Two liquids of densities `2 rho " and " rho` having their volumes in the ratio `3 : 2` are mixed together. Density of the mixture will be

Options

A. `(2 rho)/(3)`
B. `(rho)/(2)`
C. `(8 rho)/(5)`
D. `(4 rho)/(5)`

Explanation

To find the density of the mixture of two liquids with given densities and volume ratios, we can follow these steps:

### Step 1: Define the Densities and Volumes
Let:
- Density of liquid 1, \( \rho_1 = 2\rho \)
- Density of liquid 2, \( \rho_2 = \rho \)
- Volume of liquid 1, \( V_1 = 3V \) (since the volume ratio is 3:2)
- Volume of liquid 2, \( V_2 = 2V \)

### Step 2: Calculate the Masses of Each Liquid
Using the formula for mass \( m = \rho \times V \):
- Mass of liquid 1, \( m_1 = \rho_1 \times V_1 = (2\rho) \times (3V) = 6\rho V \)
- Mass of liquid 2, \( m_2 = \rho_2 \times V_2 = \rho \times (2V) = 2\rho V \)

### Step 3: Calculate the Total Mass of the Mixture
The total mass of the mixture \( m_{\text{mixture}} \) is the sum of the masses of the two liquids:
\[ m_{\text{mixture}} = m_1 + m_2 = 6\rho V + 2\rho V = 8\rho V \]

### Step 4: Calculate the Total Volume of the Mixture
The total volume of the mixture \( V_{\text{mixture}} \) is the sum of the volumes of the two liquids:
\[ V_{\text{mixture}} = V_1 + V_2 = 3V + 2V = 5V \]

### Step 5: Calculate the Density of the Mixture
The density of the mixture \( \rho_{\text{mixture}} \) is given by the formula:
\[ \rho_{\text{mixture}} = \frac{m_{\text{mixture}}}{V_{\text{mixture}}} \]
Substituting the values we calculated:
\[ \rho_{\text{mixture}} = \frac{8\rho V}{5V} = \frac{8\rho}{5} \]

### Final Answer
The density of the mixture is \( \frac{8\rho}{5} \).

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