A milk seller has a milk of rs 100 per litre.In what ratio should water be mixed in that milk, so that after selling the mixture at rs 80 per litre, he may get a profit of 50%?

Options

A. `7 : 8`
B. `7 : 9`
C. `9 : 7`
D. `7 : 5`

Answer

To solve the problem step by step, we need to determine the ratio of water to milk that the seller should mix in order to achieve a 50% profit when selling the mixture at Rs 80 per litre.

### Step 1: Determine the Cost Price of Milk
The cost price (CP) of milk is given as Rs 100 per litre.

### Step 2: Calculate the Selling Price for a 50% Profit
To find the selling price (SP) that includes a 50% profit on the cost price:
- Profit = 50% of CP
- Profit = 0.5 * 100 = Rs 50
- Therefore, SP = CP + Profit = 100 + 50 = Rs 150 per litre.

### Step 3: Set Up the Equation for the Selling Price of the Mixture
The selling price of the mixture is given as Rs 80 per litre. We need to find the cost price of the mixture (CP_mixture) that would allow the seller to sell at Rs 80 per litre and still achieve a 50% profit.

### Step 4: Relate the Selling Price and Cost Price
Since the selling price is Rs 80 and this represents a 50% profit, we can set up the equation:
- SP = 1.5 * CP_mixture
- 80 = 1.5 * CP_mixture

### Step 5: Solve for the Cost Price of the Mixture
Now, we can solve for CP_mixture:
- CP_mixture = 80 / 1.5
- CP_mixture = Rs 53.33 per litre (approximately).

### Step 6: Determine the Ratio of Milk to Water
Now we know the cost price of the mixture is Rs 53.33. We need to find the ratio of milk to water in the mixture.

Let the quantity of milk be x litres and the quantity of water be y litres. The cost price of the mixture can be expressed as:
- CP_mixture = (Cost of Milk * x + Cost of Water * y) / (x + y)
- Since water is free, the cost of water is Rs 0.

Thus, the equation simplifies to:
- 53.33 = (100 * x) / (x + y)

### Step 7: Cross-Multiply to Solve for the Ratio
Cross-multiplying gives:
- 53.33(x + y) = 100x
- 53.33x + 53.33y = 100x
- 100x - 53.33x = 53.33y
- 46.67x = 53.33y

### Step 8: Rearranging to Find the Ratio
Now, we can find the ratio of milk to water:
- x/y = 53.33/46.67
- To simplify, we can multiply both sides by 100:
- x/y = 5333/4667

### Step 9: Simplifying the Ratio
Now, we can simplify the ratio:
- The ratio of milk to water is approximately 5333:4667, which can be simplified further to approximately 8:7.

### Final Answer
The ratio of milk to water should be approximately 8:7.

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