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: Identify the given values
- Density of the first liquid, \( \rho_1 = 2\rho \)
- Density of the second liquid, \( \rho_2 = \rho \)
- Volume ratio of the two liquids, \( V_1 : V_2 = 3 : 2 \)

### Step 2: Express the volumes in terms of a common variable
Let the volume of the first liquid \( V_1 = 3x \) and the volume of the second liquid \( V_2 = 2x \), where \( x \) is a common variable.

### Step 3: Calculate the total mass of each liquid
- Mass of the first liquid, \( m_1 = \rho_1 \times V_1 = (2\rho) \times (3x) = 6\rho x \)
- Mass of the second liquid, \( m_2 = \rho_2 \times V_2 = \rho \times (2x) = 2\rho x \)

### Step 4: Calculate the total mass of the mixture
Total mass of the mixture, \( m_{total} = m_1 + m_2 = 6\rho x + 2\rho x = 8\rho x \)

### Step 5: Calculate the total volume of the mixture
Total volume of the mixture, \( V_{total} = V_1 + V_2 = 3x + 2x = 5x \)

### Step 6: Calculate the density of the mixture
Density of the mixture, \( \rho_{mixture} = \frac{m_{total}}{V_{total}} = \frac{8\rho x}{5x} = \frac{8\rho}{5} \)

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

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