A dairyman pays Rs. 6.4 per litre of milk. He adds water and sells the mixture at Rs. 8 per litre, thereby making 37.5% profit. The proportion of water to milk received by the customers is :

Options

A. `1:5`
B. `1:10`
C. `1: 20`
D. `1:12`

Answer

To solve the problem, we need to find the proportion of water to milk in the mixture sold by the dairyman. We will follow these steps:

### Step 1: Calculate the Cost Price (CP) and Selling Price (SP)
The cost price of milk is given as Rs. 6.4 per litre. The selling price of the mixture is Rs. 8 per litre.

### Step 2: Calculate the Profit
The profit percentage is given as 37.5%. We can use the formula for profit percentage to find the profit in terms of the cost price.

Profit Percentage = (Profit / Cost Price) × 100

Let the cost price of the mixture be CP_mixture. Then we have:
\[ 37.5 = \left( \frac{SP - CP_{mixture}}{CP_{mixture}} \right) \times 100 \]

### Step 3: Rearranging the Equation
From the above equation, we can express the profit:
\[ Profit = SP - CP_{mixture} \]
Substituting the values we have:
\[ Profit = 8 - CP_{mixture} \]

Now we can substitute this into the profit percentage equation:
\[ 37.5 = \left( \frac{8 - CP_{mixture}}{CP_{mixture}} \right) \times 100 \]

### Step 4: Solve for CP_mixture
Rearranging gives:
\[ 37.5 \cdot CP_{mixture} = 800 - 100 \cdot CP_{mixture} \]
Combining like terms:
\[ 137.5 \cdot CP_{mixture} = 800 \]
Thus,
\[ CP_{mixture} = \frac{800}{137.5} \]
Calculating this gives:
\[ CP_{mixture} = 5.82 \text{ (approximately)} \]

### Step 5: Calculate the Cost of Water
Since the cost of milk is Rs. 6.4 per litre and the cost of the mixture is Rs. 5.82, we can find the cost of water. The cost of water is Rs. 0 per litre, so the effective cost of the mixture is due to the dilution of milk with water.

### Step 6: Determine the Ratio of Milk to Water
Let the quantity of milk be M litres and the quantity of water be W litres. The total cost of the mixture can be expressed as:
\[ CP_{mixture} = \frac{6.4M + 0W}{M + W} \]
Setting this equal to our calculated CP_mixture:
\[ 5.82 = \frac{6.4M}{M + W} \]
Cross-multiplying gives:
\[ 5.82(M + W) = 6.4M \]
Expanding and rearranging:
\[ 5.82M + 5.82W = 6.4M \]
\[ 5.82W = 6.4M - 5.82M \]
\[ 5.82W = 0.58M \]
Thus, the ratio of milk to water is:
\[ \frac{M}{W} = \frac{5.82}{0.58} = 10 \]

### Step 7: Final Proportion of Water to Milk
The proportion of water to milk is the inverse of the milk to water ratio:
\[ \frac{W}{M} = \frac{1}{10} \]

### Conclusion
The proportion of water to milk received by the customers is 1:10.

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