A man buys milk at Rs. 5 a litre and mixes it with water. By selling the mixture at Rs.4 a litre he gains `12 1/2` % on his outlay. How much water did each litre of the mixture contain ?

Options

A. `(32)/(45)` litre
B. `(13)/ (45)` litre
C. `(32)/(13)` litre
D. None of these

Answer

To solve the problem step by step, we need to find out how much water is mixed with the milk in the mixture.

### Step 1: Understand the Cost Price and Selling Price
The man buys milk at Rs. 5 per litre. He sells the mixture at Rs. 4 per litre.

### Step 2: Calculate the Gain Percentage
The gain percentage is given as 12.5%. This means that the selling price of the mixture is more than the cost price of the milk used in the mixture.

### Step 3: Calculate the Cost Price of the Mixture
Let’s denote the cost price of the mixture as CP. The gain percentage formula is given by:

\[ \text{Gain Percentage} = \left(\frac{\text{Selling Price} - \text{Cost Price}}{\text{Cost Price}}\right) \times 100 \]

Given that the Selling Price (SP) is Rs. 4 and the Gain Percentage is 12.5%, we can set up the equation:

\[ 12.5 = \left(\frac{4 - CP}{CP}\right) \times 100 \]

### Step 4: Solve for CP
Rearranging the equation gives us:

\[ 0.125 = \frac{4 - CP}{CP} \]

Cross-multiplying:

\[ 0.125 \times CP = 4 - CP \]

Combining like terms:

\[ 0.125 CP + CP = 4 \]

\[ 1.125 CP = 4 \]

Now, solving for CP:

\[ CP = \frac{4}{1.125} = \frac{32}{9} \approx 3.56 \text{ Rs.} \]

### Step 5: Determine the Amount of Milk in the Mixture
Since the cost price of milk is Rs. 5 per litre, we need to find out how much milk is used in the mixture. Let’s denote the quantity of milk in the mixture as \( x \) litres.

The cost of \( x \) litres of milk is:

\[ \text{Cost of milk} = 5x \]

### Step 6: Set Up the Equation for the Mixture
Since the total cost of the mixture is equal to the cost of the milk used, we can set up the equation:

\[ 5x = CP \]

Substituting the value of CP we found earlier:

\[ 5x = \frac{32}{9} \]

### Step 7: Solve for x
Now, solving for \( x \):

\[ x = \frac{32}{9 \times 5} = \frac{32}{45} \text{ litres of milk} \]

### Step 8: Calculate the Total Volume of the Mixture
Let’s denote the total volume of the mixture as \( y \) litres. The selling price is Rs. 4 per litre, and the total selling price of the mixture is equal to the total cost price of the mixture:

\[ 4y = 4 \]

Thus, the total volume of the mixture is:

\[ y = 1 \text{ litre} \]

### Step 9: Calculate the Amount of Water in the Mixture
The amount of water in the mixture is given by:

\[ \text{Water} = \text{Total Volume} - \text{Volume of Milk} \]

Substituting the values:

\[ \text{Water} = 1 - \frac{32}{45} = \frac{45}{45} - \frac{32}{45} = \frac{13}{45} \text{ litres} \]

### Final Answer
Thus, the amount of water in each litre of the mixture is:

\[ \frac{13}{45} \text{ litres} \]

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