Equation
Rearranging the equation gives us:
\[ S - I = 0.50 + 0.75 \]
\[ S - I = 1.25 \]

### Step 5: Apply the Allegation Method
Now, we can apply the allegation method:
- The difference between the price of the superior variety and the mixture:
\[ S - M = 0.50 \]
- The difference between the mixture and the inferior variety:
\[ M - I = 0.75 \]

### Step 6: Set Up the Allegation Diagram
Using the differences calculated:
- The difference between the superior and the mixture is \( 0.50 \).
- The difference between the mixture and the inferior is \( 0.75 \).

### Step 7: Write the Ratios
The ratio of the quantities of superior to inferior varieties is given by the inverse of the differences:
\[ \text{Ratio} = \frac{0.75}{0.50} = \frac{75}{50} = \frac{3}{2} \]

### Step 8: Conclusion
Thus, the ratio in which the superior and inferior varieties of sugar have been mixed is:
\[ \text{Superior : Inferior} = 3 : 2 \]

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