A grocer buys kinds of rice X and Y, one (X) at the rate of a per kg and the other (Y) at the rate of रु b per kg. He mixes them and obtains a mixture of रु c per kg. What is the ratio of the variety X to that of variety Y in the mixture?

Options

A. (b+c): (c + a)
B. (b-c) : (c + a)
C. (b-c) : (c-a)
D. (b+c) : (c-a)

Answer

To find the ratio of the variety X to that of variety Y in the mixture, we can follow these steps:

### Step 1: Understand the Cost Formula
The cost of any item can be calculated using the formula:
\[ \text{Cost} = \text{Rate} \times \text{Quantity} \]

### Step 2: Define Variables
Let:
- \( X \) = quantity of rice X (in kg)
- \( Y \) = quantity of rice Y (in kg)
- \( A \) = cost of rice X (in Rs. per kg)
- \( B \) = cost of rice Y (in Rs. per kg)
- \( C \) = cost of the mixture (in Rs. per kg)

### Step 3: Calculate the Cost of Each Variety
- The cost of rice X:
\[ \text{Cost of X} = A \times X \]
- The cost of rice Y:
\[ \text{Cost of Y} = B \times Y \]

### Step 4: Calculate the Total Cost of the Mixture
The total cost of the mixture can be expressed as:
\[ \text{Cost of Mixture} = \text{Cost of X} + \text{Cost of Y} \]
This can be written as:
\[ C \times (X + Y) = A \times X + B \times Y \]

### Step 5: Rearranging the Equation
From the equation above, we can rearrange it:
\[ C(X + Y) = AX + BY \]
Expanding this gives:
\[ CX + CY = AX + BY \]

### Step 6: Isolate Terms
Rearranging the equation to isolate terms involving X and Y:
\[ CX - AX = BY - CY \]
This simplifies to:
\[ X(C - A) = Y(B - C) \]

### Step 7: Find the Ratio
Now, we can express the ratio of X to Y:
\[ \frac{X}{Y} = \frac{B - C}{C - A} \]

### Final Step: Write the Ratio
Thus, the ratio of the quantity of rice X to the quantity of rice Y in the mixture is:
\[ X : Y = (B - C) : (C - A) \]