The ratio of milk and water in a mixture is` 5 : 2`. How much fraction of the mixture must be drawn off and substituted by water so that the ratio of milk and water in the mixture becomes`2 : 1`?

Answer

To solve the problem step by step, we will follow the logic outlined in the video transcript while providing a clear mathematical approach.

### Step 1: Understand the Initial Mixture
The initial ratio of milk to water in the mixture is given as 5:2. This means for every 5 parts of milk, there are 2 parts of water.

Let’s denote:
- Milk (M) = 5 parts
- Water (W) = 2 parts

### Step 2: Calculate the Total Parts of the Initial Mixture
The total parts in the initial mixture can be calculated as:
\[ \text{Total parts} = M + W = 5 + 2 = 7 \text{ parts} \]

### Step 3: Determine the Final Ratio
We want the final ratio of milk to water to be 2:1. This means:
- For every 2 parts of milk, there will be 1 part of water.

Let’s denote the final amounts:
- Milk in final mixture = 2 parts
- Water in final mixture = 1 part

### Step 4: Equalize the Amount of Milk
Since the amount of milk remains constant during the process, we will express both the initial and final amounts of milk in terms of a common quantity.

To find a common basis, we can multiply the initial ratio by a factor to match the milk quantity in the final ratio:
- Initial milk (5 parts) can be scaled to match the final milk (2 parts).

To do this, we can multiply the initial ratio by 2.5 (to make the milk in the initial mixture equal to 10):
- Initial: \( 5 \times 2.5 = 12.5 \) (milk)
- Initial: \( 2 \times 2.5 = 5 \) (water)

Now, the initial mixture is 12.5 parts milk and 5 parts water, giving a total of:
\[ \text{Total initial mixture} = 12.5 + 5 = 17.5 \text{ parts} \]

### Step 5: Calculate the Final Mixture
To achieve the final ratio of 2:1, we can express the final amounts in terms of a common total:
- Final milk = 10 parts
- Final water = 5 parts

Thus, the total final mixture is:
\[ \text{Total final mixture} = 10 + 5 = 15 \text{ parts} \]

### Step 6: Calculate the Change in Mixture
The change in the mixture (the amount of mixture that needs to be replaced with water) can be calculated as:
\[ \text{Change} = \text{Total initial mixture} - \text{Total final mixture} = 17.5 - 15 = 2.5 \text{ parts} \]

### Step 7: Find the Fraction of the Mixture to be Drawn Off
Now, we need to find the fraction of the initial mixture that must be drawn off and replaced with water:
\[ \text{Fraction of mixture drawn off} = \frac{\text{Change}}{\text{Total initial mixture}} = \frac{2.5}{17.5} \]
To simplify:
\[ \frac{2.5}{17.5} = \frac{1}{7} \]

### Final Answer
Thus, the fraction of the mixture that must be drawn off and substituted by water is:
\[ \frac{1}{7} \]