A sample of milk diluted with water has density `1036 kg m^(-3).` if pure milk has a density `1080 kg m^(-3)` what is the percentage of water by volume in milk ?

Explanation

To find the percentage of water by volume in diluted milk, we can follow these steps:

### Step 1: Define Variables
Let:
- \( V \) = total volume of the diluted milk
- \( v \) = volume of water in the diluted milk
- \( V_m \) = volume of pure milk in the diluted milk

From the definition of volume, we have:
\[ V_m = V - v \]

### Step 2: Define Densities
We know the following densities:
- Density of diluted milk, \( \rho_d = 1036 \, \text{kg/m}^3 \)
- Density of pure milk, \( \rho_m = 1080 \, \text{kg/m}^3 \)
- Density of water, \( \rho_w = 1000 \, \text{kg/m}^3 \) (standard value)

### Step 3: Calculate Masses
The mass of the diluted milk can be expressed as:
\[ \text{Mass of diluted milk} = \rho_d \times V = 1036 \times V \]

The mass of pure milk and the mass of water can be expressed as:
- Mass of pure milk = \( \rho_m \times V_m = 1080 \times (V - v) \)
- Mass of water = \( \rho_w \times v = 1000 \times v \)

### Step 4: Set Up the Equation
The total mass of the diluted milk can be expressed as the sum of the mass of pure milk and the mass of water:
\[ \rho_d \times V = \rho_m \times (V - v) + \rho_w \times v \]
Substituting the known values:
\[ 1036V = 1080(V - v) + 1000v \]

### Step 5: Simplify the Equation
Expanding the equation:
\[ 1036V = 1080V - 1080v + 1000v \]
Combining like terms:
\[ 1036V = 1080V - 80v \]
Rearranging gives:
\[ 80v = 1080V - 1036V \]
\[ 80v = 44V \]
Thus:
\[ v = \frac{44V}{80} = \frac{11V}{20} \]

### Step 6: Calculate Volume of Pure Milk
Now, we can find the volume of pure milk:
\[ V_m = V - v = V - \frac{11V}{20} = \frac{9V}{20} \]

### Step 7: Calculate Percentage of Water by Volume
The percentage of water by volume is given by:
\[ \text{Percentage of water} = \left( \frac{v}{V} \right) \times 100 \]
Substituting for \( v \):
\[ \text{Percentage of water} = \left( \frac{\frac{11V}{20}}{V} \right) \times 100 = \left( \frac{11}{20} \right) \times 100 = 55\% \]

### Final Answer
The percentage of water by volume in the diluted milk is **55%**.

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