The ratio of milk and water in a mixture is `11 : 7`. 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` 1 : 1`?

Answer

To solve the problem step by step, we need to determine how much fraction of the mixture must be drawn off and substituted by water to achieve a 1:1 ratio of milk and water.

### Step 1: Understand the Initial Ratio
The initial ratio of milk to water is given as 11:7. This means that for every 11 parts of milk, there are 7 parts of water.

### Step 2: Calculate Total Parts
The total parts in the mixture can be calculated as:
Total parts = 11 (milk) + 7 (water) = 18 parts.

### Step 3: Assume a Total Quantity
For simplicity, let's assume the total quantity of the mixture is 18 liters. This means:
- Milk = (11/18) * 18 = 11 liters
- Water = (7/18) * 18 = 7 liters

### Step 4: Determine the New Ratio Requirement
We want the new ratio of milk to water to be 1:1. This means that in the final mixture, the amount of milk should equal the amount of water.

### Step 5: Let x be the Fraction Drawn Off
Let x be the fraction of the mixture that we will draw off. When we draw off x liters of the mixture, we will also be drawing off a proportionate amount of milk and water.

### Step 6: Calculate Amounts Drawn Off
When we draw off x liters:
- Milk drawn off = (11/18) * x liters
- Water drawn off = (7/18) * x liters

### Step 7: Calculate Remaining Quantities
After drawing off x liters, the remaining quantities will be:
- Remaining milk = 11 - (11/18)x
- Remaining water = 7 - (7/18)x

### Step 8: Substitute with Water
We will then add x liters of water back into the mixture. The new amount of water will be:
New water = Remaining water + x = (7 - (7/18)x) + x = 7 - (7/18)x + (18/18)x = 7 + (11/18)x

### Step 9: Set Up the Equation for 1:1 Ratio
For the ratio of milk to water to be 1:1, we set:
Remaining milk = New water
11 - (11/18)x = 7 + (11/18)x

### Step 10: Solve for x
Now, we solve the equation:
11 - (11/18)x = 7 + (11/18)x
=> 11 - 7 = (11/18)x + (11/18)x
=> 4 = (22/18)x
=> x = 4 * (18/22)
=> x = 4 * (9/11)
=> x = 36/11 liters

### Step 11: Calculate the Fraction
Now, to find the fraction of the mixture that must be drawn off:
Fraction drawn off = x / Total mixture = (36/11) / 18 = (36/198) = 2/11

### Final Answer
The fraction of the mixture that must be drawn off and substituted by water is **2/11**.

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