The ratio of milk and water in a mixture is `5 : 3`. 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 will follow the reasoning laid out in the video transcript.

### Step 1: Understand the initial ratio
The initial ratio of milk to water in the mixture is given as \(5:3\). This means that for every 8 parts of the mixture, 5 parts are milk and 3 parts are water.

### Step 2: Define the total volume of the mixture
Let’s assume the total volume of the mixture is 8 liters. This is a convenient number because it matches the total parts of the ratio (5 + 3 = 8).

### Step 3: Calculate the initial quantities of milk and water
From the ratio:
- Quantity of milk = \(\frac{5}{8} \times 8 = 5\) liters
- Quantity of water = \(\frac{3}{8} \times 8 = 3\) liters

### Step 4: Define the amount of mixture to be drawn off
Let \(x\) liters of the mixture be drawn off and replaced with water.

### Step 5: Calculate the quantities of milk and water removed
When \(x\) liters of the mixture is drawn off, the quantity of milk and water removed can be calculated as follows:
- Milk removed = \(\frac{5}{8}x\) liters
- Water removed = \(\frac{3}{8}x\) liters

### Step 6: Calculate the new quantities of milk and water after drawing off
After removing \(x\) liters of the mixture and adding \(x\) liters of water, the new quantities will be:
- New quantity of milk = \(5 - \frac{5}{8}x\) liters
- New quantity of water = \(3 - \frac{3}{8}x + x\) liters
- Simplifying this gives:
- New quantity of water = \(3 - \frac{3}{8}x + \frac{8}{8}x = 3 + \frac{5}{8}x\) liters

### Step 7: Set up the equation for the new ratio
We want the new ratio of milk to water to be \(1:1\). Therefore, we set up the equation:
\[ 5 - \frac{5}{8}x = 3 + \frac{5}{8}x \]

### Step 8: Solve the equation
To solve for \(x\), we first combine like terms:
\[ 5 - 3 = \frac{5}{8}x + \frac{5}{8}x \]
\[ 2 = \frac{10}{8}x \]
\[ 2 = \frac{5}{4}x \]
Multiplying both sides by \(\frac{4}{5}\):
\[ x = \frac{8}{5} \text{ liters} \]

### Step 9: Calculate the fraction of the mixture drawn off
The fraction of the mixture that was replaced is:
\[ \text{Fraction drawn off} = \frac{x}{\text{Total mixture}} = \frac{\frac{8}{5}}{8} = \frac{8}{5} \times \frac{1}{8} = \frac{1}{5} \]

### Final Answer
The fraction of the mixture that must be drawn off and substituted by water to achieve a \(1:1\) ratio of milk to water is \(\frac{1}{5}\).

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