If two liquids of same mass but densities `rho_1` and `rho_2` respectively are mixed, then the density of the mixture is:

Options

A. `p-1p_2(m_1 + m_2)/(m_1p_2 + m_2P1)`
B. `p-1p_2(m_1 + m_2)/(p_1 + p_2)`
C. `2p_1p_2/(p_1 + p_2)`
D. `(m_1p_1 + m_2p_2)/((m_1 + m_2)`

Answer

To find the density of a mixture of two liquids with the same mass but different densities, we can follow these steps:

### Step 1: Define the Variables
Let:
- Mass of each liquid = \( m \)
- Density of the first liquid = \( \rho_1 \)
- Density of the second liquid = \( \rho_2 \)

### Step 2: Calculate the Volumes of Each Liquid
Using the formula for density, which is given by:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
We can rearrange this to find the volume:
\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]
So, the volumes of the two liquids are:
- Volume of the first liquid, \( V_1 = \frac{m}{\rho_1} \)
- Volume of the second liquid, \( V_2 = \frac{m}{\rho_2} \)

### Step 3: Calculate the Total Volume of the Mixture
The total volume of the mixture \( V \) is the sum of the volumes of the two liquids:
\[ V = V_1 + V_2 = \frac{m}{\rho_1} + \frac{m}{\rho_2} \]

### Step 4: Simplify the Total Volume
Factoring out \( m \) from the total volume:
\[ V = m \left( \frac{1}{\rho_1} + \frac{1}{\rho_2} \right) \]

### Step 5: Calculate the Total Mass of the Mixture
The total mass of the mixture \( M \) is:
\[ M = m + m = 2m \]

### Step 6: Calculate the Density of the Mixture
The density \( \rho \) of the mixture is given by:
\[ \rho = \frac{\text{Total Mass}}{\text{Total Volume}} = \frac{M}{V} \]
Substituting the values we found:
\[ \rho = \frac{2m}{m \left( \frac{1}{\rho_1} + \frac{1}{\rho_2} \right)} \]

### Step 7: Simplify the Expression for Density
Cancelling \( m \) from the numerator and denominator:
\[ \rho = \frac{2}{\frac{1}{\rho_1} + \frac{1}{\rho_2}} \]
This can be rewritten as:
\[ \rho = \frac{2 \rho_1 \rho_2}{\rho_1 + \rho_2} \]

### Conclusion
Thus, the density of the mixture of the two liquids is:
\[ \rho = \frac{2 \rho_1 \rho_2}{\rho_1 + \rho_2} \]