In what ratio should milk and water be mixed so that after selling the mixture at the cost price a profit of `16 (2/3) %` is made?

Options

A. `1:2`
B. `1:6`
C. `2:3`
D. `2:5`

Answer

To solve the problem of mixing milk and water in a specific ratio to achieve a profit of 16⅔%, we will follow these steps:

### Step 1: Understand the Profit Percentage
The profit percentage given is 16⅔%. This can be converted into a fraction:
\[ 16 \frac{2}{3} \% = \frac{50}{3} \% \]
This means that for every 100 units of cost price, the selling price will be:
\[ \text{Selling Price} = \text{Cost Price} + \text{Profit} \]
\[ \text{Selling Price} = 100 + \frac{50}{3} = \frac{300 + 50}{3} = \frac{350}{3} \]

### Step 2: Calculate the Effective Cost Price
To find the effective cost price of the mixture, we can use the formula for profit:
\[ \text{Profit} = \text{Selling Price} - \text{Cost Price} \]
Given that the selling price is \(\frac{350}{3}\) and the cost price is 100, we can rearrange this to find the effective cost price of the mixture:
\[ \text{Effective Cost Price} = \frac{350}{3} - 100 = \frac{350}{3} - \frac{300}{3} = \frac{50}{3} \]

### Step 3: Set Up the Alligation
Let the cost price of milk be \(M\) and the cost price of water be \(W\). In this case, we can assume:
- Cost price of milk \(M = 100\)
- Cost price of water \(W = 0\) (since water is free)

Using the alligation method:
- The average cost price of the mixture (which is the effective cost price we calculated) is \(\frac{50}{3}\).
- The cost price of milk is \(100\).
- The cost price of water is \(0\).

### Step 4: Calculate the Ratio Using Alligation
Using the alligation rule:
\[ \text{Ratio} = \frac{(M - \text{Average})}{(\text{Average} - W)} \]
Substituting the values:
\[ \text{Ratio} = \frac{(100 - \frac{50}{3})}{(\frac{50}{3} - 0)} \]
Calculating the numerator:
\[ 100 - \frac{50}{3} = \frac{300 - 50}{3} = \frac{250}{3} \]
Calculating the denominator:
\[ \frac{50}{3} - 0 = \frac{50}{3} \]
Now substituting back into the ratio:
\[ \text{Ratio} = \frac{\frac{250}{3}}{\frac{50}{3}} = \frac{250}{50} = 5 \]
Thus, the ratio of milk to water is:
\[ \text{Milk : Water} = 5 : 1 \]

### Final Answer
The required ratio of milk to water is **5:1**.

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